This manuscript deals with the existence and uniqueness for the fourth order of Moore-Gibson-Thompson equation with, source terms, viscoelastic memory and integral condition by using Galerkin's method.
Citation: Abdelbaki Choucha, Salah Boulaaras, Djamel Ouchenane, Mohamed Abdalla, Ibrahim Mekawy. Existence and uniqueness for Moore-Gibson-Thompson equation with, source terms, viscoelastic memory and integral condition[J]. AIMS Mathematics, 2021, 6(7): 7585-7624. doi: 10.3934/math.2021442
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This manuscript deals with the existence and uniqueness for the fourth order of Moore-Gibson-Thompson equation with, source terms, viscoelastic memory and integral condition by using Galerkin's method.
Recent research on nonlinear propagation of sound in the case of high amplitude waves has shown that there is a literature on well-grounded partial differential models. (see, e.g., [1,5,7,9,10,11,12,13,16,17,18,20,21,23,24,25,26,27,28,29,30,39,49]). This highly active field of research is being carried out by a wide range of applications such as the medical and industrial use of high intensity ultrasound in lithotripsy, thermotherapy, ultrasound cleaning and ultrasound chemistry. The classical models of nonlinear acoustics are Kuznetsov's equation, the Westervelt equation, and the KZK (Kokhlov-Zabolotskaya-Kuznetsov) equation. For mathematics. Existence and singularity analysis of several types of initial boundary value problems of this second nonlinear order in evolutionary PDEs, we refer (see [19,22,31,32,33,34,35,36,37,38,40,41,42,43,44,45,46,47,48,50,51]). Focusing on the study of sound wave propagation, it should be noted that the MGT equation is one of the nonlinear sound equations describing the propagation of sound waves in gases and liquids. The behavior of sound waves depends strongly on the average property of scattering, scattering, and nonlinear effects. Arises from high-frequency ultrasound (HFU) modeling see ([16,25,41]). The original derivation dates back to [19]. This model is realized through the third order hyperbolic equation
τuttt+utt−c2Δu−bΔut=0, |
the unknown function u=u(x,t) denotes the scalar acoustic velocity, c denotes the speed of sound and τ denotes the thermal relaxation. Besides, the coefficient b=βc2 is related to the diffusively of the sound with β∈(0,τ]. In [19], W Chen and A Palmieri studied the blow-up result for the semilinear Moore- Gibson-Thompson equation with nonlinearity of derivative type in the conservative case defined as following
βuttt+utt−Δu−βΔut=|ut|p,x∈Rn,t>0. |
This paper is related to the following works (see [27,46]). Now when we talk about the (MGT) equation with memory term, we have I. Lasieka and X.Wang in [29] studied the exponential decay of energy of the temporally third order (Moore-Gibson-Thompson) equation with a memory term as follow
τuttt+αutt−c2Au−bAut−∫t0g(t−s)Aw(s)ds=0, |
where τ,α,b,c2 are physical parameters and A is a positive self-adjoint operator on a Hilbert space H. The convolution term ∫t0g(t−s)Aw(s)ds reflects the memory effects of materials due to viscoelasticity. In [13] I. Lasieka and X. Wang studied the general decay of solution of same problem above. Moore-Gibson-Thompson equation with nonlocal condition is a new posed problem. Existence and uniqueness of the generalized solution are established by using Galerkin method. This problems can be encountered in many scientific domains and many engineering models, see previous works ([20,22,31,32,33,34,35,36,37,42,43,47,48]). Mesloub and Mesloub in [40] have applied the Galerkin method to a higher dimension mixed nonlocal problem for a Boussinesq equation. While, S. Boulaaras, A. Zaraï and A. Draifia investigated the Moore-Gibson-Thompson equation with integral condition in [17]. In motivate by these outcomes, we improve the existence and uniqueness by Galerkin method of the Fourth-Order Equation of Moore-Gibson-Thompson Type with source term and integral condition, this problem was cited by the work of F. Dell'Oro and V. Pata in [24].
We define the problem as follow
{utttt+αuttt+βutt−ϱΔu−δΔut−γΔutt+∫t0h(t−σ)Δu(σ)dσ=F(x,t),u(x,0)=u0(x), ut(x,0)=u1(x), utt(x,0)=u2(x),uttt(x,0)=u3(x)∂u∂η=∫t0∫Ωu(ξ,τ)dξdτ, x∈∂Ω. | (1.1) |
The convolution term ∫t0h(t−s)Δu(s)ds reflects the memory effect of materials due to vicoelasticity, F is a given function and h is the relaxation function satisfying
(H1) h∈C1(R+,R+) is a non-increasing function satisfying
h(0)>0,∃h0>0/H(∞)<h0. | (1.2) |
where H(∞)=∫∞0h(s)ds>0, H(t)=∫t0h(s)ds and h′′>0,h‴<0.
(H2) ∃ζ>0 satisfying
h′(t)≤−ζh(t),t≥0. | (1.3) |
The impartial of this manuscript is to consider the following nonlocal mixed boundary value problem for the Moore-Gibson-Thompson (MGT) equation for all (x;t)∈QT=(0,T), where Ω⊂Rn is a bounded domain with sufficiently smooth boundary ∂Ω. solution of the posed problem.
We divide this paper into the following: In the second part, some definitions and appropriate spaces have been given. Then, we use the Galerkin's method to prove the existence, and in the fourth part we demonstrate the uniqueness.
Let V(QT) and W(QT) be the set spaces defined respectively by
V(QT)={u∈W12(QT):ut,utt∈W12(QT),u,∇u∈L2h(QT)}, |
and
W(QT)={u∈V(QT):u(x,T)=0}.L2h(QT)={u∈V(QT):∫T0h∘u(t)dt<∞}, |
where
h∘u(t)=∫Ω∫t0h(t−σ)(u(t)−u(σ))2dσdx. |
Consider the equation
(utttt,v)L2(QT)+α(uttt,v)L2(QT)+β(utt,v)L2(QT)−ϱ(Δu,v)L2(QT)−δ(Δut,v)L2(QT)−γ(Δutt,v)L2(QT)+(Δw,v)L2(QT)=(F,v)L2(QT), | (2.1) |
where
w(x,t)=∫t0h(t−σ)u(x,σ)dσ, |
and (.,.)L2(QT) defend for the inner product in L2(QT), u is supposed to be a solution of (1.1) and v∈W(QT). Upon using (2.1) and (1.1), we find
−(uttt,vt)L2(QT)−α(utt,vt)L2(QT)−β(ut,vt)L2(QT)+ϱ(∇u,∇v)L2(QT)+δ(∇ut,∇v)L2(QT)−γ(∇ut,∇vt)L2(QT)−(∇w,∇v)L2(QT)=(F,v)L2(QT)+ϱ∫T0∫∂Ωv(∫t0∫Ωu(ξ,τ)dξdτ)dsxdt+δ∫T0∫∂Ωv∫Ωu(ξ,t)dξdsxdt−δ∫T0∫∂Ωv∫Ωu0(ξ)dξdsxdt−γ∫T0∫∂Ωvt(∫t0∫Ωuτ(ξ,τ)dξdτ)dsxdt+(u3(x),v(x,0))L2(Ω)+α(u2(x),v(x,0))L2(Ω)+β(u1(x),v(x,0))L2(Ω)−γ(Δu1,v(x,0))L2(Ω)−∫T0∫∂Ωv(∫t0∫Ωw(ξ,τ)dξdτ)dsxdt. | (2.2) |
Now, we give two useful inequalities:
● Gronwall inequality: If for any t∈I, we have
y(t)≤h(t)+c∫t0y(s)ds, |
where h(t) and y(t) are two nonnegative integrable functions on the interval I with h(t) non decreasing and c is constant, then
y(t)≤h(t)exp(ct). |
● Trace inequality: When w∈W21(Ω), we have
‖w‖2L2(∂Ω)≤ε‖∇w‖2L2(Ω)+l(ε)‖w‖2L2(Ω), |
where Ω is a bounded domain in Rn with smooth boundary ∂Ω, and l(ε) is a positive constant.
Definition 1. If a function u∈V(QT) satisfies Eq (2.1) for each v∈W(QT) is called a generalized solution of problem (1.1).
Here, by using Galerkin's method, we give the existence of problem (1.1).
Theorem 1. If u0,u1,u2∈W12(Ω), u3∈L2(Ω) and F∈L2(QT), then there is at least one generalized solution in V(QT) to problem (1.1).
Proof. Let {Zk(x)}k≥1 be a fundamental system in W12(Ω), such that
(Zk,Zl)L2(Ω)=δk,l. |
First, we will find an approximate solution of the problem (1.1) in the form
uN(x,t)=N∑k=1Ck(t)Zk(x), | (3.1) |
where the constants Ck(t) are defined by the conditions
Ck(t)=(uN(x,t),Zk(x))L2(Ω), k=1,...,N, | (3.2) |
and can be determined from the relations
(uNtttt,Zl(x))L2(Ω)+α(uNttt,Zl(x))L2(Ω)+β(uNtt,Zl(x))L2(Ω)+ϱ(∇uN,∇Zl(x))L2(Ω)+δ(∇uNt,∇Zl(x))L2(Ω)+γ(∇uNtt,∇Zl(x))L2(Ω)−(∇wN,∇Zl(x))L2(Ω)=(F(x,t),Zl(x))L2(Ω)+ϱ∫∂ΩZl(x)(∫t0∫ΩuN(ξ,τ)dξdτ)dsx+δ∫∂ΩZl(x)(∫t0∫ΩuNτ(ξ,τ)dξdτ)dsx+γ∫∂ΩZl(x)(∫t0∫ΩuNττ(ξ,τ)dξdτ)dsx−∫∂ΩZl(x)(∫t0∫ΩwN(ξ,τ)dξdτ)dsx, | (3.3) |
Invoking to (3.1) in (3.3) gives for l=1,...,N.
∫ΩN∑k=1{C′′′′k(t)Zk(x)Zl(x)+αC′′′k(t)Zk(x)Zl(x)+βC′′k(t)Zk(x)Zl(x)+ϱCk(t)∇Zk(x).∇Zl(x)+δC′k(t)∇Zk(x).∇Zl(x)+γC′′k(t)∇Zk.∇Zl−(∫t0h(t−σ)Ck(σ)dσ)∇Zk(x).∇Zl(x)}dx=(F(x,t),Zl(x))L2(Ω)+ϱN∑k=1∫t0Ck(τ)(∫∂ΩZl(x)∫ΩZk(ξ)dξdsx)dτ+δN∑k=1∫t0C′k(τ)(∫∂ΩZl(x)∫ΩZk(ξ)dξdsx)dτ+γN∑k=1∫t0C′′k(τ)(∫∂ΩZl(x)∫ΩZk(ξ)dξdsx)dτ−N∑k=1∫t0∫τ0h(τ−σ)Ck(σ)(∫∂ΩZl(x)∫ΩZk(ξ)dξdsx)dσdτ. | (3.4) |
From (3.4) it follows that
N∑k=1C′′′′k(t)(Zk(x),Zl(x))L2(Ω)+αC′′′k(t)(Zk(x),Zl(x))L2(Ω)+βC′′k(t)(Zk(x),Zl(x))L2(Ω)+ϱCk(t)(∇Zk,∇Zl)L2(Ω)+δC′k(t)(∇Zk(x),∇Zl(x))L2(Ω)+γC′′k(t)(∇Zk(x),∇Zl(x))L2(Ω)−(∫t0h(t−σ)Ck(σ)dσ)(∇Zk,∇Zl)L2(Ω)}dx=(F(x,t),Zl(x))L2(Ω)+ϱN∑k=1∫t0Ck(τ)(∫∂ΩZl(x)∫ΩZk(ξ)dξdsx)dτ+δN∑k=1∫t0C′k(τ)(∫∂ΩZl(x)∫ΩZk(ξ)dξdsx)dτ+γN∑k=1∫t0(C′′k(τ)∫∂ΩZl(x)∫ΩZk(ξ)dξds)dτ−N∑k=1∫t0∫τ0h(τ−σ)Ck(σ)(∫∂ΩZl(x)∫ΩZk(ξ)dξdsx)dσdτ, l=1,...,N. | (3.5) |
Let
(Zk,Zl)L2(Ω)=δkl={1, k=l0, k≠l |
(∇Zk,∇Zl)L2(Ω)=γkl, |
∫∂ΩZl(x)∫ΩZk(ξ)dξds=χkl. |
(F(x,t),Zl(x))L2(Ω)=Fl(t). |
Then (3.5) can be written as
N∑k=1C′′′′k(t)δkl+αC′′′k(t)δkl+C′′k(t)(βδkl+γγkl)+δC′k(t)γkl+ϱCk(t)γkl−∫t0(ϱCk(τ)χkl+δC′k(τ)χkl+γC′′k(τ)χkl−h(t−τ)Ck(τ)γkl)−∫t0∫τ0h(τ−σ)Ck(σ)dσχkldσdτ=Fl(t). | (3.6) |
A differentiation with respect to t (two times), yields
N∑k=1C′′′′′′k(t)δkl+αC′′′′′k(t)δkl+C⁗k(t)(βδkl+γγkl)+C‴k(t)(δγkl−γχkl)+C″k(t)(ϱγkl−δχkl)−(ϱ+h(0))C′k(t)χkl+h(0)Ck(t)χkl=F″l(t), | (3.7) |
{N∑k=1[C′′′′k(0)δkl+αC′′′k(0)δkl+C′′k(0)(βδkl+γγkl)+δC′k(0)γkl+ϱCk(0)γkl]=Fl(0)Ck(0)=(Zk,u0)L2(Ω), C′k(0)=(Zk,u1(x))L2(Ω),C′′k(0)=(Zk,u2(x))L2(Ω),C′′′k(0)=(Zk,u3(x))L2(Ω). | (3.8) |
Thus for every n there exists a function uN(x) satisfying (3.3).
Now, we will demonstrate that the sequence uN is bounded. To do this, we multiply each equation of (3.3) by the appropriate C′k(t) summing over k from 1 to N then integrating the resultant equality with respect to t from 0 to τ, with τ≤T, yields
(uNtttt,uNt)L2(Qτ)+α(uNttt,uNt)L2(Qτ)+β(uNtt,uNt)L2(Qτ)+ϱ(∇uN,∇uNt)L2(Qτ)+δ(∇uNt,∇uNt)L2(Qτ)+γ(∇uNtt,∇uNt)L2(Qτ)−(∇wN,∇uNt)L2(Qτ)=(F,uNt)L2(Qτ)+ϱ∫τ0∫∂ΩuNt(x,t)(∫t0∫ΩuN(ξ,η)dξdη)dsxdt+δ∫τ0∫∂ΩuNt(x,t)(∫t0∫ΩuNt(ξ,η)dξdη)dsxdt+γ∫τ0∫∂ΩuNt(x,t)(∫t0∫ΩuNtt(ξ,η)dξdη)dsxdt−∫τ0∫∂ΩuNt(x,t)(∫t0∫ΩwN(ξ,η)dξdη)dsxdt, | (3.9) |
after a simplification of the LHS of (3.9), we get
(uNtttt,uNt)L2(Qτ)=−∫τ0(uNttt,uNtt)L2(Ω)dt+(uNτττ(x,τ),uNτ(x,τ))L2(Ω),−(uNttt(x,0),uNt(x,0))L2(Ω),α(uNttt,uNt)L2(Qτ)=α(uNττ(x,τ),uNτ(x,τ))L2(Ω)−(uNtt(x,0),uNt(x,0))L2(Ω)−α∫τ0‖utt(x,t)‖2L2(Ω)dt,β(uNtt,uNt)L2(Qτ)=β2‖uNτ(x,τ)‖2L2(Ω)−β2‖uNt(x,0)‖2L2(Ω),ϱ(∇uN,∇uNt)L2(Qτ)=ϱ2‖∇uN(x,τ)‖2L2(Ω)−ϱ2‖∇uN(x,0)‖2L2(Ω),δ(∇uNt,∇uNt)L2(Qτ)=δ∫τ0‖∇uNt(x,t)‖2L2(Ω)dt,γ(∇uNtt,∇uNt)L2(Qτ)=γ2‖∇uNτ(x,τ)‖2L2(Ω)−γ2‖∇uNt(x,0)‖2L2(Ω),−(∇wN,∇uNt)L2(Qτ)=12h∘∇uN(τ)−12H(τ)‖∇uN(x,τ)‖2L2(Ω)−12∫τ0h′∘∇uN(t)dt+12h(t)‖∇uN(x,t)‖2L2(Ω)dt, | (3.10) |
ϱ∫τ0∫∂ΩuNt(∫t0∫ΩuN(ξ,η)dξdη)dsxdt=ϱ∫∂ΩuN(x,τ)∫τ0∫ΩuN(ξ,t)dξdtdsx−ϱ∫∂Ω∫τ0uN(x,t)∫ΩuN(ξ,t)dξdtdsx, | (3.11) |
δ∫τ0∫∂ΩuNt(∫t0∫ΩuNt(ξ,η)dξdη)dsxdt=δ∫∂Ω∫τ0uNt(x,t)∫ΩuN(ξ,t)dξdtdsx−δ∫∂Ω∫τ0uNt(x,t)∫ΩuN(ξ,0)dξdtdsx, | (3.12) |
γ∫τ0∫∂ΩuNt(x,t)(∫t0∫ΩuNtt(ξ,η)dξdη)dsxdt=γ∫τ0∫∂ΩuNt(x,t)(∫ΩuNt(ξ,t)dξ)dsxdt−γ∫τ0∫∂ΩuNt(x,t)(∫ΩuNt(ξ,0)dξ)dsxdt. | (3.13) |
−∫τ0∫∂ΩuNt(∫t0∫ΩwN(ξ,η)dξdη)dsxdt=−∫τ0∫∂ΩuNt(∫t0∫ΩH(η)uN(ξ,η)dξdη)dsxdt+∫τ0∫∂ΩuNt(∫t0∫Ω[∫η0h(η−σ)(uN(ξ,η)−uN(ξ,σ))dσ]dξdη)dsxdt=−∫∂ΩuN(x,τ)∫τ0∫ΩH(t)uN(ξ,t)dξdtdsx+∫τ0∫∂ΩuN(x,t)∫ΩH(t)uN(ξ,t)dξdsxdt+∫∂ΩuN(x,τ)(∫τ0∫Ω[∫t0h(t−σ)(uN(ξ,t)−uN(ξ,σ))dσ]dξ)dsxdt−∫τ0∫∂ΩuN(x,t)∫Ω[∫t0h(t−σ)(uN(ξ,t)−uN(ξ,σ))dσ]dξdsxdt. | (3.14) |
Taking into account the equalities (3.10)-(3.14) in (3.9), we obtain
(uNτττ(x,τ),uNτ(x,τ))L2(Ω)+α(uNττ(x,τ),uNτ(x,τ))L2(Ω)+β2‖uNτ(x,τ)‖2L2(Ω)+ϱ2‖∇uN(x,τ)‖2L2(Ω)+γ2‖∇uNτ(x,τ)‖2L2(Ω)+12h∘∇uN(τ)−12H(τ)‖∇uN(x,τ)‖2L2(Ω)=(uNttt(x,0),uNt(x,0))L2(Ω)+α(uNtt(x,0),uNt(x,0))L2(Ω)+ϱ2‖∇uN(x,0)‖2L2(Ω)+γ2‖∇uNt(x,0)‖2L2(Ω)+∫τ0(uNttt,uNtt)L2(Ω)dt+α∫τ0‖utt(x,t)‖2L2(Ω)dt−δ∫τ0‖∇uNt(x,t)‖2L2(Ω)dt+β2‖uNt(x,0)‖2L2(Ω)+ϱ∫∂ΩuN(x,τ)∫τ0∫ΩuN(ξ,t)dξdtdsx+(F,uNt)L2(Qτ)−ϱ∫∂Ω∫τ0uN(x,t)∫ΩuN(ξ,t)dξdtdsx+δ∫∂Ω∫τ0uNt(x,t)∫ΩuN(ξ,t)dξdtdsx−δ∫∂Ω∫τ0uNt(x,t)∫ΩuN(ξ,0)dξdtdsx+γ∫τ0∫∂ΩuNt(x,t)(∫ΩuNt(ξ,t)dξ)dsxdt−γ∫τ0∫∂ΩuNt(x,t)(∫ΩuNt(ξ,0)dξ)dsxdt−12∫τ0h′∘∇uN(t)dt+12h(t)‖∇uN(x,t)‖2L2(Ω)dt−∫∂ΩuN(x,τ)∫τ0∫ΩH(t)uN(ξ,t)dξdtdsx+∫τ0∫∂ΩuN(x,t)∫ΩH(t)uN(ξ,t)dξdsxdt+∫∂ΩuN(x,τ)(∫τ0∫Ω[∫t0h(t−σ)(uN(ξ,t)−uN(ξ,σ))dσ]dξ)dsxdt−∫τ0∫∂ΩuN(x,t)∫Ω[∫t0h(t−σ)(uN(ξ,t)−uN(ξ,σ))dσ]dξdsxdt. | (3.15) |
Now, multiplying each equation of (3.3) by the appropriate C′′k(t), add them up from 1 to N and them integrate with respect to t from 0 to τ, with τ≤T, we obtain
(uNtttt,uNtt)L2(Qτ)+α(uNttt,uNtt)L2(Qτ)+β(uNtt,uNtt)L2(Qτ)+ϱ(∇uN,∇uNtt)L2(Qτ)+δ(∇uNt,∇uNtt)L2(Qτ)+γ(∇uNtt,∇uNtt)L2(Qτ)−(∇wN,∇uNtt)L2(Qτ)=(F,uNtt)L2(Qτ)+ϱ∫τ0∫∂ΩuNtt(x,t)(∫t0∫ΩuN(ξ,η)dξdη)dsxdt+δ∫τ0∫∂ΩuNtt(x,t)(∫t0∫ΩuNt(ξ,η)dξdη)dsxdt+γ∫τ0∫∂ΩuNtt(x,t)(∫t0∫ΩuNtt(ξ,η)dξdη)dsxdt−∫τ0∫∂ΩuNtt(x,t)(∫t0∫ΩwN(ξ,η)dξdη)dsxdt. | (3.16) |
With the same reasoning in (3.9), we find
(uNtttt,uNtt)L2(Qτ)=−∫τ0‖uNttt(x,t)‖2L2(Ω)dt+(uNτττ(x,τ),uNττ(x,τ))L2(Ω)−(uNttt(x,0),uNtt(x,0))L2(Ω),α(uNttt,uNtt)L2(Qτ)=α2‖uNττ(x,τ)‖2L2(Ω)−α2‖uNtt(x,0)‖2L2(Ω),β(uNtt,uNtt)L2(Qτ)=β∫τ0‖uNtt(x,t)‖2L2(Ω)dt,ϱ(∇uN,∇uNtt)L2(Qτ)=ϱ(∇uN(x,τ),∇uNτ(x,τ))L2(Qτ)−ϱ(∇uN(x,0),∇uNt(x,0))L2(Ω)−ϱ∫τ0‖∇uNt(x,t)‖2L2(Ω)dt,δ(∇uNt,∇uNtt)L2(Qτ)=δ2‖∇uNτ(x,τ)‖2L2(Ω)−δ2‖∇uNt(x,0)‖2L2(Ω),γ(∇uNtt,∇uNtt)L2(Qτ)=γ∫τ0‖∇uNtt(x,t)‖2L2(Ω)dt−(∇wN,∇uNtt)L2(Qτ)=−12{h′∘∇uN(τ)+h(τ)‖∇uN(x,τ)‖2L2(Ω)−2(∇wN(τ),∇uNτ)L2(Ω)}+12∫τ0h′′∘∇uN(t)dt−12∫τ0h′(t)‖∇uN(x,t)‖2L2(Ω)dt, | (3.17) |
ϱ∫τ0∫∂ΩuNtt(∫t0∫ΩuN(ξ,η)dξdη)dsxdt=ϱ∫∂ΩuNτ(x,τ)∫τ0∫ΩuN(ξ,t)dξdtdsx−ϱ∫∂Ω∫τ0uNt(x,t)∫ΩuN(ξ,t)dξdtdsx, | (3.18) |
δ∫τ0∫∂ΩuNtt(x,t)(∫t0∫ΩuNt(ξ,η)dξdη)dsxdt=δ∫∂ΩuNτ(x,τ)∫ΩuN(ξ,τ)dξdsx−δ∫∂ΩuNτ(x,τ)∫ΩuN(ξ,0)dξdsx−δ∫∂Ω∫τ0uNt(x,t)∫ΩuNt(ξ,t)dξdtds, | (3.19) |
γ∫τ0∫∂ΩuNtt(x,t)(∫t0∫ΩuNtt(ξ,η)dξdη)dsxdt=γ∫∂ΩuNτ(x,τ)∫ΩuNτ(ξ,τ)dξdsx−γ∫∂ΩuNτ(x,τ)∫ΩuNt(ξ,0)dξdsx−γ∫∂Ω∫τ0uNt(x,t)∫ΩuNtt(ξ,t)dξdtds, | (3.20) |
−∫τ0∫∂ΩuNtt(∫t0∫ΩwN(ξ,η)dξdη)dsxdt=−∫τ0∫∂ΩuNtt(∫t0∫ΩH(η)uN(ξ,η)dξdη)dsxdt+∫τ0∫∂ΩuNtt(∫t0∫Ω[∫η0h(η−σ)(uN(ξ,η)−uN(ξ,σ))dσ]dξdη)dsxdt=−∫∂ΩuNτ(x,τ)∫τ0∫ΩH(t)uN(ξ,t)dξdtdsx+∫τ0∫∂ΩuNt(x,t)∫ΩH(t)uN(ξ,t)dξdsxdt+∫∂ΩuNτ(x,τ)(∫τ0∫Ω[∫t0h(t−σ)(uN(ξ,t)−uN(ξ,σ))dσ]dξ)dsxdt−∫τ0∫∂ΩuNt(x,t)∫Ω[∫t0h(t−σ)(uN(ξ,t)−uN(ξ,σ))dσ]dξdsxdt. | (3.21) |
Upon using (3.17)-(3.21) into (3.16), we have
(uNτττ(x,τ),uNττ(x,τ))L2(Ω)+α2‖uNττ(x,τ)‖2L2(Ω)+δ2‖∇uNτ(x,τ)‖2L2(Ω)+ϱ(∇uN(x,τ),∇uNτ(x,τ))L2(Ω)+12h(τ)‖∇uN(x,τ)‖2L2(Ω)−12h′∘∇uN(τ)+(∇wN(τ),∇uNτ)L2(Ω)=∫τ0‖uNttt(x,t)‖2L2(Ω)dt+(uNttt(x,0),uNtt(x,0))L2(Ω)+α2‖uNtt(x,0)‖2L2(Ω)−β∫τ0‖uNtt(x,t)‖2L2(Ω)dt+ϱ(∇uN(x,0),∇uNt(x,0))L2(Ω)+ϱ∫τ0‖∇ut(x,t)‖2L2(Ω)dt+δ2‖∇uNt(x,0)‖2L2(Ω)+(F,uNtt)L2(Qτ)−γ∫τ0‖∇uNtt(x,t)‖2L2(Ω)dt+ϱ∫∂ΩuNτ(x,τ)∫τ0∫ΩuN(ξ,t)dξdtdsx−ϱ∫∂Ω∫τ0uNt(x,t)∫ΩuN(ξ,t)dξdtdsx+δ∫∂ΩuNτ(x,τ)∫ΩuN(ξ,τ)dξdsx−δ∫∂ΩuNτ(x,τ)∫ΩuN(ξ,0)dξdsx−δ∫∂Ω∫τ0uNt(x,t)∫ΩuNt(ξ,t)dξdtdsx+γ∫∂ΩuNτ(x,τ)∫ΩuNτ(ξ,τ)dξdsx−γ∫∂ΩuNτ(x,τ)∫ΩuNt(ξ,0)dξdsx−γ∫∂Ω∫τ0uNt(x,t)∫ΩuNtt(ξ,t)dξdtdsx−∫∂ΩuNτ(x,τ)∫τ0∫ΩH(t)uN(ξ,t)dξdtdsx+∫τ0∫∂ΩuNt(x,t)∫ΩH(t)uN(ξ,t)dξdsxdt+∫∂ΩuNτ(x,τ)(∫τ0∫Ω[∫t0h(t−σ)(uN(ξ,t)−uN(ξ,σ))dσ]dξ)dsxdt−∫τ0∫∂ΩuNt(x,t)∫Ω[∫t0h(t−σ)(uN(ξ,t)−uN(ξ,σ))dσ]dξdsxdt. | (3.22) |
Now, multiplying each equation of (3.3) by the appropriate C′′′k(t), add them up from 1 to N and them integrate with respect to t from 0 to τ, with τ≤T, we obtain
(uNtttt,uNttt)L2(Qτ)+α(uNttt,uNttt)L2(Qτ)+β(uNtt,uNttt)L2(Qτ)+ϱ(∇uN,∇uNttt)L2(Qτ)+δ(∇uNt,∇uNttt)L2(Qτ)+γ(∇uNtt,∇uNttt)L2(Qτ)−(∇wN,∇uNttt)L2(Qτ)=(F,uNttt)L2(Qτ)+ϱ∫τ0∫∂ΩuNttt(x,t)(∫t0∫ΩuN(ξ,η)dξdη)dsxdt+δ∫τ0∫∂ΩuNttt(x,t)(∫t0∫ΩuNt(ξ,η)dξdη)dsxdt+γ∫τ0∫∂ΩuNttt(x,t)(∫t0∫ΩuNtt(ξ,η)dξdη)dsxdt−∫τ0∫∂ΩuNttt(x,t)(∫t0∫ΩwN(ξ,η)dξdη)dsxdt. | (3.23) |
With the same reasoning in (3.9), we find
(uNtttt,uNttt)L2(Qτ)=12‖uNτττ(x,τ)‖2L2(Ω)−12‖uNttt(x,0)‖2L2(Ω)α(uNttt,uNttt)L2(Qτ)=α∫τ0‖uNttt(x,t)‖2L2(Ω),β(uNtt,uNttt)L2(Qτ)=β2‖uNττ(x,τ)‖2L2(Ω)−β2‖uNtt(x,0)‖2L2(Ω),ϱ(∇uN,∇uNttt)L2(Qτ)=ϱ(∇uN(x,τ),∇uNττ(x,τ))L2(Ω)−ϱ(∇uN(x,0),∇uNtt(x,0))L2(Ω)−ϱ∫τ0(∇uNt,∇uNtt)L2(Ω)dt,δ(∇uNt,∇uNttt)L2(Qτ)=−δ∫τ0‖∇uNtt(x,t)‖2L2(Ω)dt+δ(∇uNτ(x,τ),∇uNττ(x,τ))L2(Ω)−δ(∇uNt(x,0),∇uNtt(x,0))L2(Ω),γ(∇uNtt,∇uNttt)L2(Qτ)=γ2‖∇uNττ(x,τ)‖2L2(Ω)−γ2‖∇uNtt(x,0)‖2L2(Ω)−(∇wN,∇uNttt)L2(Qτ)=−H(τ)(∇uNττ(x,τ),∇uN(x,τ))2L2(Ω)+h(τ)(∇uNτ(x,τ),∇uN(x,τ))2L2(Ω)−12‖∇uN(x,τ)‖2L2(Ω)+∫Ω∇uNττ∫τ0h(τ−σ)(∇uN(τ)−∇uN(σ))dσdx+∫Ω∇uNτ∫τ0h′(τ−σ)(∇uN(τ)−∇uN(σ))dσdx+12h″∘∇uN(τ)+12∫τ0(h″−h‴)∘∇uN(t)dt−h(0)∫τ0‖∇uNt(x,t)‖2L2(Ω)dt, | (3.24) |
\begin{eqnarray} &&\varrho \int_{0}^{\tau }\int_{\partial \Omega }u_{ttt}^{N}\left( \int_{0}^{t}\int_{\Omega }u^{N}(\xi , \eta )d\xi d\eta \right) ds_{x}dt \end{eqnarray} | (3.25) |
\begin{eqnarray} & = &\varrho \int_{\partial \Omega }u_{\tau \tau }^{N}(x, \tau )\int_{0}^{\tau }\int_{\Omega }u^{N}\left( \xi , t\right) d\xi dtds_{x} \\ &&-\varrho \int_{\partial \Omega }\int_{0}^{\tau }u_{tt}^{N}\left( x, t\right) \int_{\Omega }u^{N}\left( \xi , t\right) d\xi dtds_{x}, \end{eqnarray} | (3.26) |
