Citation: Carlos Menéndez Villalva, Xose Luis Muiño López-Alvarez, Martín Menéndez Rodríguez, María José Modroño Freire, Olalla Quintairos Veloso, Lea Conde Guede, Sandra Vilchez Dosantos, Manuel Blanco Ramos. Blood Pressure Monitoring in Cardiovascular Disease[J]. AIMS Medical Science, 2017, 4(2): 164-191. doi: 10.3934/medsci.2017.2.164
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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) |
ϱ∫τ0∫∂ΩuNttt(∫t0∫ΩuN(ξ,η)dξdη)dsxdt | (3.25) |
=ϱ∫∂ΩuNττ(x,τ)∫τ0∫ΩuN(ξ,t)dξdtdsx−ϱ∫∂Ω∫τ0uNtt(x,t)∫ΩuN(ξ,t)dξdtdsx, | (3.26) |
δ∫τ0∫∂ΩuNttt(x,t)(∫t0∫ΩuNt(ξ,η)dξdη)dsxdt=δ∫∂ΩuNττ(x,τ)∫ΩuN(ξ,τ)dξdsx−δ∫∂ΩuNττ(x,τ)∫ΩuN(ξ,0)dξdsx−δ∫∂Ω∫τ0uNtt(x,t)∫ΩuNt(ξ,t)dξdtds, | (3.27) |
γ∫τ0∫∂ΩuNttt(x,t)(∫t0∫ΩuNtt(ξ,η)dξdη)dsxdt=γ∫∂ΩuNττ(x,τ)∫ΩuNτ(ξ,τ)dξdsx−γ∫∂ΩuNττ(x,τ)∫ΩuNt(ξ,0)dξdsx−γ∫∂Ω∫τ0uNtt(x,t)∫ΩuNtt(ξ,t)dξdtds, | (3.28) |
−∫τ0∫∂ΩuNttt(∫t0∫ΩwN(ξ,η)dξdη)dsxdt=−∫τ0∫∂ΩuNttt(∫t0∫ΩH(η)uN(ξ,η)dξdη)dsxdt+∫τ0∫∂ΩuNttt(∫t0∫Ω[∫η0h(η−σ)(uN(ξ,η)−uN(ξ,σ))dσ]dξdη)dsxdt=−∫∂ΩuNττ(x,τ)∫τ0∫ΩH(t)uN(ξ,t)dξdtdsx+∫τ0∫∂ΩuNtt(x,t)∫ΩH(t)uN(ξ,t)dξdsxdt+∫∂ΩuNττ(x,τ)(∫τ0∫Ω[∫t0h(t−σ)(uN(ξ,t)−uN(ξ,σ))dσ]dξ)dsxdt−∫τ0∫∂ΩuNtt(x,t)∫Ω[∫t0h(t−σ)(uN(ξ,t)−uN(ξ,σ))dσ]dξdsxdt. | (3.29) |
A substitution of equalities (3.24)-(3.29) in (3.23), gives
12‖uNτττ(x,τ)‖2L2(Ω)+β2‖uNττ(x,τ)‖2L2(Ω)+ϱ(∇uN(x,τ),∇uNττ(x,τ))L2(Ω)+δ(∇uNτ(x,τ),∇uNττ(x,τ))L2(Ω)+γ2‖∇uNττ(x,τ)‖2L2(Ω)−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)=(F,uNttt)L2(Qτ)+12‖uNttt(x,0)‖2L2(Ω)−α∫τ0‖uNttt(x,t)‖2L2(Ω)+ϱ(∇uN(x,0),∇uNtt(x,0))L2(Ω)+ϱ∫τ0(∇uNt,∇uNtt)L2(Ω)dt+δ∫τ0‖∇uNtt(x,t)‖2L2(Ω)dt+δ(∇uNt(x,0),∇uNtt(x,0))L2(Ω)+ϱ∫∂ΩuNττ(x,τ)∫τ0∫ΩuN(ξ,t)dξdtdsx−γ2‖∇uNtt(x,0)‖2L2(Ω)−ϱ∫∂Ω∫τ0uNtt(x,t)∫ΩuN(ξ,t)dξdtdsx−β2‖uNtt(x,0)‖2L2(Ω)+δ∫∂ΩuNττ(x,τ)∫ΩuN(ξ,τ)dξdsx−δ∫∂ΩuNττ(x,τ)∫ΩuN(ξ,0)dξdsx−δ∫∂Ω∫τ0uNtt(x,t)∫ΩuNt(ξ,t)dξdtds+δ∫∂ΩuNττ(x,τ)∫ΩuNτ(ξ,τ)dξdsx−γ∫∂ΩuNττ(x,τ)∫ΩuNt(ξ,0)dξdsx−γ∫∂Ω∫τ0uNtt(x,t)∫ΩuNtt(ξ,t)dξdtds−∫∂ΩuNττ(x,τ)∫τ0∫ΩH(t)uN(ξ,t)dξdtdsx+12∫τ0(h″−h‴)∘∇uN(t)dt−h(0)∫τ0‖∇uNt(x,t)‖2L2(Ω)dt+∫τ0∫∂ΩuNtt(x,t)∫ΩH(t)uN(ξ,t)dξdsxdt+∫∂ΩuNττ(x,τ)(∫τ0∫Ω[∫t0h(t−σ)(uN(ξ,t)−uN(ξ,σ))dσ]dξ)dsxdt−∫τ0∫∂ΩuNtt(x,t)∫Ω[∫t0h(t−σ)(uN(ξ,t)−uN(ξ,σ))dσ]dξdsxdt. | (3.30) |
Multiplying (3.15) by λ1, (3.22) by λ2, and (3.30) by λ3 such as (λ1+λ2<λ3), we get
λ1(uNτττ(x,τ),uNτ(x,τ))L2(Ω)+λ1α(uNττ(x,τ),uNτ(x,τ))L2(Ω)+λ1β2‖uNτ(x,τ)‖2L2(Ω)+λ1ϱ2‖∇uN(x,τ)‖2L2(Ω)+(λ1γ2+λ2δ2)‖∇uNτ(x,τ)‖2L2(Ω)+λ2(uNτττ(x,τ),uNττ(x,τ))L2(Ω)+(λ2α2+λ3β2)‖uNττ(x,τ)‖2L2(Ω)+λ2ϱ(∇uN(x,τ),∇uNτ(x,τ))L2(Ω)+λ32‖uNτττ(x,τ)‖2L2(Ω)+λ3ϱ(∇uN(x,τ),∇uNττ(x,τ))L2(Ω)+λ3δ(∇uNτ(x,τ),∇uNττ(x,τ))L2(Ω)+λ3γ2‖∇uNττ(x,τ)‖2L2(Ω)+λ12h∘∇uN(τ)−λ12H(τ)‖∇uN(x,τ)‖2L2(Ω)−λ22h′∘∇uN(τ)+λ2(∇wN(τ),∇uNτ)L2(Ω)−λ3H(τ)(∇uNττ(x,τ),∇uN(x,τ))2L2(Ω)+λ3h(τ)(∇uNτ(x,τ),∇uN(x,τ))2L2(Ω)−λ32‖∇uN(x,τ)‖2L2(Ω)+λ3∫Ω∇uNττ∫τ0h(τ−σ)(∇uN(τ)−∇uN(σ))dσdx+λ3∫Ω∇uNτ∫τ0h′(τ−σ)(∇uN(τ)−∇uN(σ))dσdx+