\begin{eqnarray} &&\delta \int_{0}^{\tau }\int_{\partial \Omega }u_{ttt}^{N}\left( x, t\right) \left( \int_{0}^{t}\int_{\Omega }u_{t}^{N}(\xi , \eta )d\xi d\eta \right) ds_{x}dt \\ & = &\delta \int_{\partial \Omega }u_{\tau \tau }^{N}(x, \tau )\int_{\Omega }u^{N}\left( \xi , \tau \right) d\xi ds_{x}-\delta \int_{\partial \Omega }u_{\tau \tau }^{N}(x, \tau )\int_{\Omega }u^{N}(\xi , 0)d\xi ds_{x} \\ &&-\delta \int_{\partial \Omega }\int_{0}^{\tau }u_{tt}^{N}\left( x, t\right) \int_{\Omega }u_{t}^{N}\left( \xi , t\right) d\xi dtds, \end{eqnarray} | (3.27) |
\begin{eqnarray} &&\gamma \int_{0}^{\tau }\int_{\partial \Omega }u_{ttt}^{N}\left( x, t\right) \left( \int_{0}^{t}\int_{\Omega }u_{tt}^{N}(\xi , \eta )d\xi d\eta \right) ds_{x}dt \\ & = &\gamma \int_{\partial \Omega }u_{\tau \tau }^{N}(x, \tau )\int_{\Omega }u_{\tau }^{N}\left( \xi , \tau \right) d\xi ds_{x}-\gamma \int_{\partial \Omega }u_{\tau \tau }^{N}(x, \tau )\int_{\Omega }u_{t}^{N}(\xi , 0)d\xi ds_{x} \\ &&-\gamma \int_{\partial \Omega }\int_{0}^{\tau }u_{tt}^{N}\left( x, t\right) \int_{\Omega }u_{tt}^{N}\left( \xi , t\right) d\xi dtds, \end{eqnarray} | (3.28) |
\begin{eqnarray} &&- \int_{0}^{\tau }\int_{\partial \Omega }u_{ttt}^{N}\left( \int_{0}^{t}\int_{\Omega }w^{N}(\xi , \eta )d\xi d\eta \right) ds_{x}dt \\ & = &- \int_{0}^{\tau }\int_{\partial \Omega }u_{ttt}^{N}\left( \int_{0}^{t}\int_{\Omega }H(\eta)u^{N}(\xi , \eta )d\xi d\eta \right) ds_{x}dt\\ &&+ \int_{0}^{\tau }\int_{\partial \Omega }u_{ttt}^{N}\left( \int_{0}^{t}\int_{\Omega }\bigg[\int_{0}^{\eta}h(\eta-\sigma)(u^{N}(\xi , \eta )-u^{N}(\xi , \sigma ))d\sigma\bigg] d\xi d\eta \right) ds_{x}dt\\ & = &-\int_{\partial \Omega }u_{\tau\tau}^{N}(x, \tau) \int_{0}^{\tau}\int_{\Omega }H(t)u^{N}(\xi , t )d\xi dt ds_{x}\\ &&+\int_{0}^{\tau }\int_{\partial \Omega }u_{tt}^{N}\left( x, t\right) \int\nolimits_{\Omega }H(t)u^{N}(\xi , t)d\xi ds_{x}dt\\ &&+\int_{\partial \Omega }u_{\tau\tau}^{N}(x, \tau)\left( \int_{0}^{\tau}\int_{\Omega }\bigg[\int_{0}^{t}h(t-\sigma)(u^{N}(\xi , t )-u^{N}(\xi , \sigma ))d\sigma\bigg] d\xi \right) ds_{x}dt\\ &&-\int_{0}^{\tau }\int_{\partial \Omega }u_{tt}^{N}\left( x, t\right) \int\nolimits_{\Omega }\bigg[\int_{0}^{t}h(t-\sigma)(u^{N}(\xi , t )-u^{N}(\xi , \sigma ))d\sigma\bigg]d\xi ds_{x}dt. \\ \end{eqnarray} | (3.29) |
A substitution of equalities (3.24)-(3.29) in (3.23), gives
\begin{eqnarray} &&\frac{1}{2}\left\Vert u_{\tau \tau \tau }^{N}\left( x, \tau \right) \right\Vert _{L^{2}(\Omega )}^{2}+\frac{\beta }{2}\left\Vert u_{\tau \tau }^{N}\left( x, \tau \right) \right\Vert _{L^{2}(\Omega )}^{2}+\varrho \left( \nabla u^{N}(x, \tau ), \nabla u_{\tau \tau }^{N}(x, \tau )\right) _{L^{2}\left( \Omega \right) } \\ &&+\delta \left( \nabla u_{\tau }^{N}(x, \tau ), \nabla u_{\tau \tau }^{N}(x, \tau )\right) _{L^{2}\left( \Omega \right) }+\frac{\gamma }{2} \left\Vert \nabla u_{\tau \tau }^{N}\left( x, \tau \right) \right\Vert _{L^{2}(\Omega )}^{2} \\ &&-H(\tau)( \nabla u_{\tau \tau }^{N}\left( x, \tau \right), \nabla u^{N}\left( x, \tau \right) ) _{L^{2}(\Omega )}^{2}\\ &&+h(\tau)( \nabla u_{\tau }^{N}\left( x, \tau \right), \nabla u^{N}\left( x, \tau \right) ) _{L^{2}(\Omega )}^{2}-\frac{1 }{2 }\left\Vert \nabla u^{N}\left( x, \tau\right) \right\Vert _{L^{2}(\Omega )}^{2}\\ &&+\int_{\Omega}\nabla u_{\tau\tau}^{N}\int_{0}^{\tau}h(\tau-\sigma) (\nabla u^{N}(\tau)-\nabla u^{N}(\sigma))d\sigma dx\\ &&+\int_{\Omega}\nabla u_{\tau}^{N}\int_{0}^{\tau}h'(\tau-\sigma) (\nabla u^{N}(\tau)-\nabla u^{N}(\sigma))d\sigma dx+\frac{1}{2}h''\circ\nabla u^{N}(\tau)\\ & = &\left( F, u_{ttt}^{N}\right) _{L^{2}\left( Q_{\tau }\right) }+\frac{1}{2}\left\Vert u_{ttt}^{N}\left( x, 0\right) \right\Vert _{L^{2}(\Omega )}^{2}-\alpha \int_{0}^{\tau }\left\Vert u_{ttt}^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}\\ &&+\varrho \left( \nabla u^{N}(x, 0), \nabla u_{tt}^{N}(x, 0)\right) _{L^{2}(\Omega )}+\varrho \int_{0}^{\tau }\left( \nabla u_{t}^{N}, \nabla u_{tt}^{N}\right) _{L^{2}\left( \Omega \right) }dt \\ &&+\delta \int_{0}^{\tau }\left\Vert \nabla u_{tt}^{N}(x, t)\right\Vert _{L^{2}(\Omega )}^{2}dt+\delta \left( \nabla u_{t}^{N}(x, 0), \nabla u_{tt}^{N}(x, 0)\right) _{L^{2}\left( \Omega \right) } \\ &&+\varrho \int_{\partial \Omega }u_{\tau \tau }^{N}(x, \tau )\int_{0}^{\tau }\int_{\Omega }u^{N}\left( \xi , t\right) d\xi dtds_{x}-\frac{\gamma }{2} \left\Vert \nabla u_{tt}^{N}\left( x, 0\right) \right\Vert _{L^{2}(\Omega )}^{2}\\ &&-\varrho \int_{\partial \Omega }\int_{0}^{\tau }u_{tt}^{N}\left( x, t\right) \int_{\Omega }u^{N}\left( \xi , t\right) d\xi dtds_{x}-\frac{\beta }{2}\left\Vert u_{tt}^{N}\left( x, 0\right) \right\Vert _{L^{2}(\Omega )}^{2} \\ &&+\delta \int_{\partial \Omega }u_{\tau \tau }^{N}(x, \tau )\int_{\Omega }u^{N}\left( \xi , \tau \right) d\xi ds_{x}-\delta \int_{\partial \Omega }u_{\tau \tau }^{N}(x, \tau )\int_{\Omega }u^{N}(\xi , 0)d\xi ds_{x} \\ &&-\delta \int_{\partial \Omega }\int_{0}^{\tau }u_{tt}^{N}\left( x, t\right) \int_{\Omega }u_{t}^{N}\left( \xi , t\right) d\xi dtds +\delta\int_{\partial \Omega }u_{\tau \tau }^{N}(x, \tau )\int_{\Omega }u_{\tau }^{N}\left( \xi , \tau \right) d\xi ds_{x} \\ &&-\gamma \int_{\partial \Omega }u_{\tau \tau }^{N}(x, \tau )\int_{\Omega }u_{t}^{N}(\xi , 0)d\xi ds_{x}-\gamma \int_{\partial \Omega }\int_{0}^{\tau }u_{tt}^{N}\left( x, t\right) \int_{\Omega }u_{tt}^{N}\left( \xi , t\right) d\xi dtds\\ &&-\int_{\partial \Omega }u_{\tau\tau}^{N}(x, \tau) \int_{0}^{\tau}\int_{\Omega }H(t)u^{N}(\xi , t )d\xi dt ds_{x}\\ &&+\frac{1}{2}\int_{0}^{\tau}(h''-h''')\circ\nabla u^{N}(t)dt-h(0)\int_{0}^{\tau}\left\Vert \nabla u_{t }^{N}\left( x, t \right) \right\Vert _{L^{2}(\Omega )}^{2}dt\\ &&+\int_{0}^{\tau }\int_{\partial \Omega }u_{tt}^{N}\left( x, t\right) \int\nolimits_{\Omega }H(t)u^{N}(\xi , t)d\xi ds_{x}dt\\ &&+\int_{\partial \Omega }u_{\tau\tau}^{N}(x, \tau)\left( \int_{0}^{\tau}\int_{\Omega }\bigg[\int_{0}^{t}h(t-\sigma)(u^{N}(\xi , t )-u^{N}(\xi , \sigma ))d\sigma\bigg] d\xi \right) ds_{x}dt\\ &&-\int_{0}^{\tau }\int_{\partial \Omega }u_{tt}^{N}\left( x, t\right) \int\nolimits_{\Omega }\bigg[\int_{0}^{t}h(t-\sigma)(u^{N}(\xi , t )-u^{N}(\xi , \sigma ))d\sigma\bigg]d\xi ds_{x}dt. \end{eqnarray} | (3.30) |
Multiplying (3.15) by \lambda _{1} , (3.22) by \lambda _{2} , and (3.30) by \lambda _{3} such as (\lambda_{1}+\lambda_{2} < \lambda_{3}) , we get
\begin{eqnarray} &&\lambda _{1}\left( u_{\tau \tau \tau }^{N}\left( x, \tau \right) , u_{\tau }^{N}\left( x, \tau \right) \right) _{L^{2}(\Omega )}+\lambda _{1}\alpha \left( u_{\tau \tau }^{N}\left( x, \tau \right) , u_{\tau }^{N}\left( x, \tau \right) \right) _{L^{2}(\Omega )}\\ &&+\frac{\lambda _{1}\beta }{2}\left\Vert u_{\tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+\frac{\lambda _{1}\varrho }{2}\left\Vert \nabla u^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2} \\ &&+(\frac{\lambda _{1}\gamma }{2}+\frac{\lambda _{2}\delta }{2})\left\Vert \nabla u_{\tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+\lambda _{2}\left( u_{\tau \tau \tau }^{N}\left( x, \tau \right) , u_{\tau \tau }^{N}\left( x, \tau \right) \right) _{L^{2}(\Omega )} \\ &&+(\frac{\lambda _{2}\alpha }{2}+\frac{\lambda _{3}\beta }{2})\left\Vert u_{\tau \tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+\lambda _{2}\varrho (\nabla u^{N}(x, \tau ), \nabla u_{\tau }^{N}(x, \tau ))_{L^{2}(\Omega )}\\ &&+\frac{\lambda _{3}}{2}\left\Vert u_{\tau \tau \tau }^{N}\left( x, \tau \right) \right\Vert _{L^{2}(\Omega )}^{2}+\lambda _{3}\varrho \left( \nabla u^{N}(x, \tau ), \nabla u_{\tau \tau }^{N}(x, \tau )\right) _{L^{2}\left( \Omega \right) } \\ &&+\lambda _{3}\delta \left( \nabla u_{\tau }^{N}(x, \tau ), \nabla u_{\tau \tau }^{N}(x, \tau )\right) _{L^{2}\left( \Omega \right) }+\frac{\lambda _{3}\gamma }{2} \left\Vert \nabla u_{\tau \tau }^{N}\left( x, \tau \right) \right\Vert _{L^{2}(\Omega )}^{2} \\ &&+\frac{\lambda_{1} }{2}h\circ\nabla u^{N}(\tau )-\frac{\lambda_{1}}{2}H(\tau)\left\Vert \nabla u^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}\\ &&-\frac{\lambda_{2}}{2}h^{\prime }\circ\nabla u^{N}(\tau)+\lambda_{2}(\nabla w^{N}(\tau), \nabla u_{\tau}^{N}) _{L^{2}(\Omega )}\\ &&-\lambda_{3}H(\tau)( \nabla u_{\tau \tau }^{N}\left( x, \tau \right), \nabla u^{N}\left( x, \tau \right) ) _{L^{2}(\Omega )}^{2}\\ &&+\lambda_{3}h(\tau)( \nabla u_{\tau }^{N}\left( x, \tau \right), \nabla u^{N}\left( x, \tau \right) ) _{L^{2}(\Omega )}^{2}-\frac{\lambda_{3} }{2 }\left\Vert \nabla u^{N}\left( x, \tau\right) \right\Vert _{L^{2}(\Omega )}^{2}\\ &&+\lambda_{3}\int_{\Omega}\nabla u_{\tau\tau}^{N}\int_{0}^{\tau}h(\tau-\sigma) (\nabla u^{N}(\tau)-\nabla u^{N}(\sigma))d\sigma dx\\ &&+\lambda_{3}\int_{\Omega}\nabla u_{\tau}^{N}\int_{0}^{\tau}h'(\tau-\sigma) (\nabla u^{N}(\tau)-\nabla u^{N}(\sigma))d\sigma dx+\frac{\lambda_{3}}{2}h''\circ\nabla u^{N}(\tau)\\ & = &\lambda _{1}\left( u_{ttt}^{N}\left( x, 0\right) , u_{t}^{N}\left( x, 0\right) \right) _{L^{2}(\Omega )}+\lambda _{1}\alpha \left( u_{tt}^{N}(x, 0), u_{t}^{N}(x, 0)\right) _{L^{2}(\Omega )}\\ &&+\frac{\lambda _{1}\varrho }{2}\left\Vert \nabla u^{N}(x, 0)\right\Vert _{L^{2}(\Omega )}^{2}+\frac{\lambda _{1}\beta }{2}\left\Vert u_{t}^{N}(x, 0)\right\Vert _{L^{2}(\Omega )}^{2} \\ &&+(\frac{\lambda _{1}\gamma }{2}+\frac{\lambda _{2}\delta }{2})\left\Vert \nabla u_{t}^{N}(x, 0)\right\Vert _{L^{2}(\Omega )}^{2}+\lambda _{1}\int_{0}^{\tau }\left( u_{ttt}^{N}, u_{tt}^{N}\right) _{L^{2}(\Omega )}dt \\ &&+(\lambda _{1}\alpha -\lambda _{2}\beta )\int_{0}^{\tau }\left\Vert u_{tt}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt+(\lambda _{2}\varrho -\lambda _{1}\delta )\int_{0}^{\tau }\left\Vert \nabla u_{t}^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt \\ &&+(\lambda _{2}-\lambda _{3}\alpha )\int_{0}^{\tau }\left\Vert u_{ttt}^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt+\lambda _{2}\left( u_{ttt}^{N}\left( x, 0\right) , u_{tt}^{N}\left( x, 0\right) \right) _{L^{2}(\Omega )}\\ &&+(\frac{\lambda _{2}\alpha }{2}-\frac{\lambda _{3}\beta }{2} )\left\Vert u_{tt}^{N}(x, 0)\right\Vert _{L^{2}(\Omega )}^{2}\frac{ \lambda _{3}}{2}\left\Vert u_{ttt}^{N}\left( x, 0\right) \right\Vert _{L^{2}(\Omega )}^{2} \\ &&+\lambda _{2}\varrho (\nabla u^{N}(x, 0), \nabla u_{t}^{N}(x, 0))_{L^{2}(\Omega )}+(\lambda _{3}\delta -\lambda _{2}\gamma )\int_{0}^{\tau }\left\Vert \nabla u_{tt}^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt \\ &&+\lambda _{3}\varrho \left( \nabla u^{N}(x, 0), \nabla u_{tt}^{N}(x, 0)\right) _{L^{2}(\Omega )}+\lambda _{3}\varrho \int_{0}^{\tau }\left( \nabla u_{t}^{N}, \nabla u_{tt}^{N}\right) _{L^{2}\left( \Omega \right) }dt\\ &&+\lambda _{3}\delta \left( \nabla u_{t}^{N}(x, 0), \nabla u_{tt}^{N}(x, 0)\right) _{L^{2}\left( \Omega \right) }-\frac{\lambda _{3}\gamma }{2}\left\Vert \nabla u_{tt}^{N}\left( x, 0\right) \right\Vert _{L^{2}(\Omega )}^{2} \\ &&+\lambda _{1}\varrho \int_{\partial \Omega }u^{N}(x, \tau )\int_{0}^{\tau }\int_{\Omega }u^{N}\left( \xi , t\right) d\xi dtds_{x}\\ &&-\lambda _{1}\varrho \int_{\partial \Omega }\int_{0}^{\tau }u^{N}\left( x, t\right) \int_{\Omega }u^{N}\left( \xi , t\right) d\xi dtds_{x} \\ &&+(\lambda _{1}\delta -\lambda _{2}\varrho )\int_{\partial \Omega }\int_{0}^{\tau }u_{t}^{N}\left( x, t\right) \int_{\Omega }u^{N}\left( \xi , t\right) d\xi dtds_{x}\\ &&-\lambda _{1}\delta \int_{\partial \Omega }\int_{0}^{\tau }u_{t}^{N}\left( x, t\right) \int_{\Omega }u^{N}(\xi , 0)d\xi dtds_{x} \\ &&+(\lambda _{1}\gamma -\lambda _{2}\delta )\int_{0}^{\tau }\int_{\partial \Omega }u_{t}^{N}\left( x, t\right) \left( \int\nolimits_{\Omega }u_{t}^{N}\left( \xi , t\right) d\xi \right) ds_{x}dt\\ &&-\lambda _{1}\gamma \int_{0}^{\tau }\int_{\partial \Omega }u_{t}^{N}\left( x, t\right) \left( \int\nolimits_{\Omega }u_{t}^{N}(\xi , 0)d\xi \right) ds_{x}dt \\ &&+\lambda _{2}\varrho \int_{\partial \Omega }u_{\tau }^{N}(x, \tau )\int_{0}^{\tau }\int_{\Omega }u^{N}\left( \xi , t\right) d\xi dtds_{x}\\ &&+\lambda _{2}\delta \int_{\partial \Omega }u_{\tau }^{N}(x, \tau )\int_{\Omega }u^{N}\left( \xi , \tau \right) d\xi ds_{x} -\lambda _{2}\delta \int_{\partial \Omega }u_{\tau }^{N}(x, \tau )\int_{\Omega }u^{N}(\xi , 0)d\xi ds_{x}\\ &&+\lambda _{2}\gamma \int_{\partial \Omega }u_{\tau }^{N}(x, \tau )\int_{\Omega }u_{\tau }^{N}\left( \xi , \tau \right) d\xi ds_{x}-\lambda _{2}\gamma \int_{\partial \Omega }u_{\tau }^{N}(x, \tau )\int_{\Omega }u_{t}^{N}(\xi , 0)d\xi ds_{x}\\ &&-\lambda _{2}\gamma \int_{\partial \Omega }\int_{0}^{\tau }u_{t}^{N}\left( x, t\right) \int_{\Omega }u_{tt}^{N}\left( \xi , t\right) d\xi dtds_{x} \\ &&+\lambda _{3}\varrho \int_{\partial \Omega }u_{\tau \tau }^{N}(x, \tau )\int_{0}^{\tau }\int_{\Omega }u^{N}\left( \xi , t\right) d\xi dtds_{x}\\ &&-\lambda _{3}\varrho \int_{\partial \Omega }\int_{0}^{\tau }u_{tt}^{N}\left( x, t\right) \int_{\Omega }u^{N}\left( \xi , t\right) d\xi dtds_{x} \\ &&+\lambda _{3}\delta \int_{\partial \Omega }u_{\tau \tau }^{N}(x, \tau )\int_{\Omega }u^{N}\left( \xi , \tau \right) d\xi ds_{x}-\lambda _{3}\delta \int_{\partial \Omega }u_{\tau \tau }^{N}(x, \tau )\int_{\Omega }u^{N}(\xi , 0)d\xi ds_{x} \\ &&-\lambda _{3}\delta \int_{\partial \Omega }\int_{0}^{\tau }u_{tt}^{N}\left( x, t\right) \int_{\Omega }u_{t}^{N}\left( \xi , t\right) d\xi dtds\\ &&+\lambda _{3}\gamma \int_{\partial \Omega }u_{\tau \tau }^{N}(x, \tau )\int_{\Omega }u_{\tau }^{N}\left( \xi , \tau \right) d\xi ds_{x}-\lambda _{3}\gamma \int_{\partial \Omega }u_{\tau \tau }^{N}(x, \tau )\int_{\Omega }u_{t}^{N}(\xi , 0)d\xi ds_{x}\\ &&-\lambda _{3}\gamma \int_{\partial \Omega }\int_{0}^{\tau }u_{tt}^{N}\left( x, t\right) \int_{\Omega }u_{tt}^{N}\left( \xi , t\right) d\xi dtds\\ &&+\lambda_{1}\left( F, u_{t}^{N}\right) _{L^{2}\left( Q_{\tau }\right) }+\lambda_{2}\left( F, u_{tt}^{N}\right) _{L^{2}\left( Q_{\tau }\right) }+\lambda_{3}\left( F, u_{ttt}^{N}\right) _{L^{2}\left( Q_{\tau }\right) }\\ &&+\frac{\lambda_{1} }{2}\int_{0}^{\tau}h'\circ\nabla u^{N}(t )dt-\frac{\lambda_{1}}{2}h(t)\left\Vert \nabla u^{N}(x, t)\right\Vert _{L^{2}(\Omega )}^{2}dt\\ &&-\lambda_{1}\int_{\partial \Omega }u^{N}(x, \tau) \int_{0}^{\tau}\int_{\Omega }H(t)u^{N}(\xi , t )d\xi dt ds_{x}\\ &&+\lambda_{1}\int_{0}^{\tau }\int_{\partial \Omega }u^{N}\left( x, t\right) \int\nolimits_{\Omega }H(t)u^{N}(\xi , t)d\xi ds_{x}dt\\ &&+\lambda_{1}\int_{\partial \Omega }u^{N}(x, \tau)\left( \int_{0}^{\tau}\int_{\Omega }\bigg[\int_{0}^{t}h(t-\sigma)(u^{N}(\xi , t )-u^{N}(\xi , \sigma ))d\sigma\bigg] d\xi \right) ds_{x}dt\\ &&-\lambda_{1}\int_{0}^{\tau }\int_{\partial \Omega }u^{N}\left( x, t\right) \int\nolimits_{\Omega }\bigg[\int_{0}^{t}h(t-\sigma)(u^{N}(\xi , t )-u^{N}(\xi , \sigma ))d\sigma\bigg]d\xi ds_{x}dt\\ &&-\lambda_{3}\int_{\partial \Omega }u_{\tau\tau}^{N}(x, \tau) \int_{0}^{\tau}\int_{\Omega }H(t)u^{N}(\xi , t )d\xi dt ds_{x}\\ &&-\frac{\lambda_{3}}{2}\int_{0}^{\tau}(h''-h''')\circ\nabla u^{N}(t)dt+\lambda_{3}h(0)\int_{0}^{\tau}\left\Vert \nabla u_{t }^{N}\left( x, t \right) \right\Vert _{L^{2}(\Omega )}^{2}dt\\ &&+\lambda_{3}\int_{0}^{\tau }\int_{\partial \Omega }u_{tt}^{N}\left( x, t\right) \int\nolimits_{\Omega }H(t)u^{N}(\xi , t)d\xi ds_{x}dt\\ &&+\lambda_{3}\int_{\partial \Omega }u_{\tau\tau}^{N}(x, \tau)\left( \int_{0}^{\tau}\int_{\Omega }\bigg[\int_{0}^{t}h(t-\sigma)(u^{N}(\xi , t )-u^{N}(\xi , \sigma ))d\sigma\bigg] d\xi \right) ds_{x}dt\\ &&-\lambda_{3}\int_{0}^{\tau }\int_{\partial \Omega }u_{tt}^{N}\left( x, t\right) \int\nolimits_{\Omega }\bigg[\int_{0}^{t}h(t-\sigma)(u^{N}(\xi , t )-u^{N}(\xi , \sigma ))d\sigma\bigg]d\xi ds_{x}dt\\ &&-\lambda_{2}\int_{\partial \Omega }u_{\tau}^{N}(x, \tau) \int_{0}^{\tau}\int_{\Omega }H(t)u^{N}(\xi , t )d\xi dt ds_{x}\\ &&+\lambda_{2}\int_{0}^{\tau }\int_{\partial \Omega }u_{t}^{N}\left( x, t\right) \int\nolimits_{\Omega }H(t)u^{N}(\xi , t)d\xi ds_{x}dt\\ &&+\lambda_{2}\int_{\partial \Omega }u_{\tau}^{N}(x, \tau)\left( \int_{0}^{\tau}\int_{\Omega }\bigg[\int_{0}^{t}h(t-\sigma)(u^{N}(\xi , t )-u^{N}(\xi , \sigma ))d\sigma\bigg] d\xi \right) ds_{x}dt\\ &&-\lambda_{2}\int_{0}^{\tau }\int_{\partial \Omega }u_{t}^{N}\left( x, t\right) \int\nolimits_{\Omega }\bigg[\int_{0}^{t}h(t-\sigma)(u^{N}(\xi , t )-u^{N}(\xi , \sigma ))d\sigma\bigg]d\xi ds_{x}dt\\ &&-\frac{\lambda_{2}}{2}\int_{0}^{\tau }h''\circ\nabla u^{N}(t)dt+\frac{\lambda_{2}}{2}\int_{0}^{\tau }h'(t)\left\Vert \nabla u^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt. \end{eqnarray} | (3.31) |
We can estimate all the terms in the RHS of (3.31) as follows
\begin{equation} \begin{array}{l} \lambda _{1}\varrho \int_{\partial \Omega }u^{N}(x, \tau )\int_{0}^{\tau }\int_{\Omega }u^{N}\left( \xi , t\right) d\xi dtds_{x} \\ \leq \frac{\lambda _{1}\varrho }{2\varepsilon _{1}}\left( \varepsilon \left\Vert \nabla u^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+l(\varepsilon )\left\Vert u^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}\right) \\ +\frac{\lambda _{1}\varrho }{2}\varepsilon _{1}T\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \int_{0}^{\tau }\left\Vert u^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt, \end{array} \end{equation} | (3.32) |
\begin{equation} \begin{array}{l} -\lambda _{1}\varrho \int_{\partial \Omega }\int_{0}^{\tau }u^{N}\left( x, t\right) \int_{\Omega }u^{N}\left( \xi , t\right) d\xi dtds_{x} \\ \leq \frac{\lambda _{1}\varrho }{2}\varepsilon \int_{0}^{\tau }\left\Vert \nabla u^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt \\ +\frac{\lambda _{1}\varrho }{2}\left( l(\varepsilon )+\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \right) \int_{0}^{\tau }\left\Vert u^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt, \end{array} \end{equation} | (3.33) |
\begin{equation} \begin{array}{l} (\lambda _{1}\delta -\lambda _{2}\varrho )\int_{\partial \Omega }\int_{0}^{\tau }u_{t}^{N}\left( x, t\right) \int_{\Omega }u^{N}\left( \xi , t\right) d\xi dtds_{x} \\ \leq \frac{(\lambda _{1}\delta +\lambda _{2}\varrho )}{2}\left( \varepsilon \int_{0}^{\tau }\left\Vert \nabla u_{t}^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt+l(\varepsilon )\int_{0}^{\tau }\left\Vert u_{t}^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt\right) \\ +\frac{(\lambda _{1}\delta +\lambda _{2}\varrho )}{2}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \int_{0}^{\tau }\left\Vert u^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt, \end{array} \end{equation} | (3.34) |
\begin{equation} \begin{array}{l} -\lambda _{1}\delta \int_{\partial \Omega }\int_{0}^{\tau }u_{t}^{N}\left( x, t\right) \int_{\Omega }u^{N}(\xi , 0)d\xi dtds_{x} \\ \leq \frac{\lambda _{1}\delta }{2}\left( \varepsilon \int_{0}^{\tau }\left\Vert \nabla u_{t}^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt+l(\varepsilon )\int_{0}^{\tau }\left\Vert u_{t}^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt\right) \\ +\frac{\lambda _{1}\delta }{2}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert T\left\Vert u^{N}(x, 0)\right\Vert _{L^{2}(\Omega )}^{2}, \end{array} \end{equation} | (3.35) |
\begin{equation} \begin{array}{l} \lambda _{2}\varrho \int_{\partial \Omega }u_{\tau }^{N}(x, \tau )\int_{0}^{\tau }\int_{\Omega }u^{N}\left( \xi , t\right) d\xi dtds_{x} \\ \leq \frac{\lambda _{2}\varrho }{2}\left( \frac{\varepsilon }{\varepsilon _{2}}\left\Vert \nabla u_{\tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+\frac{l(\varepsilon )}{\varepsilon _{2}}\left\Vert u_{\tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}\right) \\ +\frac{\lambda _{2}\varrho }{2}\varepsilon _{2}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert T\int_{0}^{\tau }\left\Vert u^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt, \end{array} \end{equation} | (3.36) |
\begin{equation} \begin{array}{l} \lambda _{2}\delta \int_{\partial \Omega }u_{\tau }^{N}(x, \tau )\int_{\Omega }u^{N}\left( \xi , \tau \right) d\xi ds_{x} \\ \leq \frac{\lambda _{2}\delta }{2\varepsilon _{3}}\left( \varepsilon \left\Vert \nabla u_{\tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+l(\varepsilon )\left\Vert u_{\tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}\right) \\ +\frac{\lambda _{2}\delta }{2}\varepsilon _{3}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \left\Vert u^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}, \end{array} \end{equation} | (3.37) |
\begin{equation} \begin{array}{l} -\lambda _{2}\delta \int_{\partial \Omega }u_{\tau }^{N}(x, \tau )\int_{\Omega }u^{N}(\xi , 0)d\xi ds_{x} \\ \leq \frac{\lambda _{2}\delta }{2\varepsilon _{4}}\left( \varepsilon \left\Vert \nabla u_{\tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+l(\varepsilon )\left\Vert u_{\tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}\right) \\ +\frac{\lambda _{2}\delta }{2}\varepsilon _{4}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \left\Vert u^{N}(x, 0)\right\Vert _{L^{2}(\Omega )}^{2}, \end{array} \end{equation} | (3.38) |
\begin{equation} \begin{array}{l} (\lambda _{1}\gamma -\lambda _{2}\delta )\int_{\partial \Omega }\int_{0}^{\tau }u_{t}^{N}\left( x, t\right) \int_{\Omega }u_{t}^{N}\left( \xi , t\right) d\xi dtds_{x} \\ \leq \frac{(\lambda _{1}\gamma +\lambda _{2}\delta )}{2}\varepsilon \int_{0}^{\tau }\left\Vert \nabla u_{t}^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt \\ +\frac{(\lambda _{1}\gamma +\lambda _{2}\delta )}{2}\left( l(\varepsilon )+\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \right) \int_{0}^{\tau }\left\Vert u_{t}^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt, \end{array} \end{equation} | (3.39) |
\begin{equation} \begin{array}{l} -\lambda _{1}\gamma \int_{0}^{\tau }\int_{\partial \Omega }u_{t}^{N}\left( x, t\right) \left( \int\nolimits_{\Omega }u_{t}^{N}(\xi , 0)d\xi \right) ds_{x}dt \\ \leq \frac{\lambda _{1}\gamma }{2}\left( \varepsilon \int_{0}^{\tau }\left\Vert \nabla u_{t}^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt+l(\varepsilon )\int_{0}^{\tau }\left\Vert u_{t}^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt\right) \\ +\frac{\lambda _{1}\gamma }{2}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert T\left\Vert u_{t}^{N}(x, 0)\right\Vert _{L^{2}(\Omega )}^{2}, \end{array} \end{equation} | (3.40) |
\begin{equation} \begin{array}{l} \lambda _{2}\gamma \int_{\partial \Omega }u_{\tau }^{N}(x, \tau )\int_{\Omega }u_{\tau }^{N}\left( \xi , \tau \right) d\xi ds_{x} \\ \leq \frac{\lambda _{2}\gamma }{2\varepsilon _{5}}\left( \varepsilon \left\Vert \nabla u_{\tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+l(\varepsilon )\left\Vert u_{\tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}\right) \\ +\frac{\lambda _{2}\gamma }{2}\varepsilon _{5}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \left\Vert u_{\tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}, \end{array} \end{equation} | (3.41) |
\begin{equation} \begin{array}{l} -\lambda _{2}\gamma \int_{\partial \Omega }u_{\tau }^{N}(x, \tau )\int_{\Omega }u_{t}^{N}(\xi , 0)d\xi ds_{x} \\ \leq \frac{\lambda _{2}\gamma }{2\varepsilon _{6}}\left( \varepsilon \left\Vert \nabla u_{\tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+l(\varepsilon )\left\Vert u_{\tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}\right) \\ +\frac{\lambda _{2}\gamma }{2}\varepsilon _{6}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \left\Vert u_{t}^{N}(x, 0)\right\Vert _{L^{2}(\Omega )}^{2}, \end{array} \end{equation} | (3.42) |
\begin{eqnarray} &&-\lambda _{2}\gamma \int_{\partial \Omega }\int_{0}^{\tau }u_{t}^{N}\left( x, t\right) \int_{\Omega }u_{tt}^{N}\left( \xi , t\right) d\xi dtds_{x} \\ &\leq &\frac{\lambda _{2}\gamma }{2}\varepsilon \int_{0}^{\tau }\left\Vert \nabla u_{t}^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt \\ &&+\frac{\lambda _{2}\gamma }{2}l(\varepsilon )\int_{0}^{\tau }\left\Vert u_{t}^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt \\ &&+\frac{\lambda _{2}\gamma }{2}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \int_{0}^{\tau }\left\Vert u_{tt}^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt, \end{eqnarray} | (3.43) |
\begin{eqnarray} &&\lambda _{3}\varrho \int_{\partial \Omega }u_{\tau \tau }^{N}(x, \tau )\int_{0}^{\tau }\int_{\Omega }u^{N}\left( \xi , t\right) d\xi dtds_{x} \\ &\leq &\frac{\lambda _{3}\varrho }{2}\left( \frac{\varepsilon }{\varepsilon _{7}}\left\Vert \nabla u_{\tau \tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+\frac{l(\varepsilon )}{\varepsilon _{7}}\left\Vert u_{\tau \tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}\right) \\ &&+\frac{\lambda _{3}\varrho }{2}\varepsilon _{7}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert T\int_{0}^{\tau }\left\Vert u^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt, \end{eqnarray} | (3.44) |
and
\begin{eqnarray} &&-\lambda _{3}\varrho \int_{\partial \Omega }\int_{0}^{\tau }u_{tt}^{N}\left( x, t\right) \int_{\Omega }u^{N}\left( \xi , t\right) d\xi dtds_{x} \\ &\leq &\frac{\lambda _{3}\varrho }{2}\left( \varepsilon \int_{0}^{\tau }\left\Vert \nabla u_{tt}^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt+l(\varepsilon )\int_{0}^{\tau }\left\Vert u_{tt}^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt\right) \\ &&+\frac{\lambda _{3}\varrho }{2}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \int_{0}^{\tau }\left\Vert u^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt, \end{eqnarray} | (3.45) |
\begin{eqnarray} &&\lambda _{3}\delta \int_{\partial \Omega }u_{\tau \tau }^{N}(x, \tau )\int_{\Omega }u^{N}\left( \xi , \tau \right) d\xi ds_{x} \\ &\leq &\frac{\lambda _{3}\delta }{2\varepsilon _{8}}\left( \varepsilon \left\Vert \nabla u_{\tau \tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+l(\varepsilon )\left\Vert u_{\tau \tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}\right) \\ &&+\frac{\lambda _{3}\delta }{2}\varepsilon _{8}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \left\Vert u^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}, \end{eqnarray} | (3.46) |
\begin{eqnarray} &&-\lambda _{3}\delta \int_{\partial \Omega }u_{\tau \tau }^{N}(x, \tau )\int_{\Omega }u^{N}(\xi , 0)d\xi ds_{x} \\ &\leq &\frac{\lambda _{3}\delta }{2\varepsilon _{9}}\left( \varepsilon \left\Vert \nabla u_{\tau \tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+l(\varepsilon )\left\Vert u_{\tau \tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}\right) \\ &&+\frac{\lambda _{3}\delta }{2}\varepsilon _{9}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \left\Vert u^{N}(x, 0)\right\Vert _{L^{2}(\Omega )}^{2}, \end{eqnarray} | (3.47) |
\begin{eqnarray} &&-\lambda _{3}\delta \int_{\partial \Omega }\int_{0}^{\tau }u_{tt}^{N}\left( x, t\right) \int_{\Omega }u_{t}^{N}\left( \xi , t\right) d\xi dtds_{x} \\ &\leq &\frac{\lambda _{3}\delta }{2}\left( \varepsilon \int_{0}^{\tau }\left\Vert \nabla u_{tt}^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt+l(\varepsilon )\int_{0}^{\tau }\left\Vert u_{tt}^{N}(x, t)\right\Vert _{L^{2}(\Omega )}^{2}dt\right) \\ &&+\frac{\lambda _{3}\delta }{2}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \int_{0}^{\tau }\left\Vert u_{t}^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt, \end{eqnarray} | (3.48) |
\begin{eqnarray} &&\lambda _{3}\gamma \int_{\partial \Omega }u_{\tau \tau }^{N}(x, \tau )\int_{\Omega }u_{\tau }^{N}\left( \xi , \tau \right) d\xi ds_{x} \\ &\leq &\frac{\lambda _{3}\gamma }{2\varepsilon _{10}}\left( \varepsilon \left\Vert \nabla u_{\tau \tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+l(\varepsilon )\left\Vert u_{\tau \tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}\right) \\ &&+\frac{\lambda _{3}\gamma }{2}\varepsilon _{10}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \left\Vert u_{\tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}, \end{eqnarray} | (3.49) |
\begin{eqnarray} &&-\lambda _{3}\gamma \int_{\partial \Omega }u_{\tau \tau }^{N}(x, \tau )\int_{\Omega }u_{t}^{N}(\xi , 0)d\xi ds_{x} \\ &\leq &\frac{\lambda _{3}\gamma }{2\varepsilon _{11}}\left( \varepsilon \left\Vert \nabla u_{\tau \tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+l(\varepsilon )\left\Vert u_{\tau \tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}\right) \\ &&+\frac{\lambda _{3}\gamma }{2}\varepsilon _{11}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \left\Vert u_{t}^{N}(x, 0)\right\Vert _{L^{2}(\Omega )}^{2}, \end{eqnarray} | (3.50) |
\begin{eqnarray} &&-\lambda _{3}\gamma \int_{\partial \Omega }\int_{0}^{\tau }u_{tt}^{N}\left( x, t\right) \int_{\Omega }u_{tt}^{N}\left( \xi , t\right) d\xi dtds_{x} \\ &\leq &\frac{\lambda _{3}\gamma }{2}\varepsilon \int_{0}^{\tau }\left\Vert \nabla u_{tt}^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt \\ &&+\frac{\lambda _{3}\gamma }{2}\left( l(\varepsilon )+\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \right) \int_{0}^{\tau }\left\Vert u_{tt}^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt, \end{eqnarray} | (3.51) |
\begin{eqnarray} &&-\frac{\lambda _{1}}{2}\left\Vert u_{\tau \tau \tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}-\frac{\lambda _{1}}{2}\left\Vert u_{\tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2} \\ &\leq &\lambda _{1}\left( u_{\tau \tau \tau }^{N}(x, \tau ), u_{\tau }^{N}(x, \tau )\right) _{L^{2}(\Omega )}, \end{eqnarray} | (3.52) |
\begin{eqnarray} &&-\frac{\lambda _{2}}{2}\left\Vert u_{\tau \tau \tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}-\frac{\lambda _{2}}{2}\left\Vert u_{\tau \tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2} \\ &\leq &\lambda _{2}\left( u_{\tau \tau \tau }^{N}(x, \tau ), u_{\tau \tau }^{N}(x, \tau )\right) _{L^{2}(\Omega )}, \end{eqnarray} | (3.53) |
\begin{eqnarray} &&-\frac{\lambda _{1}\alpha }{2}\left\Vert u_{\tau \tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}-\frac{\lambda _{1}\alpha }{2}\left\Vert u_{\tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2} \\ &\leq &\lambda _{1}\alpha \left( u_{\tau \tau }^{N}(x, \tau ), u_{\tau }^{N}(x, \tau )\right) _{L^{2}(\Omega )}, \end{eqnarray} | (3.54) |
\begin{eqnarray} &&-\frac{\lambda _{2}\varrho \varepsilon _{12}}{2}\left\Vert \nabla u^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}-\frac{\lambda _{2}\varrho }{ 2\varepsilon _{12}}\left\Vert \nabla u_{\tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2} \\ &\leq &\lambda _{2}\varrho \left( \nabla u^{N}(x, \tau ), \nabla u_{\tau }^{N}(x, \tau )\right) _{L^{2}(\Omega )}, \end{eqnarray} | (3.55) |
\begin{eqnarray} &&-\frac{\lambda _{2}\varrho \varepsilon _{13}}{2}\left\Vert \nabla u^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}-\frac{\lambda _{2}\varrho }{ 2\varepsilon _{13}}\left\Vert \nabla u_{\tau \tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2} \\ &\leq &\lambda _{3}\varrho \left( \nabla u^{N}(x, \tau ), \nabla u_{\tau \tau }^{N}(x, \tau )\right) _{L^{2}(\Omega )}, \end{eqnarray} | (3.56) |
\begin{eqnarray} &&-\frac{\lambda _{3}\delta \varepsilon _{14}}{2}\left\Vert \nabla u_{\tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}-\frac{\lambda _{3}\delta }{2\varepsilon _{14}}\left\Vert \nabla u_{\tau \tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2} \\ &\leq &\lambda _{3}\delta \left( \nabla u_{\tau }^{N}(x, \tau ), \nabla u_{\tau \tau }^{N}(x, \tau )\right) _{L^{2}(\Omega )}, \end{eqnarray} | (3.57) |
\begin{eqnarray} &&\lambda _{1}\left( u_{ttt}^{N}(x, 0), u_{t}^{N}(x, 0)\right) _{L^{2}(\Omega )} \\ &\leq &\frac{\lambda _{1}}{2}\left\Vert u_{ttt}^{N}(x, 0)\right\Vert _{L^{2}(\Omega )}^{2}+\frac{\lambda _{1}}{2}\left\Vert u_{t}^{N}(x, 0)\right\Vert _{L^{2}(\Omega )}^{2} \end{eqnarray} | (3.58) |
\begin{eqnarray} &&\lambda _{1}\alpha \left( u_{tt}^{N}(x, 0), u_{t}^{N}(x, 0)\right) _{L^{2}(\Omega )} \\ &\leq &\frac{\lambda _{1}\alpha }{2}\left\Vert u_{tt}^{N}(x, 0)\right\Vert _{L^{2}(\Omega )}^{2}+\frac{\lambda _{1}\alpha }{2}\left\Vert u_{t}^{N}(x, 0)\right\Vert _{L^{2}(\Omega )}^{2}, \end{eqnarray} | (3.59) |
\begin{eqnarray} &&\lambda _{2}\left( u_{ttt}^{N}(x, 0), u_{tt}^{N}(x, 0)\right) _{L^{2}(\Omega )} \\ &\leq &\frac{\lambda _{2}}{2}\left\Vert u_{ttt}^{N}(x, 0)\right\Vert _{L^{2}(\Omega )}^{2}+\frac{\lambda _{2}}{2}\left\Vert u_{tt}^{N}(x, 0)\right\Vert _{L^{2}(\Omega )}^{2}, \end{eqnarray} | (3.60) |
\begin{eqnarray} &&\lambda _{2}\varrho \left( \nabla u^{N}(x, 0), \nabla u_{t}^{N}(x, 0)\right) _{L^{2}(\Omega )} \\ &\leq &\frac{\lambda _{2}}{2}\varrho \left\Vert \nabla u^{N}(x, 0)\right\Vert _{L^{2}(\Omega )}^{2}+\frac{\lambda _{2}}{2}\varrho \left\Vert \nabla u_{t}^{N}(x, 0)\right\Vert _{L^{2}(\Omega )}^{2}, \end{eqnarray} | (3.61) |
\begin{eqnarray} &&\lambda _{3}\varrho \left( \nabla u^{N}(x, 0), \nabla u_{tt}^{N}(x, 0)\right) _{L^{2}(\Omega )} \\ &\leq &\frac{\lambda _{3}}{2}\varrho \left\Vert \nabla u^{N}(x, 0)\right\Vert _{L^{2}(\Omega )}^{2}+\frac{\lambda _{3}}{2}\varrho \left\Vert \nabla u_{tt}^{N}(x, 0)\right\Vert _{L^{2}(\Omega )}^{2}, \end{eqnarray} | (3.62) |
\begin{eqnarray} &&\lambda _{3}\delta \left( \nabla u_{t}^{N}(x, 0), \nabla u_{tt}^{N}(x, 0)\right) _{L^{2}(\Omega )} \\ &\leq &\frac{\lambda _{3}}{2}\delta \left\Vert \nabla u_{t}^{N}(x, 0)\right\Vert _{L^{2}(\Omega )}^{2}+\frac{\lambda _{3}}{2}\delta \left\Vert \nabla u_{tt}^{N}(x, 0)\right\Vert _{L^{2}(\Omega )}^{2}, \end{eqnarray} | (3.63) |
\begin{equation} \lambda _{1}\int_{0}^{\tau }\left( u_{ttt}^{N}, u_{tt}^{N}\right) _{L^{2}(\Omega )}dt\leq \frac{\lambda _{1}}{2}\int_{0}^{\tau }\left\Vert u_{ttt}^{N}(x, t)\right\Vert _{L^{2}(\Omega )}^{2}dt+\frac{\lambda _{1}}{2} \int_{0}^{\tau }\left\Vert u_{tt}^{N}(x, t)\right\Vert _{L^{2}(\Omega )}^{2}dt, \end{equation} | (3.64) |
\begin{equation} \lambda _{3}\varrho \int_{0}^{\tau }\left( \nabla u_{t}^{N}, \nabla u_{tt}^{N}\right) _{L^{2}\left( \Omega \right) }dt\leq \frac{\lambda _{3}\varrho }{2}\int_{0}^{\tau }\left\Vert \nabla u_{t}^{N}(x, t)\right\Vert _{L^{2}(\Omega )}^{2}dt+\frac{\lambda _{3}\varrho }{2}\int_{0}^{\tau }\left\Vert u_{tt}^{N}(x, t)\right\Vert _{L^{2}(\Omega )}^{2}dt, \end{equation} | (3.65) |
\begin{equation} \begin{array}{l} \lambda _{1} \int_{\partial \Omega }u^{N}(x, \tau )\int_{0}^{\tau }\int_{\Omega }H(t)u^{N}\left( \xi , t\right) d\xi dtds_{x} \\ \leq \frac{\lambda _{1}h_{0} }{2}\left( \varepsilon \left\Vert \nabla u^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+l(\varepsilon )\left\Vert u^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}\right) \\ +\frac{\lambda _{1}h_{0} }{2}T\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \int_{0}^{\tau }\left\Vert u^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt, \end{array} \end{equation} | (3.66) |
\begin{equation} \begin{array}{l} -\lambda _{1} \int_{\partial \Omega }\int_{0}^{\tau }u^{N}\left( x, t\right) \int_{\Omega }H(t)u^{N}\left( \xi , t\right) d\xi dtds_{x} \\ \leq \frac{\lambda _{1}h_{0} }{2}\varepsilon \int_{0}^{\tau }\left\Vert \nabla u^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt \\ +\frac{\lambda _{1}h_{0} }{2}\left( l(\varepsilon )+\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \right) \int_{0}^{\tau }\left\Vert u^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt, \end{array} \end{equation} | (3.67) |
\begin{equation} \begin{array}{l} \lambda _{1} \int_{\partial \Omega }u^{N}(x, \tau )\int_{0}^{\tau }\int_{\Omega }\int_{0}^{t}h(t-\sigma)(u^{N}\left( \xi , t\right)-u^{N}\left( \xi , \sigma\right))d\sigma d\xi dtds_{x} \\ \leq \frac{\lambda _{1}h_{0} }{2}\left( \varepsilon \left\Vert \nabla u^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+l(\varepsilon )\left\Vert u^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}\right) \\ +\frac{\lambda _{1} }{2}T\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \int_{0}^{\tau }h\circ u^{N}(t )dt, \end{array} \end{equation} | (3.68) |
\begin{equation} \begin{array}{l} -\lambda _{1} \int_{\partial \Omega }\int_{0}^{\tau }u^{N}\left( x, t\right) \int_{\Omega }\int_{0}^{t}h(t-\sigma)(u^{N}\left( \xi , t\right)-u^{N}\left( \xi , \sigma\right))d\sigma d\xi dtds_{x} \\ \leq \frac{\lambda _{1}h_{0} }{2}\varepsilon \int_{0}^{\tau }\left\Vert \nabla u^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt+\frac{\lambda _{1}h_{0} }{2}l(\varepsilon) \int_{0}^{\tau }\left\Vert u^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt \\ +\frac{\lambda _{1}}{2}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \int_{0}^{\tau }h\circ u^{N}(t )dt, \end{array} \end{equation} | (3.69) |
\begin{equation} \begin{array}{l} \lambda _{2} \int_{\partial \Omega }u_{\tau}^{N}(x, \tau )\int_{0}^{\tau }\int_{\Omega }H(t)u^{N}\left( \xi , t\right) d\xi dtds_{x} \\ \leq \frac{\lambda _{2}h_{0} }{2}\left( \varepsilon \left\Vert \nabla u_{\tau}^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+l(\varepsilon )\left\Vert u_{\tau}^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}\right) \\ +\frac{\lambda _{2}h_{0} }{2}T\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \int_{0}^{\tau }\left\Vert u^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt, \end{array} \end{equation} | (3.70) |
\begin{equation} \begin{array}{l} -\lambda _{2} \int_{\partial \Omega }\int_{0}^{\tau }u_{\tau}^{N}\left( x, t\right) \int_{\Omega }H(t)u^{N}\left( \xi , t\right) d\xi dtds_{x} \\ \leq \frac{\lambda _{2}h_{0} }{2}\varepsilon \int_{0}^{\tau }\left\Vert \nabla u_{t}^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt+\frac{\lambda _{2}h_{0} }{2} l(\varepsilon )\int_{0}^{\tau }\left\Vert u_{t}^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt \\ +\frac{\lambda _{2}h_{0} }{2}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \int_{0}^{\tau }\left\Vert u^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt, \end{array} \end{equation} | (3.71) |
\begin{equation} \begin{array}{l} \lambda _{2} \int_{\partial \Omega }u_{\tau}^{N}(x, \tau )\int_{0}^{\tau }\int_{\Omega }\int_{0}^{t}h(t-\sigma)(u^{N}\left( \xi , t\right)-u^{N}\left( \xi , \sigma\right))d\sigma d\xi dtds_{x} \\ \leq \frac{\lambda _{2}h_{0} }{2}\left( \varepsilon \left\Vert \nabla u_{\tau}^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+l(\varepsilon )\left\Vert u_{\tau}^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}\right) \\ +\frac{\lambda _{2} }{2}T\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \int_{0}^{\tau }h\circ u^{N}(t )dt, \end{array} \end{equation} | (3.72) |
\begin{equation} \begin{array}{l} -\lambda _{2} \int_{\partial \Omega }\int_{0}^{\tau }u_{t}^{N}\left( x, t\right) \int_{\Omega }\int_{0}^{t}h(t-\sigma)(u^{N}\left( \xi , t\right)-u^{N}\left( \xi , \sigma\right))d\sigma d\xi dtds_{x} \\ \leq \frac{\lambda _{2}h_{0} }{2}\varepsilon \int_{0}^{\tau }\left\Vert \nabla u_{t}^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt+\frac{\lambda _{2}}{2} h_{0}l(\varepsilon )\int_{0}^{\tau }\left\Vert u_{t}^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt \\ +\frac{\lambda _{2}}{2}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \int_{0}^{\tau }h\circ u^{N}(t )dt, \end{array} \end{equation} | (3.73) |
\begin{equation} \begin{array}{l} \lambda _{3} \int_{\partial \Omega }u_{\tau\tau}^{N}(x, \tau )\int_{0}^{\tau }\int_{\Omega }H(t)u^{N}\left( \xi , t\right) d\xi dtds_{x} \\ \leq \frac{\lambda _{3}h_{0} }{2\varepsilon _{18}}\left( \varepsilon \left\Vert \nabla u_{\tau\tau}^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+l(\varepsilon )\left\Vert u_{\tau\tau}^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}\right) \\ +\frac{\lambda _{3}h_{0} }{2}\varepsilon _{18}T\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \int_{0}^{\tau }\left\Vert u^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt, \end{array} \end{equation} | (3.74) |
\begin{equation} \begin{array}{l} -\lambda _{3} \int_{\partial \Omega }\int_{0}^{\tau }u_{\tau\tau}^{N}\left( x, t\right) \int_{\Omega }H(t)u^{N}\left( \xi , t\right) d\xi dtds_{x} \\ \leq \frac{\lambda _{3}h_{0} }{2}\varepsilon \int_{0}^{\tau }\left\Vert \nabla u_{tt}^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt+\frac{\lambda _{3}h_{0} }{2} l(\varepsilon )\int_{0}^{\tau }\left\Vert u_{tt}^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt \\ +\frac{\lambda _{3}h_{0} }{2}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \int_{0}^{\tau }\left\Vert u^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt, \end{array} \end{equation} | (3.75) |
\begin{equation} \begin{array}{l} \lambda _{3} \int_{\partial \Omega }u_{\tau\tau}^{N}(x, \tau )\int_{0}^{\tau }\int_{\Omega }\int_{0}^{t}h(t-\sigma)(u^{N}\left( \xi , t\right)-u^{N}\left( \xi , \sigma\right))d\sigma d\xi dtds_{x} \\ \leq \frac{\lambda _{3}h_{0} }{2\varepsilon _{15}}\left( \varepsilon \left\Vert \nabla u_{\tau\tau}^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+l(\varepsilon )\left\Vert u_{\tau\tau}^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}\right) \\ +\frac{\lambda _{3} }{2}\varepsilon _{15}T\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \int_{0}^{\tau }h\circ u^{N}(t )dt, \end{array} \end{equation} | (3.76) |
\begin{equation} \begin{array}{l} -\lambda _{3} \int_{\partial \Omega }\int_{0}^{\tau }u_{tt}^{N}\left( x, t\right) \int_{\Omega }\int_{0}^{t}h(t-\sigma)(u^{N}\left( \xi , t\right)-u^{N}\left( \xi , \sigma\right))d\sigma d\xi dtds_{x} \\ \leq \frac{\lambda _{3}h_{0} }{2}\varepsilon \int_{0}^{\tau }\left\Vert \nabla u_{tt}^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt+\frac{\lambda _{3}}{2} h_{0}l(\varepsilon )\int_{0}^{\tau }\left\Vert u_{tt}^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt \\ +\frac{\lambda _{3}}{2}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \int_{0}^{\tau }h\circ u^{N}(t )dt, \end{array} \end{equation} | (3.77) |
\begin{eqnarray} -\lambda_{3}H(\tau)(\nabla u_{\tau\tau}^{N}, \nabla u^{N})_{L^{2}(\Omega )}^{2}&\geq& -\frac{\lambda _{3}h_{0}}{2\varepsilon_{16}}\left\Vert\nabla u_{\tau\tau}^{N}\left( x, \tau\right) \right\Vert _{L^{2}(\Omega )}^{2}\\ &&-\frac{\lambda _{3}h_{0}\varepsilon_{16}}{2}\left\Vert\nabla u^{N}\left( x, \tau\right) \right\Vert _{L^{2}(\Omega )}^{2}, \end{eqnarray} | (3.78) |
\begin{eqnarray} -\lambda_{3}h(\tau)(\nabla u_{\tau}^{N}, \nabla u^{N})_{L^{2}(\Omega )}^{2}&\geq& -\frac{\lambda _{3}h(0)}{2}\left\Vert\nabla u_{\tau}^{N}\left( x, \tau\right) \right\Vert _{L^{2}(\Omega )}^{2}\\ &&-\frac{\lambda _{3}h(0)}{2}\left\Vert\nabla u^{N}\left( x, \tau\right) \right\Vert _{L^{2}(\Omega )}^{2}, \end{eqnarray} | (3.79) |
\begin{eqnarray} && \lambda_{3}\int_{\Omega}\nabla u_{\tau\tau}^{N}\bigg[\int_{0}^{\tau}h(\tau-\sigma)(\nabla u^{N}(\tau)-\nabla u^{N}(\sigma)d\sigma\bigg]dx\\ &\geq &-\frac{\lambda _{3}h_{0}}{2\varepsilon_{17}}\left\Vert\nabla u_{\tau\tau}^{N}\left( x, \tau\right) \right\Vert _{L^{2}(\Omega )}^{2}-\frac{\lambda _{3}\varepsilon_{17}}{2}h\circ\nabla u^{N}(\tau), \end{eqnarray} | (3.80) |
\begin{eqnarray} && -\lambda_{3}\int_{\Omega}\nabla u_{\tau}^{N}\bigg[\int_{0}^{\tau}h'(\tau-\sigma)(\nabla u^{N}(\tau)-\nabla u^{N}(\sigma)d\sigma\bigg]dx\\ &\geq&-\frac{\lambda _{3}h_{0}}{2}\left\Vert\nabla u_{\tau}^{N}\left( x, \tau\right) \right\Vert _{L^{2}(\Omega )}^{2}+\frac{\lambda _{3}}{2}h'\circ\nabla u^{N}(\tau), \end{eqnarray} | (3.81) |
\begin{eqnarray} &&\lambda_{2}\int_{\Omega}\nabla u_{\tau}^{N}\bigg[\int_{0}^{\tau}h(\tau-\sigma)\nabla u^{N}(\sigma)d\sigma\bigg]dx\geq-\frac{\lambda _{2}}{2}h\circ\nabla u^{N}(\tau)\\ &&-\frac{\lambda _{2}(h_{0}+1)}{2}\left\Vert\nabla u_{\tau}^{N}\left( x, \tau\right) \right\Vert _{L^{2}(\Omega )}^{2} -\frac{\lambda _{2}h_{0}}{2}\left\Vert\nabla u^{N}\left( x, \tau\right) \right\Vert _{L^{2}(\Omega )}^{2}, \end{eqnarray} | (3.82) |
\begin{eqnarray} \lambda_{1}(F, u_{t}^{N})_{L^{2}(Q_{\tau})}&\leq&\frac{\lambda_{1}}{2}\int_{0}^{\tau }\left\Vert F\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt+\frac{ \lambda_{1}}{2}\int_{0}^{\tau }\left\Vert u_{t}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt \\ \lambda_{2}(F, u_{tt}^{N})_{L^{2}(Q_{\tau})}&\leq&\frac{\lambda_{2}}{2}\int_{0}^{\tau }\left\Vert F\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt+\frac{\lambda_{2}}{2} \int_{0}^{\tau }\left\Vert u_{tt}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt\\ \lambda_{3}(F, u_{ttt}^{N})_{L^{2}(Q_{\tau})}&\leq&\frac{\lambda_{3}}{2}\int_{0}^{\tau }\left\Vert F\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt+\frac{\lambda_{3}}{2} \int_{0}^{\tau }\left\Vert u_{ttt}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt.\\ \end{eqnarray} | (3.83) |
Substituting (3.32)-(3.83) into (3.31) and make use of the following inequality
\begin{eqnarray*} m_{1}\left\Vert u^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}&\leq& m_{1}\left\Vert u^{N}(x, t)\right\Vert _{L^{2}(Q_{\tau })}^{2}+m_{1}\left\Vert u_{t}^{N}(x, t)\right\Vert _{L^{2}(Q_{\tau })}^{2}\notag\\ &&+m_{1}\left\Vert u^{N}(x, 0)\right\Vert _{L^{2}(\Omega )}^{2}\notag\\ m_{2}\left\Vert u_{\tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}&\leq& m_{2}\left\Vert u_{t}^{N}(x, t)\right\Vert _{L^{2}(Q_{\tau })}^{2}+m_{2}\left\Vert u_{tt}^{N}(x, t)\right\Vert _{L^{2}(Q_{\tau })}^{2}\notag\\ &&+m_{2}\left\Vert u_{t}^{N}(x, 0)\right\Vert _{L^{2}(\Omega )}^{2}\notag\\ m_{3}\left\Vert u_{\tau \tau }^{N}\left( x, \tau \right) \right\Vert _{L^{2}(\Omega )}^{2}&\leq& m_{3}\left\Vert u_{tt}^{N}(x, t)\right\Vert _{L^{2}(Q_{\tau })}^{2}+m_{3}\left\Vert u_{ttt}^{N}(x, t)\right\Vert _{L^{2}(Q_{\tau })}^{2}\notag\\ &&+m_{3}\left\Vert u_{tt}^{N}(x, 0)\right\Vert _{L^{2}(\Omega )}^{2}\notag\\ m_{4}\left\Vert \nabla u^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}&\leq& m_{4}\left\Vert \nabla u^{N}(x, t)\right\Vert _{L^{2}(Q_{\tau })}^{2}+m_{4}\left\Vert \nabla u_{t}^{N}(x, t)\right\Vert _{L^{2}(Q_{\tau })}^{2}\notag\\ &&+m_{4}\left\Vert \nabla u^{N}(x, 0)\right\Vert _{L^{2}(\Omega )}^{2}\notag\\ m_{5}\left\Vert \nabla u_{\tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}&\leq& m_{5}\left\Vert \nabla u_{t}^{N}(x, t)\right\Vert _{L^{2}(Q_{\tau })}^{2}+m_{5}\left\Vert \nabla u_{tt}^{N}(x, t)\right\Vert _{L^{2}(Q_{\tau })}^{2}\notag\\ &&+m_{5}\left\Vert \nabla u_{t}^{N}(x, 0)\right\Vert _{L^{2}(\Omega )}^{2}\notag\\ m_{6}h\circ u^{N}(\tau)&\leq& m_{6}\left\Vert u_{t}^{N}(x, t)\right\Vert _{L^{2}(Q_{\tau })}^{2}+m_{6}\int_{0}^{\tau}h\circ u^{N}(t)dt\notag\\ m_{7}h\circ\nabla u^{N}(\tau)&\leq&m_{7}\left\Vert \nabla u_{t}^{N}(x, t)\right\Vert _{L^{2}(Q_{\tau })}^{2}+m_{7}\int_{0}^{\tau}h\circ\nabla u^{N}(t)dt\notag\\ -m_{8}h'\circ\nabla u^{N}(\tau)&\leq&m_{8}\left\Vert \nabla u_{t}^{N}(x, t)\right\Vert _{L^{2}(Q_{\tau })}^{2}-m_{8}\int_{0}^{\tau}h'\circ\nabla u^{N}(t)dt, \end{eqnarray*} |
where
\begin{eqnarray*} m_{1}& = &\frac{\lambda_{1}\varrho}{\varepsilon_{1}}l(\varepsilon)+\frac{\lambda _{2}\delta }{2}\varepsilon _{3}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert +\frac{\lambda _{3}\delta }{2}\varepsilon _{8}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert+\lambda_{1}h_{0}l(\varepsilon), \notag\\ m_{2}& = &\frac{\lambda _{2}\varrho }{2}\frac{l(\varepsilon )}{\varepsilon _{2}}+ \frac{\lambda _{2}\delta }{2}\frac{l(\varepsilon )}{\varepsilon _{3}}+\frac{ \lambda _{2}\delta }{2}\frac{l(\varepsilon )}{\varepsilon _{4}}+\frac{ \lambda _{2}\gamma }{2}\left( \frac{l(\varepsilon )}{\varepsilon _{5}} +\varepsilon _{5}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \right)\notag\\ && +\frac{\lambda _{2}\gamma }{2}\frac{l(\varepsilon )}{ \varepsilon _{6}}+\frac{\lambda _{3}\gamma }{2}\varepsilon _{10}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert +\frac{\lambda _{1}(1+\alpha) }{2}+\lambda_{2}h_{0}l(\varepsilon), \notag\\ m_{3}& = &\frac{\lambda _{3}\varrho }{2}\frac{l(\varepsilon )}{\varepsilon _{7}}+ \frac{\lambda _{3}\delta }{2}\frac{l(\varepsilon )}{\varepsilon _{8}}+\frac{ \lambda _{3}\delta }{2}\frac{l(\varepsilon )}{\varepsilon _{9}}+\frac{ \lambda _{3}\gamma }{2}\frac{l(\varepsilon )}{\varepsilon _{10}}+\frac{ \lambda _{3}\gamma }{2}\frac{l(\varepsilon )}{\varepsilon _{11}}+\frac{ \lambda _{2}}{2}\notag\\ &&+\frac{\lambda _{1}\alpha }{2}+\frac{\lambda_{3}h_{0}}{2\varepsilon_{18}}l(\varepsilon)+\frac{\lambda_{3}}{2\varepsilon_{15}}l(\varepsilon), \notag\\ m_{4}& = &\frac{\lambda _{1}h_{0} }{2\varepsilon _{1}}\varepsilon +\frac{ \lambda _{2}\varrho }{2}\varepsilon _{12}+\frac{\lambda _{2}\varrho }{2} \varepsilon _{13}+\lambda_{1}h_{0}\varepsilon+\frac{\lambda_{3}}{2}+ \frac{\lambda _{3}h_{0}}{2}\varepsilon _{16}\notag\\ && +\frac{ \lambda _{3}h(0) }{2}+\frac{\lambda _{2}h_{0} }{2} +\frac{\lambda_{1}\varrho}{2\varepsilon_{1}}\varepsilon, \notag\\ m_{5}& = &\frac{\lambda _{2}\varrho }{2}\frac{\varepsilon }{\varepsilon _{2}}+ \frac{\lambda _{2}\delta }{2}\frac{\varepsilon }{\varepsilon _{3}}+\frac{ \lambda _{2}\delta }{2}\frac{\varepsilon }{\varepsilon _{4}}+\frac{\lambda _{2}\gamma }{2}\frac{\varepsilon }{\varepsilon _{5}}+\frac{\lambda _{2}\gamma }{2}\frac{\varepsilon }{\varepsilon _{6}}+\frac{\lambda _{2}\varrho }{2\varepsilon _{12}}+\frac{\lambda _{3}\delta \varepsilon _{14} }{2}\notag\\ &&+\lambda_{2}h_{0}\varepsilon+\frac{\lambda_{3}(h_{0}+h(0)) }{2}+ \frac{\lambda _{2}(h_{0}+1) }{2}, \notag\\ m_{7}& = &\frac{\lambda _{2}\varepsilon_{17} }{2}+ \frac{\lambda _{2} }{2} , \quad m_{8} = \frac{\lambda _{3}}{2}, \quad m_{6} = 1, \label{80} \end{eqnarray*} |
we have
\begin{equation} \begin{array}{l} \frac{\lambda _{1}\varrho }{2\varepsilon _{1}}l(\varepsilon )\left\Vert u^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+\frac{\lambda _{1}\beta }{2} \left\Vert u_{\tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+\left( \frac{\lambda _{2}\alpha }{2}+\frac{\lambda _{3}\beta }{2}\right) \left\Vert u_{\tau \tau }^{N}\left( x, \tau \right) \right\Vert _{L^{2}(\Omega )}^{2} \\ +\left\{ \frac{\lambda _{3}}{2}-\frac{\lambda _{1}}{2}-\frac{\lambda _{2}}{2} \right\} \left\Vert u_{\tau \tau \tau }^{N}\left( x, \tau \right) \right\Vert _{L^{2}(\Omega )}^{2}+\frac{\lambda _{1}\varrho }{2}\left\Vert \nabla u^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2} \\ +\left\{ \frac{\lambda _{1}\gamma }{2}+\frac{\lambda _{2}\delta }{2}\right\} \left\Vert \nabla u_{\tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+h\circ u^{N}(\tau)+\frac{\lambda_{1}}{2}h\circ \nabla u^{N}(\tau)-\frac{\lambda_{2}}{2}h'\circ \nabla u^{N}(\tau) \\ +\bigg\{\frac{\lambda _{3}\gamma }{2} -\frac{\lambda _{3}\varrho }{2}\frac{\varepsilon }{\varepsilon _{7}} -\frac{\lambda _{3}\delta }{2}\frac{\varepsilon }{\varepsilon _{8}}-\frac{ \lambda _{3}\delta }{2}\frac{\varepsilon }{\varepsilon _{9}}-\frac{\lambda _{3}\gamma }{2}\frac{\varepsilon }{\varepsilon _{10}}-\frac{\lambda _{3}\gamma }{2}\frac{\varepsilon }{\varepsilon _{11}}-\frac{\lambda _{2}\varrho }{2\varepsilon _{13}}-\frac{\lambda _{3}\delta }{2\varepsilon _{14}}\\ -\frac{\lambda_{3}h_{0} }{2}\frac{\varepsilon }{\varepsilon _{16}}-\frac{\lambda _{3}h_{0} }{2}\frac{\varepsilon }{\varepsilon _{17}}-\frac{\lambda _{3}h_{0} }{2\varepsilon _{18}}-\frac{\lambda _{3}h_{0} }{2\varepsilon _{15}}\bigg\} \left\Vert \nabla u_{\tau \tau }^{N}\left( x, \tau \right) \right\Vert _{L^{2}(\Omega )}^{2} \\ \leq \gamma_{7} \left\Vert u^{N}(x, 0)\right\Vert _{L^{2}(\Omega )}^{2} +\left\{ \frac{\lambda _{2}}{2}+\frac{\lambda _{1}\alpha }{2}+\left( \frac{ \lambda _{2}\alpha }{2}-\frac{\lambda _{3}\beta }{2}\right) +m_{3}\right\} \left\Vert u_{tt}^{N}(x, 0)\right\Vert _{L^{2}(\Omega )}^{2}\\ +\left\{ \frac{ \lambda _{1}}{2}+\frac{\lambda _{2}}{2}+\frac{\lambda _{3}}{2}\right\} \left\Vert u_{ttt}^{N}(x, 0)\right\Vert _{L^{2}(\Omega )}^{2} +\left\{ \frac{\lambda _{1}\varrho }{2}+\frac{\lambda _{2}\varrho }{2}+\frac{ \lambda _{3}\varrho }{2}+m_{4}\right\} \left\Vert \nabla u^{N}(x, 0)\right\Vert _{L^{2}(\Omega )}^{2}\\ +\gamma _{8} \left\Vert u_{t}^{N}(x, 0)\right\Vert _{L^{2}(\Omega )}^{2}+\left\{ \frac{\lambda _{2}\varrho }{2}+\frac{\lambda _{3}\delta }{2}+\frac{\lambda _{1}\gamma }{2}+ \frac{\lambda _{2}\delta }{2}+m_{5}\right\} \left\Vert \nabla u_{t}^{N}(x, 0)\right\Vert _{L^{2}(\Omega )}^{2}\\ +\left\{ \frac{\lambda _{3}\varrho }{2}+\frac{3\lambda _{3}\delta }{2}-\frac{ \lambda _{3}\gamma }{2}-\lambda_{2}\gamma\right\} \left\Vert \nabla u_{tt}^{N}(x, 0)\right\Vert _{L^{2}(\Omega )}^{2}+\left( \gamma _{1}+m_{1}\right) \int_{0}^{\tau }\left\Vert u^{N}(x, t)\right\Vert _{L^{2}(\Omega )}^{2}dt \\ +\left( \gamma _{2}+m_{1}+m_{2}\right) \int_{0}^{\tau }\left\Vert u_{t}^{N}(x, t)\right\Vert _{L^{2}(\Omega )}^{2}dt \\ +\left\{ \frac{\lambda _{1}}{2}+\lambda _{2}-\lambda _{3}\alpha +m_{3}\right\} \int_{0}^{\tau }\left\Vert u_{ttt}^{N}(x, t)\right\Vert _{L^{2}(\Omega )}^{2}dt-m_{8}\int_{0}^{\tau }h'\circ \nabla u^{N}(t)dt \\ +\left\{ \gamma_{6} +m_{4}\right\} \int_{0}^{\tau }\left\Vert \nabla u^{N}(x, t)\right\Vert _{L^{2}(\Omega )}^{2}dt+\left( \gamma _{3}+m_{2}+m_{3}\right) \int_{0}^{\tau }\left\Vert u_{tt}^{N}(x, t)\right\Vert _{L^{2}(\Omega )}^{2}dt \\ +\left( \gamma _{4}+m_{4}+m_{5}+m_{7}+m_{8}\right) \int_{0}^{\tau }\left\Vert \nabla u_{t}^{N}(x, t)\right\Vert _{L^{2}(\Omega )}^{2}dt\\ +\left( \gamma _{5}+m_{5}\right) \int_{0}^{\tau }\left\Vert \nabla u_{tt}^{N}(x, t)\right\Vert _{L^{2}(\Omega )}^{2}dt\\ +\int_{0}^{\tau }h\circ u^{N}(t)dt+m_{7}\int_{0}^{\tau }h\circ \nabla u^{N}(t)dt+\frac{\lambda_{1}+\lambda_{2}+\lambda_{3}}{2}\int_{0}^{\tau }\left\Vert F\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt, \end{array} \end{equation} | (3.84) |
where
\begin{eqnarray*} \gamma _{1} & = &\frac{\lambda _{1}\varrho }{2}\varepsilon _{1}T\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert +\frac{\lambda _{1}\varrho }{2}\left( l(\varepsilon )+\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \right) +\left( \frac{\lambda _{1}\delta +\lambda _{2}\varrho }{2}\right) \left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \\ &&+\frac{\lambda _{2}\varrho }{2} \varepsilon _{2}T\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert +\frac{\lambda _{3}\varrho }{2}\varepsilon _{7}T\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert +\frac{\lambda _{3}\varrho }{2}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert++\frac{\lambda _{1}h_{0} }{2} l(\varepsilon )\notag\\ &&+\bigg[\frac{\lambda _{3}h_{0}}{2}\varepsilon _{18}+\frac{(\lambda _{3}+\lambda _{2}+\lambda _{1})h_{0}}{2} +\frac{(\lambda _{1}+\lambda _{2})h_{0}T}{2}\bigg]\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert\notag\\ \gamma _{2}& = &\left( \frac{\lambda _{1}\delta +\lambda _{2}\varrho }{2}\right) l(\varepsilon )+\frac{\lambda _{1}\delta }{2}l(\varepsilon )+\left( \frac{ \lambda _{1}\gamma +\lambda _{2}\delta }{2}\right) \left( l(\varepsilon )+\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \right)\notag\\ && +\frac{\lambda _{1}\gamma }{2}l(\varepsilon )+\frac{\lambda _{2}\gamma }{2}l(\varepsilon )+\frac{\lambda _{3}\delta }{2}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert+\lambda_{2}h_{0}l(\varepsilon), \end{eqnarray*} |
\begin{eqnarray*} \gamma _{3}& = &\frac{\lambda _{2}\gamma }{2}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert +\frac{\lambda _{3}\varrho }{2} l(\varepsilon )+\frac{\lambda _{3}\delta }{2}l(\varepsilon )+\frac{\lambda _{3}\gamma }{2}\left( l(\varepsilon )+\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \right) +\frac{\lambda _{1}}{2}\notag\\ &&+\left( \lambda _{1}\alpha -\lambda _{2}\beta \right)+\frac{\lambda _{3}\varrho}{2}+\lambda _{3}h_{0}l(\varepsilon), \notag\\ \gamma _{4}& = &\left( \frac{\lambda _{1}\delta +\lambda _{2}\varrho }{2}\right) \varepsilon +\frac{\lambda _{1}\delta }{2}\varepsilon +\left( \frac{\lambda _{1}\gamma +\lambda _{2}\delta }{2}\right) \varepsilon +\frac{\lambda _{1}\gamma }{2}\varepsilon +\frac{\lambda _{2}\gamma }{2}\varepsilon +\frac{ \lambda _{3}\varrho }{2}\notag\\ &&+\left( \lambda _{2}\varrho -\lambda _{1}\delta \right)+h(0)\lambda_{3}+\lambda_{3}h_{0}\varepsilon, \notag\\ \gamma _{5}& = &\frac{\lambda _{3}\delta }{2}\varepsilon +\frac{\lambda _{3}\gamma }{2}\varepsilon +\frac{\lambda _{3}\varrho }{2}+\left( \lambda _{3}\delta -\lambda _{2}\gamma \right)+\lambda_{3}h_{0}\varepsilon, \notag\\ \gamma _{6}& = &\frac{\lambda _{1}\varrho }{2}\varepsilon +\lambda_{1}h_{0}\varepsilon, \notag\\ \gamma _{7}& = & \frac{\lambda _{1}\delta }{2}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert T+\frac{\lambda _{2}\delta }{2} \varepsilon _{4}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert +\frac{\lambda _{3}\delta }{2}\varepsilon _{9}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert +m_{1}, \notag\\ \gamma _{8}& = &\frac{\lambda _{1}\gamma }{2}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert T+\frac{\lambda _{2}\gamma }{2} \varepsilon _{6}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert +\frac{\lambda _{3}\gamma }{2}\varepsilon _{11}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert +\frac{\lambda _{1}}{2}+ \frac{\lambda _{1}\alpha }{2}+\frac{\lambda _{1}\beta }{2}+m_{2}. \end{eqnarray*} |
Choosing \varepsilon _{7}, \ \varepsilon _{8}, \ \varepsilon _{9}, \ \varepsilon _{10}, \ \varepsilon _{11}, \ \varepsilon _{13} , \varepsilon _{14}, \varepsilon _{15}, \ \varepsilon _{16}, \ \varepsilon _{17} and \varepsilon _{18} sufficiently large
\begin{equation} \begin{array}{l} \beta_{0}: = \frac{\lambda _{3}\gamma }{2}-\frac{\lambda _{3}\varrho }{2}\frac{\varepsilon }{\varepsilon _{7}}-\frac{ \lambda _{3}\delta }{2}\frac{\varepsilon }{\varepsilon _{8}}-\frac{\lambda _{3}\delta }{2}\frac{\varepsilon }{\varepsilon _{9}}-\frac{\lambda _{3}\gamma }{2}\frac{\varepsilon }{\varepsilon _{10}}-\frac{\lambda _{3}\gamma }{2}\frac{\varepsilon }{\varepsilon _{11}}-\frac{\lambda _{3}\delta }{2\varepsilon _{14}}\\ \quad -\frac{\lambda _{2}\varrho }{2\varepsilon _{13}}-\frac{\lambda_{3}h_{0} }{2}\frac{\varepsilon }{\varepsilon _{16}}-\frac{\lambda _{3} }{2}\frac{\varepsilon }{\varepsilon _{17}}-\frac{\lambda _{3}h_{0} }{2\varepsilon _{18}}-\frac{\lambda _{3} }{2\varepsilon _{15}} > 0, \end{array} \end{equation} | (3.85) |
the relation (3.84) reduces to
\begin{eqnarray} &&\bigg\{ \left\Vert u^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+\left\Vert \nabla u^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+\left\Vert u_{\tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2} \\ &&+\left\Vert \nabla u_{\tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2} +\left\Vert u_{\tau \tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+\left\Vert \nabla u_{\tau \tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}\\ &&+\left\Vert u_{\tau \tau \tau }^{N}(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+h\circ\nabla u^{N}(\tau)+h\circ u^{N}(\tau)-h'\circ\nabla u^{N}(\tau)\bigg\} \end{eqnarray} | (3.86) |
\begin{eqnarray} &\leq& D\int_{0}^{\tau }\bigg\{ \left\Vert u^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}+\left\Vert \nabla u^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}+\left\Vert u_{t}^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2} \\ && +\left\Vert \nabla u_{t}^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}+\left\Vert u_{tt}^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}+\left\Vert \nabla u_{tt}^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}\\ &&+\left\Vert u_{ttt}^{N}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}+h\circ\nabla u^{N}(t)+h\circ u^{N}(t)-h'\circ\nabla u^{N}(t)+\left\Vert F \right\Vert _{L^{2}(\Omega )}^{2} \bigg\} dt \\ &&+D\bigg\{ \left\Vert u^{N}(x, 0)\right\Vert _{W_{2}^{1}(\Omega )}^{2}+\left\Vert u_{t}^{N}(x, 0)\right\Vert _{W_{2}^{1}(\Omega )}^{2} +\left\Vert u_{tt}^{N}(x, 0)\right\Vert _{W_{2}^{1}(\Omega )}^{2}\\ &&+\left\Vert u_{ttt}^{N}(x, 0)\right\Vert _{L^{2}(\Omega )}^{2}+h\circ\nabla u^{N}(0)+h\circ u^{N}(0)-h'\circ\nabla u^{N}(0)\bigg\}, \end{eqnarray} | (3.87) |
where
\begin{equation} D: = \frac{ \begin{array}{c} \max \left\{ \frac{\lambda _{1}\delta }{2}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert T+\frac{\lambda _{2}\delta }{2} \varepsilon _{4}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert +\frac{\lambda _{3}\delta }{2}\varepsilon _{9}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert+m_{1}\right.\\ , \left.\frac{\lambda _{1}\gamma }{2}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert T+\frac{\lambda _{2}\gamma }{2}\varepsilon _{6}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \right. \\ +\frac{\lambda _{3}\gamma }{2}\varepsilon _{11}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert +\frac{\lambda _{1}}{2}+\frac{\lambda _{1}\alpha }{2}+\frac{\lambda _{1}\beta }{2}+m_{2}, \frac{\lambda _{2}}{2}+ \frac{\lambda _{1}\alpha }{2}+\frac{\lambda _{2}\alpha }{2}-\frac{\lambda _{3}\beta }{2}+m_{3}, \\ \frac{\lambda_{1}+\lambda_{2}+\lambda_{3}}{2}, \frac{\lambda _{1}\varrho }{2}+\frac{\lambda _{2}\varrho }{2}+\frac{\lambda _{3}\varrho }{2}+m_{4}, \frac{\lambda _{2}\varrho }{2}+\frac{\lambda _{3}\delta }{2}+\frac{\lambda _{1}\gamma }{2}+\frac{\lambda _{2}\delta }{2} +m_{5}, \\ \gamma _{1}+m_{1}, \gamma _{2}+m_{1}+m_{2}, \gamma _{3}+m_{2}+m_{3}, \frac{ \lambda _{1}}{2}+\lambda _{2}-\lambda _{3}\alpha +m_{3}, \\ \left.\frac{\lambda _{3}\varrho }{2}+\frac{\lambda _{3}\delta }{2}-\frac{ \lambda _{3}\gamma }{2}, \gamma_{6}+m_{4}, \gamma _{4}+m_{4}+m_{5}, \gamma _{5}+m_{5}, m_{7}, m_{8}, 1\right\} \end{array} }{ \begin{array}{c} \min \left\{ \frac{\lambda _{1}\varrho }{2\varepsilon _{1}}l(\varepsilon ), \frac{\lambda _{1}\beta }{2}, \frac{\lambda _{2}\alpha }{2}+\frac{\lambda _{3}\beta }{2}, \frac{\lambda _{3}}{2}-\frac{\lambda _{1}}{2}-\frac{\lambda _{2}}{2}, \frac{\lambda _{1}\varrho }{2}, \frac{\lambda _{1}\gamma }{2}+\frac{\lambda _{2}\delta }{2}, 1, \frac{\lambda_{1}}{2}, \frac{\lambda_{2}}{2}, \beta_{0}\right\} \end{array} }. \end{equation} | (3.88) |
Applying the Gronwall inequality to (3.87) and then integrate from 0 to \tau appears that
\begin{equation} \begin{array}{c} \left\Vert u^{N}\left( x, t\right) \right\Vert _{W_{2}^{1}\left( Q_{\tau }\right) }^{2}+\left\Vert u_{t}^{N}\left( x, t\right) \right\Vert _{W_{2}^{1}\left( Q_{\tau }\right) }^{2}+\left\Vert u_{tt}^{N}\left( x, t\right) \right\Vert _{W_{2}^{1}\left( Q_{\tau }\right) }^{2} +\Vert u^{N}\left( x, t\right)\Vert_{h}\\ \leq De^{DT}\bigg\{ \left\Vert u_{0}\left( x\right) \right\Vert _{W_{2}^{1}(\Omega )}^{2}+\left\Vert u_{1}\left( x\right) \right\Vert _{W_{2}^{1}(\Omega )}^{2}+\left\Vert u_{2}\left( x\right) \right\Vert _{L^{2}(\Omega )}^{2}\\ \quad +\left\Vert u_{3}\left( x\right) \right\Vert _{L^{2}(\Omega )}^{2}+\left\Vert F \right\Vert _{L^{2}(\Omega )}^{2}\bigg\} . \end{array}. \end{equation} | (3.89) |
We deduce from (3.89) that
\begin{equation} \left\Vert u^{N}\left( x, t\right) \right\Vert _{W_{2}^{1}\left( Q_{\tau }\right) }^{2}+\left\Vert u_{t}^{N}\left( x, t\right) \right\Vert _{W_{2}^{1}\left( Q_{\tau }\right) }^{2}+\left\Vert u_{tt}^{N}\left( x, t\right) \right\Vert _{W_{2}^{1}\left( Q_{\tau }\right) }^{2}+\Vert u^{N}\left( x, t\right)\Vert_{h}\leq A, \end{equation} | (3.90) |
where
\begin{equation*} \Vert u^{N}\left( x, t\right)\Vert_{h}: = \int_{0}^{\tau}\bigg(h\circ\nabla u^{N}(t)+h\circ u^{N}(t)-h'\circ\nabla u^{N}(t)\bigg)dt. \end{equation*} |
Therefore the sequence \left\{ u^{N}\right\} _{N\geq 1} is bounded in V\left(Q_{T}\right), and we can extract from it a subsequence for which we use the same notation which converges weakly in V\left(Q_{T}\right) \ to a limit function u\left(x, t\right) we have to show that u\left(x, t\right) is a generalized solution of (1.1). Since u^{N}\left(x, t\right) \rightarrow u\left(x, t\right) in L^{2}\left(Q_{T}\right) and u^{N}(x, 0)\rightarrow \zeta \left(x\right) in L^{2}(\Omega) , then u(x, 0) = \zeta \left(x\right).
Now to prove that (2.1) holds, we multiply each of the relations (3.5) by a function p_{l}\left(t\right) \in W_{2}^{1}(0, T), \ p_{l}\left(t\right) = 0, then add up the obtained equalities ranging from l = 1\ to l = N, and integrate over t on (0, T).\ If we let \eta ^{N} = \sum\limits_{k = 1}^{N}p_{k}\left(t\right) Z_{k}\left(x\right), then we have
\begin{equation} \begin{array}{l} -(u_{ttt}^{N}, \eta _{t}^{N})_{L^{2}\left( Q_{T}\right) }-\alpha (u_{tt}^{N}, \eta _{t}^{N})_{L^{2}\left( Q_{T}\right) }-\beta (u_{t}^{N}, \eta _{t}^{N})_{L^{2}\left( Q_{T}\right) }+\varrho (\nabla u^{N}, \nabla \eta ^{N})_{L^{2}\left( Q_{T}\right) } \\ +\delta (\nabla u_{t}^{N}, \nabla \eta ^{N})_{L^{2}\left( Q_{T}\right) }-\gamma \left( \nabla u_{t}^{N}, \nabla \eta _{t}^{N}\right) _{L^{2}\left( Q_{T}\right) }- (\nabla w^{N}, \nabla \eta ^{N})_{L^{2}\left( Q_{T}\right) } \\ = \varrho \int\nolimits_{\partial \Omega }\int\nolimits_{0}^{T}\eta ^{N}\left( x, t\right) \left( \int\nolimits_{0}^{t}\int\nolimits_{\Omega }u^{N}\left( \xi , \tau \right) d\xi d\tau \right) dtds_{x}\\ +\delta\int\nolimits_{\partial \Omega }\int\nolimits_{0}^{T}\eta ^{N}\left( x, t\right) \int\nolimits_{\Omega }u^{N}\left( \xi , t\right) d\xi dtds_{x} \\ -\delta \int\nolimits_{\partial \Omega }\int\nolimits_{0}^{T}\eta ^{N}\left( x, t\right) \int\nolimits_{\Omega }u^{N}(\xi , 0)d\xi dtds_{x}-\gamma \int\nolimits_{0}^{T}\int\nolimits_{\partial \Omega }\eta _{t}^{N}\left( \int\nolimits_{\Omega }u^{N}\left( \xi , t\right) d\xi \right) ds_{x}dt \\ +\gamma \int\nolimits_{0}^{T}\int\nolimits_{\partial \Omega }\eta _{t}^{N}\left( \int\nolimits_{\Omega }u^{N}\left( \xi , 0\right) d\xi \right) ds_{x}dt-\gamma \left( \Delta u_{t}^{N}(x, 0), \eta ^{N}(0)\right) _{L^{2}(\Omega )} \\ -\int\nolimits_{\partial \Omega }\int\nolimits_{0}^{T}\eta ^{N}\left( x, t\right) \left( \int\nolimits_{0}^{t}\int\nolimits_{\Omega }w^{N}\left( \xi , \tau \right) d\xi d\tau \right) dtds_{x}+\left( F, \eta _{t}^{N}\right) _{L^{2}\left( Q_{T}\right)}\\ +\left( u_{ttt}^{N}(x, 0), \eta ^{N}(0)\right) _{L^{2}(\Omega )}+\alpha \left( u_{tt}^{N}(x, 0), \eta ^{N}(0)\right) _{L^{2}(\Omega )}+\beta \left( u_{tt}^{N}(x, 0), \eta ^{N}(0)\right) _{L^{2}(\Omega )}, \end{array} \end{equation} | (3.91) |
for all \eta ^{N} of the form \sum\limits_{k = 1}^{N}p_{l}\left(t\right) Z_{k}\left(x\right).
Since
\begin{equation*} \int\nolimits_{0}^{t}\int\nolimits_{\Omega }(\left( u^{N}\left( \xi , \tau \right) -u\left( \xi , \tau \right) \right) d\xi d\tau \leq \sqrt{T\left\vert \Omega \right\vert }\left\Vert u^{N}-u\right\Vert _{L^{2}\left( Q_{T}\right) }, \end{equation*} |
\begin{eqnarray*} &&\int\nolimits_{0}^{T}\eta ^{N}\left( x, t\right) \int\nolimits_{\Omega }\left( u_{t}^{N}\left( \xi , t\right) -u_{t}\left( \xi , t\right) \right) d\xi dt\notag\\ &\leq& \sqrt{\left\vert \Omega \right\vert }\left( \int\nolimits_{0}^{T}(\eta ^{N}\left( x, t\right) )^{2}dt\right) ^{1/2}\left\Vert u_{t}^{N}-u_{t}\right\Vert _{L^{2}\left( Q_{T}\right) }, \end{eqnarray*} |
\begin{eqnarray*} &&\int\nolimits_{0}^{T}\eta ^{N}\left( x, t\right) \int\nolimits_{\Omega }(u\left( ^{N}(\xi , 0)-u(\xi , 0)\right) d\xi dt \\ &\leq &\sqrt{\left\vert \Omega \right\vert }\left( \int\nolimits_{0}^{T}(\eta ^{N}\left( x, t\right) )^{2}dt\right) ^{1/2}\left\Vert u^{N}(x, 0)-u(x, 0)\right\Vert _{L^{2}\left( Q_{T}\right) }, \end{eqnarray*} |
and
\begin{equation*} \left\Vert u^{N}-u\right\Vert _{L^{2}\left( Q_{T}\right) }\rightarrow 0, \rm{ \ as }N\rightarrow \infty , \end{equation*} |
therefore we have
\begin{eqnarray*} &&\varrho \int\nolimits_{\partial \Omega }\int\nolimits_{0}^{T}\eta ^{N}\left( x, t\right) \int\nolimits_{0}^{t}\int\nolimits_{\Omega }u^{N}\left( \xi , \tau \right) d\xi d\tau dtds_{x} \\ &\rightarrow &\varrho \int\nolimits_{\partial \Omega }\int\nolimits_{0}^{T}\eta \left( x, t\right) \int\nolimits_{0}^{t}\int\nolimits_{\Omega }u\left( \xi , \tau \right) d\xi d\tau dtds_{x}, \end{eqnarray*} |
\begin{eqnarray*} &&\delta \int\nolimits_{\partial \Omega }\int\nolimits_{0}^{T}\eta ^{N}\left( x, t\right) \int\nolimits_{\Omega }u^{N}\left( \xi , t\right) d\xi dtds_{x} \\ &\rightarrow &\delta \int\nolimits_{\partial \Omega }\int\nolimits_{0}^{T}\eta \left( x, t\right) \int\nolimits_{\Omega }u\left( \xi , t\right) d\xi dtds_{x}, \end{eqnarray*} |
\begin{eqnarray*} &&-\delta \int\nolimits_{\partial \Omega }\int\nolimits_{0}^{T}\eta ^{N}\left( x, t\right) \int\nolimits_{\Omega }u^{N}(\xi , 0)d\xi dtds \\ &\rightarrow &-\delta \int\nolimits_{\partial \Omega }\int\nolimits_{0}^{T}\eta \left( x, t\right) \int\nolimits_{\Omega }u(\xi , 0)d\xi dtds, \end{eqnarray*} |
\begin{eqnarray*} &&-\gamma \int\nolimits_{0}^{T}\int\nolimits_{\partial \Omega }\eta _{t}^{N}\left( \int\nolimits_{\Omega }u^{N}\left( \xi , t\right) d\xi \right) ds_{x}dt \\ &\rightarrow &-\gamma \int\nolimits_{0}^{T}\int\nolimits_{\partial \Omega }\eta _{t}\left( \int\nolimits_{\Omega }u\left( \xi , t\right) d\xi \right) ds_{x}dt, \end{eqnarray*} |
\begin{eqnarray*} &&\gamma \int\nolimits_{0}^{T}\int\nolimits_{\partial \Omega }\eta _{t}^{N}\left( \int\nolimits_{\Omega }u^{N}\left( \xi , 0\right) d\xi \right) ds_{x}dt \\ &\rightarrow &\gamma \int\nolimits_{0}^{T}\int\nolimits_{\partial \Omega }\eta _{t}\left( \int\nolimits_{\Omega }u\left( \xi , 0\right) d\xi \right) ds_{x}dt. \end{eqnarray*} |
\begin{eqnarray*} && \int\nolimits_{\partial \Omega }\int\nolimits_{0}^{T}\eta ^{N}\left( x, t\right) \int\nolimits_{0}^{t}\int\nolimits_{\Omega }w^{N}\left( \xi , \tau \right) d\xi d\tau dtds_{x} \\ &\rightarrow &\varrho \int\nolimits_{\partial \Omega }\int\nolimits_{0}^{T}\eta \left( x, t\right) \int\nolimits_{0}^{t}\int\nolimits_{\Omega }w\left( \xi , \tau \right) d\xi d\tau dtds_{x}. \end{eqnarray*} |
Thus, the limit function u satisfies (2.1) for every \eta ^{N} = \sum\limits_{k = 1}^{N}p_{l}\left(t\right) Z_{k}\left(x\right). We denote by \mathbb{Q}_{N} the totality of all functions of the form \eta ^{N} = \sum\limits_{k = 1}^{N}p_{l}\left(t\right) Z_{k}\left(x\right), with p_{l}\left(t\right) \in W_{2}^{1}(0, T), p_{l}\left(t\right) = 0.
But \cup _{l = 1}^{N} \mathbb{Q}_{N} is dense in W\left(Q_{T}\right) , then relation (2.1) holds for all u \in W\left(Q_{T}\right).\ Thus we have shown that the limit function u\left(x, t\right) is a generalized solution of problem (1.1) in V\left(Q_{T}\right).
Theorem 2. The problem (1.1) cannot have more than one generalized solution in V\left(Q_{T}\right).
Proof. Suppose that there exist two different generalized solutions u_{1}\in V\left(Q_{T}\right) and u_{2}\in V\left(Q_{T}\right) \ for the problem (1.1). Then, U = u_{1}-u_{2} solves
\begin{equation} \left\{ \begin{array}{l} U_{tttt}+\alpha U_{ttt}+\beta U_{tt}-\varrho \Delta U-\delta \Delta U_{t}-\gamma \Delta U_{tt}+ \int_{0}^{t}h(t-\sigma)\Delta u(\sigma)d\sigma = 0, \\ \\ U(x, 0) = U_{t}(x, 0) = U_{tt}(x, 0) = U_{ttt}(x, 0) = 0 \\ \\ \dfrac{\partial u}{\partial \eta } = \int_{0}^{t}\int_{\Omega }u\left( \xi , \tau \right) d\xi d\tau , \ \ \ x\in \partial \Omega . \end{array} \right. \end{equation} | (4.1) |
and (2.1) gives
\begin{eqnarray} &&-(U_{ttt}, v_{t})_{L^{2}\left( Q_{T}\right) }-\alpha (U_{tt}, v_{t})_{L^{2}\left( Q_{T}\right) }-\beta (U_{t}, v_{t})_{L^{2}\left( Q_{T}\right) }+\varrho (\nabla U, \nabla v)_{L^{2}\left( Q_{T}\right) } \\ &&+\delta (\nabla U_{t}, \nabla v)_{L^{2}\left( Q_{T}\right) }-\gamma \left( \nabla U_{t}, \nabla v_{t}\right) _{L^{2}\left( Q_{T}\right) }-(\nabla W, \nabla v)_{L^{2}\left( Q_{T}\right) } \\ & = &\varrho \int_{0}^{T}\int_{\partial \Omega }v\left( \int_{0}^{t}\int_{\Omega }u\left( \xi , \tau \right) d\xi d\tau \right) ds_{x}dt+\delta \int_{0}^{T}\int_{\partial \Omega }v\int_{\Omega }U\left( \xi , t\right) d\xi ds_{x}dt \\ &&-\gamma \int_{0}^{T}\int_{\partial \Omega }v_{t}\left( \int_{\Omega }U_{\tau }\left( \xi , t\right) d\xi dt\right) ds_{x}dt\\ &&-\int_{0}^{T}\int_{\partial \Omega }v\left( \int_{0}^{t}\int_{\Omega }W\left( \xi , \tau \right) d\xi d\tau \right) ds_{x}dt, \end{eqnarray} | (4.2) |
where
\begin{equation*} W(x, t): = \int_{0}^{t}h(t-\sigma)\Delta U(\sigma)d\sigma. \end{equation*} |
Consider the function
\begin{equation} v\left( x, t\right) = \left\{ \begin{array}{ll} \int_{t}^{\tau }U(x, s)ds, & 0\leq t\leq \tau , \\ 0, & \tau \leq t\leq T. \end{array} \right. \end{equation} | (4.3) |
It is obvious that v\in W\left(Q_{T}\right) and v_{t}\left(x, t\right) = -U\left(x, t\right) for all t\in \left[0, \tau \right].\ Integration by parts in the left hand side of (4.2) gives
\begin{equation} -(U_{ttt}, v_{t})_{L^{2}\left( Q_{T}\right) } = (U_{\tau \tau }(x, \tau ), U(x, \tau ))_{L^{2}\left( \Omega \right) }-\frac{1}{2}\left\Vert U_{\tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}, \end{equation} | (4.4) |
\begin{equation} -\alpha \left( U_{tt}, v_{t}\right) _{L^{2}\left( Q_{T}\right) } = \alpha \left( U_{\tau }(x, \tau ), U(x, \tau )\right) _{L^{2}(\Omega )}-\alpha \int_{0}^{\tau }\left\Vert U_{t}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt, \end{equation} | (4.5) |
\begin{equation} -\beta \left( U_{t}, v_{t}\right) _{L^{2}\left( Q_{T}\right) } = \frac{\beta }{2 }\left\Vert U(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}, \end{equation} | (4.6) |
\begin{equation} \varrho \left( \nabla U, \nabla v\right) _{L^{2}\left( Q_{T}\right) } = \frac{ \varrho }{2}\left\Vert \nabla v(x, 0)\right\Vert _{L^{2}(\Omega )}^{2}, \end{equation} | (4.7) |
\begin{equation} \delta \left( \nabla U_{t}, \nabla v\right) _{L^{2}\left( Q_{T}\right) } = \delta \int_{0}^{\tau }\left\Vert \nabla v_{t}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt, \end{equation} | (4.8) |
\begin{equation} -\gamma \left( \nabla U_{t}, \nabla v_{t}\right) _{L^{2}\left( Q_{T}\right) } = \frac{\gamma }{2}\left\Vert \nabla U(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}, \end{equation} | (4.9) |
\begin{eqnarray} - \left( \nabla W, \nabla v\right) _{L^{2}\left( Q_{T}\right) }&\leq& h_{0}\int_{0}^{\tau}\left\Vert \nabla v(x, t )\right\Vert _{L^{2}(\Omega )}^{2}dt+\frac{h_{0}}{2}\int_{0}^{\tau}\left\Vert \nabla U(x, t )\right\Vert _{L^{2}(\Omega )}^{2}dt\\ &&+\frac{1}{2}\int_{0}^{\tau}h\circ \nabla U(t )dt. \end{eqnarray} | (4.10) |
Plugging (4.4)-(4.10) into (4.2) we get
\begin{eqnarray} &&(U_{\tau \tau }(x, \tau ), U(x, \tau ))_{L^{2}\left( \Omega \right) }+\alpha \left( U_{\tau }(x, \tau ), U(x, \tau )\right) _{L^{2}(\Omega )}+\frac{\beta }{2 }\left\Vert U(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2} \\ &&+\frac{\varrho }{2}\left\Vert \nabla v(x, 0)\right\Vert _{L^{2}(\Omega )}^{2}+\frac{\gamma }{2}\left\Vert \nabla U(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}-\frac{1}{2}\left\Vert U_{\tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2} \\ &\leq&\alpha \int_{0}^{\tau }\left\Vert U_{t}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt-\delta \int_{0}^{\tau }\left\Vert \nabla v_{t}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt+h_{0}\int_{0}^{\tau}\left\Vert \nabla v(x, t )\right\Vert _{L^{2}(\Omega )}^{2}dt\\ &&+\frac{h_{0}}{2}\int_{0}^{\tau}\left\Vert \nabla U(x, t )\right\Vert _{L^{2}(\Omega )}^{2}dt+\frac{1}{2}\int_{0}^{\tau}h\circ \nabla U(t )dt \\ &&+\varrho \int_{0}^{T}\int_{\partial \Omega }v\left( \int_{0}^{t}\int_{\Omega }U\left( \xi , \tau \right) d\xi d\tau \right) ds_{x}dt \\ &&+\delta \int_{0}^{T}\int_{\partial \Omega }v\int_{\Omega }U\left( \xi , t\right) d\xi ds_{x}dt-\gamma \int\nolimits_{0}^{T}\int\nolimits_{\partial \Omega }v_{t}\left( \int\nolimits_{\Omega }U\left( \xi , t\right) d\xi \right) dsdt\\ &&-\int_{0}^{T}\int_{\partial \Omega }v\left( \int_{0}^{t}\int_{\Omega }W\left( \xi , \tau \right) d\xi d\tau \right) ds_{x}dt . \end{eqnarray} | (4.11) |
Now since
\begin{equation*} v^{2}\left( x, t\right) = \left( \int_{t}^{\tau }U(x, s)ds\right) ^{2}\leq \tau \int_{0}^{\tau }U^{2}(x, s)ds, \end{equation*} |
then
\begin{equation} \left\Vert v\right\Vert _{L^{2}\left( Q_{\tau }\right) }^{2}\leq \tau ^{2}\left\Vert U\right\Vert _{L^{2}\left( Q_{\tau }\right) }^{2}\leq T^{2}\left\Vert U\right\Vert _{L^{2}\left( Q_{\tau }\right) }^{2}. \end{equation} | (4.12) |
Using the trace inequality, the RHS of (4.11) can be estimated as follows
\begin{eqnarray} &&\varrho \int_{0}^{T}\int_{\partial \Omega }v\left( \int_{0}^{t}\int_{\Omega }U\left( \xi , \tau \right) d\xi d\tau \right) ds_{x}dt \\ &\leq &\frac{\varrho }{2}T^{2}\left\{ l(\varepsilon )+\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \right\} \int_{0}^{\tau }\left\Vert U\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt +\frac{\varrho }{2}\varepsilon \int_{0}^{\tau }\left\Vert \nabla v\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt, \end{eqnarray} | (4.13) |
and
\begin{eqnarray} &&\delta \int_{0}^{T}\int_{\partial \Omega }v\int_{\Omega }U\left( \xi , t\right) d\xi ds_{x}dt \\ &\leq &\frac{\delta }{2}\left\{ T^{2}l(\varepsilon )+\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \right\} \int_{0}^{\tau }\left\Vert U\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt +\frac{\delta }{2}\varepsilon \int_{0}^{\tau }\left\Vert \nabla v\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt, \end{eqnarray} | (4.14) |
and
\begin{eqnarray} &&-\gamma \int\nolimits_{0}^{T}\int\nolimits_{\partial \Omega }v_{t}\left( \int\nolimits_{\Omega }U\left( \xi , t\right) d\xi \right) dsdt \\ & = &\gamma \int\nolimits_{0}^{\tau }\int\nolimits_{\partial \Omega }v\left( \int\nolimits_{\Omega }U_{t}\left( \xi , t\right) d\xi \right) dsdt \\ &\leq &\frac{\gamma \left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert }{2}\left\Vert U_{t}\right\Vert _{L^{2}(Q_{\tau })}^{2}+\frac{ \gamma T^{2}}{2}\varepsilon \left\Vert \nabla v\right\Vert _{L^{2}(Q_{\tau })}^{2}+\frac{\gamma }{2}l(\varepsilon )T^{2}\left\Vert U\right\Vert _{L^{2}(Q_{\tau })}^{2}. \end{eqnarray} | (4.15) |
\begin{eqnarray} &&- \int_{0}^{T}\int_{\partial \Omega }v\left( \int_{0}^{t}\int_{\Omega }W\left( \xi , \tau \right) d\xi d\tau \right) ds_{x}dt \\ & = & -\int_{0}^{T}\int_{\partial \Omega }v\left( \int_{0}^{t}\int_{\Omega }H(\tau)U\left( \xi , \tau \right) d\xi d\tau \right) ds_{x}dt\\ &&+ \int_{0}^{T}\int_{\partial \Omega }v\left( \int_{0}^{t}\int_{\Omega }\bigg[\int_{0}^{\tau}h(\tau-\sigma)(U\left( \xi , \tau \right)-U\left( \xi , \sigma \right))d\sigma\bigg] d\xi d\tau \right) ds_{x}dt\\ &\leq &\frac{h_{0} }{2}T^{2}\left\{ l(\varepsilon )+\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \right\} \int_{0}^{\tau }\left\Vert U\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt +\frac{h_{0} }{2}\varepsilon \int_{0}^{\tau }\left\Vert \nabla v\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt\\ &&+\frac{1}{2} l(\varepsilon ) \int_{0}^{\tau }\left\Vert U\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt+\frac{1}{2}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert\int_{0}^{\tau }h\circ U(t)dt \\ && +\frac{1}{2}\varepsilon \int_{0}^{\tau }\left\Vert \nabla v\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt. \end{eqnarray} | (4.16) |
Combining the relations (4.13)-(4.16) and (4.11) we get
\begin{equation} \begin{array}{l} (U_{\tau \tau }(x, \tau ), U(x, \tau ))_{L^{2}\left( \Omega \right) }+\alpha \left( U_{\tau }(x, \tau ), U(x, \tau )\right) _{L^{2}(\Omega )}+\frac{\beta }{2 }\left\Vert U(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2} \\ +\frac{\varrho }{2}\left\Vert \nabla v(x, 0)\right\Vert _{L^{2}(\Omega )}^{2}+ \frac{\gamma }{2}\left\Vert \nabla U(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}-\frac{1}{2}\left\Vert U_{\tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2} \\ \leq \bigg\{ \frac{\varrho }{2}T^{2}\left( l(\varepsilon )+\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \right) +\frac{\delta }{2} \left( T^{2}l(\varepsilon )+\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \right) +\frac{\gamma }{2}l(\varepsilon )T^{2}\\ \quad +\frac{h_{0}}{2}T^{2}(l(\varepsilon )+\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert)+\frac{1}{2}l(\varepsilon) \bigg\} \int_{0}^{\tau }\left\Vert U\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt \\ +\left( \alpha +\frac{\gamma \left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert }{2}\right) \int_{0}^{\tau }\left\Vert U_{t}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt+\frac{1}{2} \int_{0}^{\tau }\left\Vert U_{tt}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt \\ +\bigg\{\left( \frac{\varrho +\delta +\gamma +h_{0}}{2}\right) \varepsilon+h_{0}\bigg\} \int_{0}^{\tau }\left\Vert \nabla v\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt+h_{0}\int_{0}^{\tau}\left\Vert \nabla v(x, t )\right\Vert _{L^{2}(\Omega )}^{2}dt\\ +\frac{h_{0}}{2}\int_{0}^{\tau}\left\Vert \nabla U(x, t )\right\Vert _{L^{2}(\Omega )}^{2}dt+\frac{1}{2}\int_{0}^{\tau}h\circ \nabla U(t )dt+\frac{1}{2}\vert \Omega\vert\vert \partial \Omega \vert\int_{0}^{\tau}h\circ U(t )dt. \end{array} \end{equation} | (4.17) |
Next, multiplying the differential equation in (4.1) by U_{ttt} and integrating over Q_{\tau } = \Omega \times (0, \tau), \ we obtain
\begin{eqnarray} &&(U_{tttt}, U_{ttt})_{L^{2}\left( Q_{\tau }\right) }+\alpha (U_{ttt}, U_{ttt})_{L^{2}\left( Q_{\tau }\right) }+\beta (U_{tt}, U_{ttt})_{L^{2}\left( Q_{\tau }\right) }-\varrho (\Delta U, U_{ttt})_{L^{2}\left( Q_{\tau }\right) } \\ &&-\delta (\Delta U_{t}, U_{ttt})_{L^{2}\left( Q_{\tau }\right) }-\gamma (\Delta U_{t}, U_{ttt})_{L^{2}\left( Q_{\tau }\right) }+ (\Delta W, U_{ttt})_{L^{2}\left( Q_{\tau }\right) } = 0. \end{eqnarray} | (4.18) |
An integration by parts in (4.18) yields
\begin{equation} (U_{tttt}, U_{ttt})_{L^{2}\left( Q_{\tau }\right) } = \frac{1}{2}\left\Vert U_{\tau \tau \tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}, \end{equation} | (4.19) |
\begin{equation} \alpha \left( U_{ttt}, U_{ttt}\right) _{L^{2}\left( Q_{\tau }\right) } = \alpha \int_{0}^{\tau }\left\Vert U_{ttt}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt, \end{equation} | (4.20) |
\begin{equation} \beta \left( U_{tt}, U_{ttt}\right) _{L^{2}\left( Q_{\tau }\right) } = \frac{ \beta }{2}\left\Vert U_{\tau \tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}, \end{equation} | (4.21) |
\begin{eqnarray} -\varrho \left( \Delta U, U_{ttt}\right) _{L^{2}\left( Q_{\tau }\right) } & = &\varrho \left( \nabla U(x, \tau ), \nabla U_{\tau \tau }(x, \tau )\right) _{L^{2}(\Omega )}-\frac{\varrho }{2}\left\Vert \nabla U_{\tau }\left( x, \tau \right) \right\Vert _{L^{2}(\Omega )}^{2} \\ &&-\varrho \int_{\partial \Omega }U_{\tau \tau }(x, \tau )\left( \int_{0}^{\tau }\int_{\Omega }U(\xi , \eta )d\xi d\eta \right) ds_{x} \\ &&+\varrho \int_{\partial \Omega }\int_{0}^{\tau }U_{tt}\left( x, t\right) \int_{\Omega }U\left( \xi , t\right) d\xi dtds_{x}, \end{eqnarray} | (4.22) |
\begin{eqnarray} -\delta \left( \Delta U_{t}, U_{ttt}\right) _{L^{2}\left( Q_{\tau }\right) } & = &\delta \left( \nabla U_{\tau }(x, \tau ), \nabla U_{\tau \tau }(x, \tau )\right) _{L^{2}(\Omega )}-\delta \int_{0}^{\tau }\left\Vert \nabla U_{tt}(x, )\right\Vert _{L^{2}(\Omega )}^{2}dt \\ &&-\delta \int_{\partial \Omega }U_{\tau \tau }(x, \tau )\int_{\Omega }U\left( \xi , \tau \right) d\xi ds_{x} \\ &&+\delta \int_{0}^{\tau }\int_{\partial \Omega }U_{tt}\left( x, t\right) \int_{\Omega }U_{t}\left( \xi , t\right) d\xi ds_{x}dt, \end{eqnarray} | (4.23) |
\begin{eqnarray} -\gamma \left( \Delta U_{tt}, U_{ttt}\right) _{L^{2}\left( Q_{\tau }\right) } & = &\frac{\gamma }{2}\left\Vert \nabla U_{\tau \tau }\left( x, \tau \right) \right\Vert _{L^{2}(\Omega )}^{2}-\gamma \int_{\partial \Omega }U_{\tau \tau }\left( x, \tau \right) \int_{\Omega }U_{\tau }\left( \xi , \tau \right) d\xi ds_{x} \\ &&+\gamma \int_{0}^{\tau }\int_{\partial \Omega }U_{tt}\left( x, t\right) \int_{\Omega }U_{tt}\left( \xi , t\right) d\xi ds_{x}dt. \end{eqnarray} | (4.24) |
\begin{eqnarray} \left( \Delta W, U_{ttt}\right) _{L^{2}\left( Q_{\tau }\right) } & = &-H(\tau) \left( \nabla U(x, \tau ), \nabla U_{\tau \tau }(x, \tau )\right) _{L^{2}(\Omega )}\\ &&+\int_{\Omega}\nabla U_{\tau\tau}\int_{0}^{\tau}h(\tau-\sigma)(\nabla U(\tau)-\nabla U(\sigma))d\sigma dx \\ &&-\int_{0}^{\tau}(\nabla U_{tt}, \int_{0}^{t}h'(t-\sigma)(\nabla U(t)-\nabla U(\sigma))d\sigma)_{L^{2}(\Omega )}dt\\ &&+\int_{0}^{\tau}h(t)(\nabla U_{tt}, \nabla U(t))_{L^{2}(\Omega )}dt\\ &&+\int_{\partial \Omega }U_{\tau \tau }(x, \tau )\left( \int_{0}^{\tau }\int_{\Omega }W(\xi , \eta )d\xi d\eta \right) ds_{x} \\ &&- \int_{\partial \Omega }\int_{0}^{\tau }U_{tt}\left( x, t\right) \int_{\Omega }W\left( \xi , t\right) d\xi dtds_{x}, \end{eqnarray} | (4.25) |
Substitution (4.19)-(4.25) into (4.18) we get the equality
\begin{equation} \begin{array}{l} \frac{1}{2}\left\Vert U_{\tau \tau \tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+\frac{\beta }{2}\left\Vert U_{\tau \tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+\varrho \left( \nabla U(x, \tau ), \nabla U_{\tau \tau }(x, \tau )\right) _{L^{2}(\Omega )} \\ +\delta \left( \nabla U_{\tau }(x, \tau ), \nabla U_{\tau \tau }(x, \tau )\right) _{L^{2}(\Omega )}+\frac{\gamma }{2}\left\Vert \nabla U_{\tau \tau }\left( x, \tau \right) \right\Vert _{L^{2}(\Omega )}^{2}-\frac{\varrho }{2} \left\Vert \nabla U_{\tau }\left( x, \tau \right) \right\Vert _{L^{2}(\Omega )}^{2} \\ -H(\tau) \left( \nabla U(x, \tau ), \nabla U_{\tau \tau }(x, \tau )\right) _{L^{2}(\Omega )}\\ +\int_{\Omega}\nabla U_{\tau\tau}\int_{0}^{\tau}h(\tau-\sigma)(\nabla U(\tau)-\nabla U(\sigma))d\sigma dx\\ = -\alpha \int_{0}^{\tau }\left\Vert U_{ttt}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt+\delta \int_{0}^{\tau }\left\Vert \nabla U_{tt}(x, )\right\Vert _{L^{2}(\Omega )}^{2}dt \\ +\varrho \int_{\partial \Omega }U_{\tau \tau }(x, \tau )\left( \int_{0}^{\tau }\int_{\Omega }U(\xi , \eta )d\xi d\eta \right) ds_{x}-\varrho \int_{\partial \Omega }\int_{0}^{\tau }U_{tt}\left( x, t\right) \int_{\Omega }U\left( \xi , t\right) d\xi dtds_{x} \\ +\delta \int_{\partial \Omega }U_{\tau \tau }(x, \tau )\int_{\Omega }U\left( \xi , \tau \right) d\xi ds_{x}-\delta \int_{0}^{\tau }\int_{\partial \Omega }U_{tt}\left( x, t\right) \int_{\Omega }U_{t}\left( \xi , t\right) d\xi ds_{x}dt \\ +\gamma \int_{\partial \Omega }U_{\tau \tau }\left( x, \tau \right) \int_{\Omega }U_{\tau }\left( \xi , \tau \right) d\xi ds_{x}-\gamma \int_{0}^{\tau }\int_{\partial \Omega }U_{tt}\left( x, t\right) \int_{\Omega }U_{tt}\left( \xi , t\right) d\xi ds_{x}dt\\ -\int_{0}^{\tau}(\nabla U_{tt}, \int_{0}^{t}h'(t-\sigma)(\nabla U(t)-\nabla U(\sigma))d\sigma)_{L^{2}(\Omega )}dt\\ +\int_{0}^{\tau}h(t)(\nabla U_{tt}, \nabla U(t))_{L^{2}(\Omega )}dt +\int_{\partial \Omega }U_{\tau \tau }(x, \tau )\left( \int_{0}^{\tau }\int_{\Omega }W(\xi , \eta )d\xi d\eta \right) ds_{x} \\ - \int_{\partial \Omega }\int_{0}^{\tau }U_{tt}\left( x, t\right) \int_{\Omega }W\left( \xi , t\right) d\xi dtds_{x}. \end{array} \end{equation} | (4.26) |
The right hand side of (4.26) can be bounded as follows
\begin{eqnarray} &&\varrho \int_{\partial \Omega }U_{\tau \tau }(x, \tau )\left( \int_{0}^{\tau }\int_{\Omega }U\left( \xi , \eta \right) d\xi d\eta \right) ds_{x} \\ &\leq &\frac{\varrho }{2\varepsilon _{1}^{\prime }}\left( \varepsilon \left\Vert \nabla U_{\tau \tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+l(\varepsilon )\left\Vert U_{\tau \tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}\right) \\ &&+\frac{\varrho }{2}\varepsilon _{1}^{\prime }T\left\vert \partial \Omega \right\vert \left\vert \Omega \right\vert \int_{0}^{\tau }\left\Vert U\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt, \end{eqnarray} | (4.27) |
\begin{eqnarray} &&-\varrho \int_{\partial \Omega }\int_{0}^{\tau }U_{tt}\left( x, t\right) \int_{\Omega }U\left( \xi , t\right) d\xi dtds_{x} \\ &\leq &\frac{\varrho }{2}\int_{0}^{\tau }\left\{ \varepsilon \left\Vert \nabla U_{tt}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}+l(\varepsilon )\left\Vert U_{tt}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}\right\} dt \\ &&+\frac{\varrho }{2}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \int_{0}^{\tau }\left\Vert U\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt, \end{eqnarray} | (4.28) |
\begin{eqnarray} &&\delta \int_{\partial \Omega }U_{\tau \tau }(x, \tau )\int_{\Omega }U\left( \xi , \tau \right) d\xi ds_{x} \\ &\leq &\frac{\delta }{2\varepsilon _{2}^{\prime }}\left( \varepsilon \left\Vert \nabla U_{\tau \tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+l(\varepsilon )\left\Vert U_{\tau \tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}\right) \\ &&+\frac{\delta }{2}\varepsilon _{2}^{\prime }T\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \left\Vert U(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}, \end{eqnarray} | (4.29) |
\begin{eqnarray} &&-\delta \int_{0}^{\tau }\int_{\partial \Omega }U_{tt}\left( x, t\right) \int_{\Omega }U_{t}\left( \xi , t\right) d\xi ds_{x}dt \\ &\leq &\frac{\delta }{2}\varepsilon \int_{0}^{\tau }\left\Vert \nabla U_{tt}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt+\frac{\delta }{2} l(\varepsilon )\int_{0}^{\tau }\left\Vert U_{tt}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt \\ &&+\frac{\delta }{2}T\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \int_{0}^{\tau }\left\Vert U_{t}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt, \end{eqnarray} | (4.30) |
\begin{eqnarray} &&\gamma \int_{\partial \Omega }U_{\tau \tau }\left( x, \tau \right) \int_{\Omega }U_{\tau }\left( \xi , \tau \right) d\xi ds_{x} \\ &\leq &\frac{\gamma }{2\varepsilon _{3}^{\prime }}\left( \varepsilon \left\Vert \nabla U_{\tau \tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+l(\varepsilon )\left\Vert U_{\tau \tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}\right) \\ &&+\frac{\gamma }{2}\varepsilon _{3}^{\prime }T\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \left\Vert U_{\tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}, \end{eqnarray} | (4.31) |
\begin{eqnarray} &&-\gamma \int_{0}^{\tau }\int_{\partial \Omega }U_{tt}\left( x, t\right) \int_{\Omega }U_{tt}\left( \xi , t\right) d\xi ds_{x}dt \\ &\leq &\frac{\gamma }{2}l(\varepsilon )\int_{0}^{\tau }\left\Vert U_{tt}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt+\frac{\gamma }{2} \varepsilon \int_{0}^{\tau }\left\Vert \nabla U_{tt}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt \\ &&+\frac{\gamma }{2}T\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \int_{0}^{\tau }\left\Vert U_{tt}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt, \end{eqnarray} | (4.32) |
\begin{eqnarray} && \int_{\partial \Omega }U_{\tau \tau }(x, \tau )\left( \int_{0}^{\tau }\int_{\Omega }W\left( \xi , \eta \right) d\xi d\eta \right) ds_{x} \\ &\leq &(\frac{h_{0}}{2\varepsilon _{6}^{\prime }}+\frac{1}{2\varepsilon _{7}^{\prime }})\bigg( \varepsilon \left\Vert \nabla U_{\tau \tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+l(\varepsilon )\left\Vert U_{\tau \tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}\bigg) \\ &&+\frac{h_{0}}{2}\varepsilon _{6}^{\prime }T\left\vert \partial \Omega \right\vert \left\vert \Omega \right\vert \int_{0}^{\tau }\left\Vert U\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt+\frac{1}{2}\varepsilon _{7}^{\prime }\left\vert \partial \Omega \right\vert \left\vert \Omega \right\vert \int_{0}^{\tau }h\circ U(t)dt, \end{eqnarray} | (4.33) |
\begin{eqnarray} &&-\int_{\partial \Omega }\int_{0}^{\tau }U_{tt}\left( x, t\right) \int_{\Omega }W\left( \xi , t\right) d\xi dtds_{x} \\ &\leq &\frac{h_{0}+1 }{2}\int_{0}^{\tau }\left\{ \varepsilon \left\Vert \nabla U_{tt}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}+l(\varepsilon )\left\Vert U_{tt}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}\right\} dt \\ &&+\frac{h_{0} }{2}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \int_{0}^{\tau }\left\Vert U\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt+\frac{1}{2}\left\vert \partial \Omega \right\vert \left\vert \Omega \right\vert \int_{0}^{\tau }h\circ U(t)dt. \end{eqnarray} | (4.34) |
\begin{eqnarray} &&\int_{\Omega}\nabla U_{\tau\tau}\int_{0}^{\tau}h(\tau-\sigma)(\nabla U(\tau)-\nabla U(\sigma))d\sigma dx\\ &\geq&-\frac{1}{2\varepsilon'_{8}}h_{0} \left\Vert \nabla U_{\tau\tau}\left( x, \tau\right) \right\Vert _{L^{2}(\Omega )}^{2}-\frac{1}{2}\varepsilon'_{8}h\circ\nabla U(\tau), \end{eqnarray} | (4.35) |
\begin{eqnarray} &&-H(\tau) \left( \nabla U(x, \tau ), \nabla U_{\tau \tau }(x, \tau )\right) _{L^{2}(\Omega )} \\ &\geq &-\frac{1}{2\varepsilon'_{9}}h_{0} \left\Vert \nabla U_{\tau\tau}\left( x, \tau\right) \right\Vert _{L^{2}(\Omega )}^{2}-\frac{1}{2}\varepsilon'_{9}h_{0}\left\Vert\nabla U\left( x, \tau\right) \right\Vert _{L^{2}(\Omega )}^{2}, \end{eqnarray} | (4.36) |
\begin{eqnarray} &&\int_{0}^{\tau}h(t)\nabla U_{tt}(x, t)\nabla U(x, t)dt\\ &\leq&\frac{h(0)}{2}\int_{0}^{\tau} \left\Vert \nabla U_{tt}\left( x, \tau\right) \right\Vert _{L^{2}(\Omega )}^{2}dt+\frac{h(0)}{2}\int_{0}^{\tau} \left\Vert \nabla U\left( x, \tau\right) \right\Vert _{L^{2}(\Omega )}^{2}dt, \end{eqnarray} | (4.37) |
\begin{eqnarray} &&\int_{0}^{\tau}\nabla U_{tt}\int_{0}^{t}h'(t-\sigma)(\nabla U(t)-\nabla U(\sigma))d\sigma dx\\ &\leq&-\frac{h(t)-h(0)}{2}\int_{0}^{\tau} \left\Vert \nabla U_{tt}\left( x, \tau\right) \right\Vert _{L^{2}(\Omega )}^{2}dt-\frac{1}{2}\int_{0}^{\tau}h'\circ\nabla U(t)dt. \end{eqnarray} | (4.38) |
So, combining inequalities (4.27)-(4.38) and equality (4.26) we obtain
\begin{equation} \begin{array}{l} \frac{1}{2}\left\Vert U_{\tau \tau \tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+\left\{ \frac{\beta }{2}-\frac{\varrho }{2\varepsilon _{1}^{\prime }} l(\varepsilon )-\frac{\delta }{2\varepsilon _{2}^{\prime }}l(\varepsilon )- \frac{\gamma }{2\varepsilon'_{3}} l(\varepsilon )-(\frac{h_{0}}{2\varepsilon _{6}^{\prime }}+\frac{1}{2\varepsilon _{7}^{\prime }})l(\varepsilon)\right\} \left\Vert U_{\tau \tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2} \\ -\frac{\gamma }{2}\varepsilon _{3}^{\prime }T\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \left\Vert U_{\tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}-\frac{\delta }{2}\varepsilon _{2}^{\prime }T\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \left\Vert U(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2} \\ +\left\{ \frac{\gamma }{2}-\frac{\varrho }{2\varepsilon _{1}^{\prime }} \varepsilon -\frac{\delta }{2\varepsilon _{2}^{\prime }}\varepsilon -\frac{ \gamma }{2\varepsilon _{3}^{\prime }}\varepsilon+(\frac{h_{0}}{2\varepsilon _{6}^{\prime }}+\frac{1}{2\varepsilon _{7}^{\prime }})\varepsilon-(\frac{1}{2\varepsilon _{8}^{\prime }}+\frac{1}{2\varepsilon _{9}^{\prime }})h_{0} \right\} \left\Vert \nabla U_{\tau \tau }\left( x, \tau \right) \right\Vert _{L^{2}(\Omega )}^{2}\\ -\frac{\varepsilon'_{8}}{2}h\circ \nabla U(\tau)-\frac{\varepsilon'_{9}}{2}h_{0}\left\Vert \nabla U\left( x, \tau \right) \right\Vert _{L^{2}(\Omega )}^{2}-\frac{ \varrho }{2}\left\Vert \nabla U_{\tau }\left( x, \tau \right) \right\Vert _{L^{2}(\Omega )}^{2} \\ +\varrho \left( \nabla U(x, \tau ), \nabla U_{\tau \tau }(x, \tau )\right) _{L^{2}(\Omega )}+\delta \left( \nabla U_{\tau }(x, \tau ), \nabla U_{\tau \tau }(x, \tau )\right) _{L^{2}(\Omega )} \\ \leq -\alpha \int_{0}^{\tau }\left\Vert U_{ttt}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt+\left\{ \frac{\varrho }{2}l(\varepsilon )+\frac{ \delta }{2}l(\varepsilon )+\frac{\gamma }{2}l(\varepsilon )+\frac{\gamma }{2} T\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert+(\frac{h_{0}+1}{0})l(\varepsilon) \right\} \int_{0}^{\tau }\left\Vert U_{tt}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt \\ +\left\{ \frac{\varrho }{2}\varepsilon _{1}^{\prime }T\left\vert \partial \Omega \right\vert \left\vert \Omega \right\vert +(\frac{\varrho }{2}+\frac{h_{0}}{2}(1+T\varepsilon'_{6})) \left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \right\} \int_{0}^{\tau }\left\Vert U\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt+\frac{\delta }{2}T\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \int_{0}^{\tau }\left\Vert U_{t}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt \\ +\left\{ \delta +\frac{\varrho }{2}\varepsilon +\frac{\delta }{2}\varepsilon +\frac{\gamma }{2}\varepsilon+\frac{h_{0}+1}{2}\varepsilon+\frac{3h(0)}{2} \right\} \int_{0}^{\tau }\left\Vert \nabla U_{tt}(x, )\right\Vert _{L^{2}(\Omega )}^{2}dt\\ -\frac{1}{2}\int_{0}^{\tau }h'\circ \nabla U(t)dt-\frac{h(0)}{2}\int_{0}^{\tau }\left\Vert \nabla U\left( x, \tau \right) \right\Vert _{L^{2}(\Omega )}^{2}dt+\frac{1+\varepsilon'_{7}}{2}\left\vert \partial \Omega \right\vert \left\vert \Omega \right\vert\int_{0}^{\tau }h'\circ U(t)dt. \end{array} \end{equation} | (4.39) |
Adding side to side (4.17) and (4.39), we obtain
\begin{equation} \begin{array}{l} \left\{ \frac{\beta }{2}-\frac{\delta }{2}\varepsilon _{2}^{\prime }T\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert-\frac{1+\alpha}{2} \right\} \left\Vert U(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+\frac{1}{ 2}\left\Vert U_{\tau \tau \tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}\\ +\left\{- \frac{1+\alpha}{2}-\frac{\gamma }{2}\varepsilon _{3}^{\prime }T\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \right\} \left\Vert U_{\tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2} \\ +\left\{ \frac{\beta }{2}-\frac{\varrho }{2\varepsilon _{1}^{\prime }} l(\varepsilon )-l(\varepsilon )\frac{\delta }{2\varepsilon _{2}^{\prime }}- \frac{\gamma }{2\varepsilon _{3}^{\prime }}l(\varepsilon )-(\frac{h_{0}}{2\varepsilon _{6}^{\prime }}+\frac{1}{2\varepsilon _{7}^{\prime }})-\frac{1}{2}\right\} \left\Vert U_{\tau \tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2} \\ +(U_{\tau \tau }(x, \tau ), U(x, \tau ))_{L^{2}\left( \Omega \right) }+\alpha \left( U_{\tau }(x, \tau ), U(x, \tau )\right) _{L^{2}(\Omega )}+\frac{\varrho }{2}\left\Vert \nabla v(x, 0)\right\Vert _{L^{2}(\Omega )}^{2} \\ +\varrho \left( \nabla U(x, \tau ), \nabla U_{\tau \tau }(x, \tau )\right) _{L^{2}(\Omega )}+\delta \left( \nabla U_{\tau }(x, \tau ), \nabla U_{\tau \tau }(x, \tau )\right) _{L^{2}(\Omega )} \\ +(\frac{\gamma }{2}-\frac{\varepsilon'_{9}}{2}h_{0})\left\Vert \nabla U(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}-\frac{\varrho }{2}\left\Vert \nabla U_{\tau }\left( x, \tau \right) \right\Vert _{L^{2}(\Omega )}^{2}-\frac{\varepsilon'_{8}}{2}h\circ \nabla U(\tau)\\ +\left\{ \frac{\gamma }{2}-\frac{\varrho }{ 2\varepsilon _{1}^{\prime }}\varepsilon -\frac{\delta }{2\varepsilon _{2}^{\prime }}-\frac{\gamma }{2\varepsilon _{3}^{\prime }}\varepsilon-(\frac{h_{0}}{2\varepsilon _{6}^{\prime }}+\frac{1}{2\varepsilon _{7}^{\prime }})\varepsilon-(\frac{1}{2\varepsilon _{8}^{\prime }}+\frac{1}{2\varepsilon _{9}^{\prime }})h_{0} \right\} \left\Vert \nabla U_{\tau \tau }\left( x, \tau \right) \right\Vert _{L^{2}(\Omega )}^{2} \\ \leq \left\{ \frac{\varrho }{2}\varepsilon _{1}^{\prime }T\left\vert \partial \Omega \right\vert \left\vert \Omega \right\vert +\frac{\varrho }{2} \left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert +\frac{ \varrho }{2}T^{2}\left( l(\varepsilon )+\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \right) +\frac{\delta }{2}\left( T^{2}l(\varepsilon )+\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \right) \right. \\ \left. +\frac{\gamma }{2}l(\varepsilon )T^{2}+\frac{h_{0}}{2}T^{2}(l(\varepsilon )+\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert)+\frac{1}{2}l(\varepsilon)\right\} \int_{0}^{\tau }\left\Vert U\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt \\ +\left( \alpha +\frac{\gamma \left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert }{2}+\frac{\delta }{2}T\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \right) \int_{0}^{\tau }\left\Vert U_{t}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt \\ +\left\{ \frac{1}{2}+l(\varepsilon )\frac{\varrho }{2}+\frac{\delta }{2} l(\varepsilon )+\frac{\gamma }{2}l(\varepsilon )+\frac{\gamma }{2} T\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \right\} \int_{0}^{\tau }\left\Vert U_{tt}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt \\ +\left\{ \frac{\delta }{2}\varepsilon +\frac{\gamma }{2}\varepsilon +\varepsilon \frac{\varrho }{2}+\delta+\frac{h_{0}+1}{2}\varepsilon+\frac{3h(0)}{2} \right\} \int_{0}^{\tau }\left\Vert \nabla U_{tt}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt\\ -\alpha \int_{0}^{\tau }\left\Vert U_{ttt}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt+\bigg(h_{0}+\left( \frac{\varrho +\delta +\gamma+h_{0} }{2}\right) \varepsilon\bigg) \int_{0}^{\tau }\left\Vert \nabla v\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt\\ +\frac{h_{0}+h(0)}{2}\int_{0}^{\tau}\left\Vert \nabla U(x, t )\right\Vert _{L^{2}(\Omega )}^{2}dt+\frac{1}{2}\int_{0}^{\tau}h\circ \nabla U(t )dt+\frac{1}{2}\vert \Omega\vert\vert \partial \Omega \vert\int_{0}^{\tau}h\circ U(t )dt\\ -\frac{1}{2}\int_{0}^{\tau }h'\circ \nabla U(t)dt+\frac{1+\varepsilon'_{7}}{2}\left\vert \partial \Omega \right\vert \left\vert \Omega \right\vert\int_{0}^{\tau }h'\circ U(t)dt. \end{array} \end{equation} | (4.40) |
Now to deal with the last term on the right hand side of (4.40) , we define the function \theta \left(x, t\right) by the relation
\begin{equation*} \theta \left( x, t\right) : = \int_{0}^{t}U(x, s)ds. \end{equation*} |
Hence using (4.12) it follows that
\begin{equation} v\left( x, t\right) = \theta (x, \tau )-\theta \left( x, t\right) , \ \nabla v(x, 0) = \nabla \theta (x, \tau ), \end{equation} | (4.41) |
and
\begin{eqnarray} \left\Vert \nabla v\right\Vert _{L^{2}\left( Q_{\tau }\right) }^{2} & = &\left\Vert \nabla \theta (x, \tau )-\nabla \theta \left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2} \\ &\leq &2\left( \tau \left\Vert \nabla \theta (x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+\left\Vert \nabla \theta \left( x, t\right) \right\Vert _{L^{2}\left( Q_{\tau }\right) }^{2}\right) . \end{eqnarray} | (4.42) |
And make use of the following inequality
\begin{equation} -\frac{\alpha }{2}\left\Vert U_{\tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}-\frac{\alpha }{2}\left\Vert U(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}\leq \alpha (U_{\tau }(x, \tau ), U(x, \tau ))_{L^{2}(\Omega )}, \end{equation} | (4.43) |
\begin{equation} -\frac{1}{2}\left\Vert U_{\tau \tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}-\frac{1}{2}\left\Vert U(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}\leq (U_{\tau \tau }(x, \tau ), U(x, \tau ))_{L^{2}\left( \Omega \right) }, \end{equation} | (4.44) |
\begin{equation} -\frac{\varrho }{2\varepsilon _{4}^{\prime }}\left\Vert \nabla U_{\tau \tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}-\frac{\varrho }{2}\varepsilon _{4}^{\prime }\left\Vert \nabla U(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}\leq \varrho \left( \nabla U(x, \tau ), \nabla U_{\tau \tau }(x, \tau )\right) _{L^{2}(\Omega )}, \end{equation} | (4.45) |
\begin{equation} -\frac{\delta }{2\varepsilon _{5}^{\prime }}\left\Vert \nabla U_{\tau \tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}-\frac{\delta }{2}\varepsilon _{5}^{\prime }\left\Vert \nabla U_{\tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}\leq \delta \left( \nabla U_{\tau }(x, \tau ), \nabla U_{\tau \tau }(x, \tau )\right) _{L^{2}(\Omega )}. \end{equation} | (4.46) |
\begin{eqnarray} m_{1}\left\Vert U(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}&\leq& m_{1}\left\Vert U(x, t)\right\Vert _{L^{2}(Q_{\tau })}^{2}+m_{1}\left\Vert U_{t}(x, t)\right\Vert _{L^{2}(Q_{\tau })}^{2} , \\ m_{2}\left\Vert U_{\tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}&\leq& m_{2}\left\Vert U_{t}(x, t)\right\Vert _{L^{2}(Q_{\tau })}^{2}+m_{2}\left\Vert U_{tt}(x, t)\right\Vert _{L^{2}(Q_{\tau })}^{2}, \\ m_{3}\left\Vert U_{\tau \tau }\left( x, \tau \right) \right\Vert _{L^{2}(\Omega )}^{2}&\leq& m_{3}\left\Vert U_{tt}(x, t)\right\Vert _{L^{2}(Q_{\tau })}^{2}+m_{3}\left\Vert U_{ttt}(x, t)\right\Vert _{L^{2}(Q_{\tau })}^{2} , \\ m_{4}\left\Vert \nabla U(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}&\leq& m_{4}\left\Vert \nabla U(x, t)\right\Vert _{L^{2}(Q_{\tau })}^{2}+m_{4}\left\Vert \nabla U_{t}(x, t)\right\Vert _{L^{2}(Q_{\tau })}^{2}, \end{eqnarray} | (4.47) |
\begin{eqnarray} m_{5}\left\Vert \nabla U_{\tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}&\leq& m_{5}\left\Vert \nabla U_{t}(x, t)\right\Vert _{L^{2}(Q_{\tau })}^{2}+m_{5}\left\Vert \nabla U_{tt}(x, t)\right\Vert _{L^{2}(Q_{\tau })}^{2}, \\ m_{6}h\circ \nabla U (\tau)&\leq&m_{6}\left\Vert \nabla U_{t}(x, t)\right\Vert _{L^{2}(Q_{\tau })}^{2}+ m_{6}\int_{0}^{\tau}h\circ\nabla U (t)dt\\ m_{7}h\circ U (\tau)&\leq&m_{7}\left\Vert U_{t}(x, t)\right\Vert _{L^{2}(Q_{\tau })}^{2}+ m_{7}\int_{0}^{\tau}h\circ U (t)dt\\ -m_{8}h'\circ \nabla U (\tau)&\leq&m_{8}\left\Vert \nabla U_{t}(x, t)\right\Vert _{L^{2}(Q_{\tau })}^{2}- m_{8}\int_{0}^{\tau}h'\circ\nabla U (t)dt. \end{eqnarray} | (4.48) |
Let
\begin{equation} \left\{ \begin{array}{l} m_{1}: = \frac{1+\alpha}{2}+\frac{\delta }{2}\varepsilon _{2}^{\prime }T\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert, \\ \\ m_{2}: = 1+\frac{\gamma }{2}\varepsilon _{3}^{\prime }T\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert +\frac{\alpha }{2} \\ \\ m_{3}: = \bigg(\frac{\varrho }{2\varepsilon _{1}^{\prime }}+\frac{\delta }{2\varepsilon _{2}^{\prime }}+\frac{\gamma }{ 2\varepsilon _{3}^{\prime }}+\frac{h_{0} }{2\varepsilon _{6}^{\prime }}+\frac{1 }{2\varepsilon _{7}^{\prime }}\bigg)l(\varepsilon )+\frac{1}{2} \\ \\ m_{4}: = \frac{\varrho }{2}\varepsilon _{4}^{\prime }+\frac{h_{0}}{2} \\ m_{5}: = 1+\frac{\varrho }{2}+\frac{\delta }{2\varepsilon _{5}^{\prime }}\\ m_{6}: = \frac{1}{2}\varepsilon _{8}^{\prime }+1 , \quad m_{7}: = 1, \quad m_{8}: = 1, \end{array} \right. \end{equation} | (4.49) |
choosing \varepsilon _{1}^{\prime }, \ \varepsilon _{2}^{\prime }, \varepsilon _{3}^{\prime }, \varepsilon _{4}^{\prime } , \varepsilon _{5}^{\prime }, \ \varepsilon _{6}^{\prime }, \varepsilon _{7}^{\prime }, \varepsilon _{8}^{\prime } and \varepsilon _{9}^{\prime } sufficiently large
\begin{equation} \alpha_{0}: = \frac{\gamma }{2}-\frac{\varrho }{ 2\varepsilon _{1}^{\prime }}\varepsilon -\frac{\delta }{2\varepsilon _{2}^{\prime }}-\frac{\gamma }{2\varepsilon _{3}^{\prime }}\varepsilon-\frac{\varrho }{2\varepsilon _{4}^{\prime }}-\frac{\delta }{2\varepsilon _{5}^{\prime }}\varepsilon-(\frac{h_{0}}{2\varepsilon _{6}^{\prime }}+\frac{1}{2\varepsilon _{7}^{\prime }})\varepsilon-(\frac{1}{2\varepsilon _{8}^{\prime }}+\frac{1}{2\varepsilon _{9}^{\prime }})h_{0} > 0. \end{equation} | (4.50) |
Since \tau is arbitrary we get that \alpha_{1}: = \frac{\varrho }{2}-2\tau \bigg(h_{0}+\varepsilon \frac{\left(\varrho +\delta +\gamma+h_{0} \right)}{2}\bigg) > 0, \ thus inequality (4.40) takes the form
\begin{equation} \begin{array}{l} \frac{\beta }{2}\left\Vert U(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+\frac{ 1}{2}\left\Vert U_{\tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+\frac{ \beta }{2}\left\Vert U_{\tau \tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+\frac{1}{2}\left\Vert U_{\tau \tau \tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2} \\ +\left\{ \frac{\varrho }{2}-2\tau(h_{0}+ \varepsilon\frac{\left( \varrho +\delta +\gamma+h_{0} \right)}{2}) \right\} \left\Vert \nabla \theta (x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+\frac{\gamma }{2}\left\Vert \nabla U(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2} \\ +\left\Vert \nabla U_{\tau }\left( x, \tau \right) \right\Vert _{L^{2}(\Omega )}^{2}+\alpha_{0} \left\Vert \nabla U_{\tau \tau }\left( x, \tau \right) \right\Vert _{L^{2}(\Omega )}^{2} +h\circ \nabla U(\tau)+h\circ U(\tau)-h'\circ \nabla U(\tau)\\ \leq \left\{ \gamma _{1}^{\prime }+m_{1}\right\} \int_{0}^{\tau }\left\Vert U\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt+\left( \gamma _{2}^{\prime }+m_{1}+m_{2}+m_{7}\right) \int_{0}^{\tau }\left\Vert U_{t}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt \\ +\left\{ \gamma _{3}^{\prime }+m_{2}+m_{3}\right\} \int_{0}^{\tau }\left\Vert U_{tt}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt+\left( m_{3}-\alpha \right) \int_{0}^{\tau }\left\Vert U_{ttt}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt \\ +(2h_{0}+ \varepsilon\left( \varrho +\delta +\gamma+h_{0} \right)) \int_{0}^{\tau }\left\Vert \nabla \theta \left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt+(\gamma _{4}^{\prime } +m_{5}) \int_{0}^{\tau }\left\Vert \nabla U_{tt}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt \\ +(m_{4}+\frac{h_{0}+h(0)}{2})\int_{0}^{\tau }\left\Vert \nabla U\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt\\ +\left( m_{4}+m_{5}+m_{6}+m_{8}\right) \int_{0}^{\tau }\left\Vert \nabla U_{t}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt+(\frac{1}{2}+m_{6})\int_{0}^{\tau }h\circ \nabla U(t)dt \\ +(\gamma'_{5}+m_{7})\int_{0}^{\tau }h\circ U(t)dt-(\frac{1}{2}+m_{8})\int_{0}^{\tau }h'\circ \nabla U(t)dt, \end{array} \end{equation} | (4.51) |
where
\begin{equation} \left\{ \begin{array}{l} \gamma _{1}^{\prime }: = \frac{\varrho }{2}\varepsilon _{1}^{\prime }T\left\vert \partial \Omega \right\vert \left\vert \Omega \right\vert + \frac{\varrho }{2}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert +\frac{\varrho }{2}T^{2}\left( l(\varepsilon )+\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \right)\\ \quad +\frac{\delta }{2} \left( T^{2}l(\varepsilon )+\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \right) +\frac{\gamma }{2}l(\varepsilon )T^{2} \\ \\ \gamma _{2}^{\prime }: = \alpha +\frac{\gamma \left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert }{2}+\frac{\delta }{2}T\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \\ \gamma _{3}^{\prime }: = \frac{1}{2}+l(\varepsilon )\frac{\varrho }{2}+\frac{ \delta }{2}l(\varepsilon )+\frac{\gamma }{2}l(\varepsilon )+\frac{\gamma }{2} T\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert\\ \gamma _{4}^{\prime }: = \frac{\delta }{2}\varepsilon +\frac{\gamma }{2}\varepsilon +\varepsilon \frac{\varrho }{2}+\delta \\ \gamma _{5}^{\prime }: = (1+\frac{1}{2}\varepsilon'_{7})\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \end{array} \right. \end{equation} | (4.52) |
We obtain
\begin{eqnarray} &&\left\Vert U(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+\left\Vert U_{\tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+\left\Vert U_{\tau \tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+\left\Vert U_{\tau \tau \tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2} \\ &&+\left\Vert \nabla U(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+\left\Vert \nabla U_{\tau }\left( x, \tau \right) \right\Vert _{L^{2}(\Omega )}^{2}+\left\Vert \nabla U_{\tau \tau }\left( x, \tau \right) \right\Vert _{L^{2}(\Omega )}^{2} \\ &&+\left\Vert \nabla \theta (x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+h\circ \nabla U(\tau)+h\circ U(\tau)-h'\circ \nabla U(\tau)\\ &\leq &D\int_{0}^{\tau }\bigg\{ \left\Vert U\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}+\left\Vert U_{t}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}+\left\Vert U_{tt}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}+h\circ U(t) \\ &&+\left\Vert \nabla U\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}+\left\Vert \nabla U_{t}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}+\left\Vert \nabla U_{tt}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}-h'\circ \nabla U(t) \\ && +\left\Vert U_{ttt}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}+\left\Vert \nabla \theta \left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}+h\circ \nabla U(t)\bigg\} dt, \end{eqnarray} | (4.53) |
where
\begin{equation} D: = \frac{ \begin{array}{c} \max \left\{ (\gamma _{1}^{\prime }+m_{1}), \left( \gamma _{2}^{\prime }+m_{1}+m_{2}+m_{7}\right), \gamma _{3}^{\prime }+m_{2}+m_{3}, m_{3}-\alpha , \right. \\ \left. m_{4}+m_{5}+m_{6}+m_{8}, \gamma _{4}^{\prime } +m_{5}, (2h_{0}+\varepsilon \left( \varrho +\delta +\gamma+h_{0} \right)), \right. \\ \left.m_{4}+\frac{h_{0}+h(0)}{2}, \frac{1}{2}+m_{6}, \gamma'_{5}+m_{7}, \frac{1}{2}+m_{8} \right\} \end{array} }{ \begin{array}{c} \min \left\{ \frac{\beta }{2}, \frac{1}{2}, , \frac{\gamma }{2}, \alpha_{0} , \alpha_{1} \right\} \end{array} }. \end{equation} | (4.54) |
Further, applying Gronwall's lemma to (4.53), we deduce that
\begin{equation} \begin{array}{l} \left\Vert U(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+\left\Vert U_{\tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+\left\Vert U_{\tau \tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+\left\Vert U_{\tau \tau \tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2} \\ +\left\Vert \nabla U(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+\left\Vert \nabla U_{\tau }\left( x, \tau \right) \right\Vert _{L^{2}(\Omega )}^{2}+\left\Vert \nabla U_{\tau \tau }\left( x, \tau \right) \right\Vert _{L^{2}(\Omega )}^{2}\\ +\left\Vert \nabla \theta (x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+h\circ \nabla U(\tau)+h\circ U(\tau)-h'\circ \nabla U(\tau) \leq 0, \forall \tau \in \left[ 0, \alpha_{2}\right] . \end{array} \end{equation} | (4.55) |
where \alpha_{2}: = \frac{\varrho }{4h_{0}+2\varepsilon \left(\varrho +\delta +\gamma+h_{0} \right) } .
Proceeding in the same way for the intervals \tau \in \left[(m-1)\alpha_{2}, m\alpha_{2}\right] \ to cover the whole interval \left[0, T\right], and thus proving that U(x, \tau) = 0 , for all \tau in \left[0, T\right].\ Thus, the uniqueness is proved.
Study of sound wave propagation, it should be noted that the Moore-Gibson-Thomson equation is one of the nonlinear sound equations that describes the propagation of sound waves in gases and liquids. The behavior of sound waves depends strongly on the average scattering, scattering and nonlinear effects. Arises from high-frequency ultrasound (HFU) modeling (see [16,25,41]). In this work, we have studied the solvability of the nonlocal mixed boundary value problem for the fourth order of Moore-Gibson-Thompson equation with source and memory terms. Galerkin's method was the main used tool for proving the solvability of the given non local problem. In the next work, we will try to using the same method with Hall-MHD equations which are nonlinear partial differential equation that arises in hydrodynamics and some physical applications (see for example [2,3,4,6]) by using some famous algorithms (see [8,14,15]).
The fourth author extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through research groups program under grant (R.G.P-2/1/42).
This work does not have any conflicts of interest.
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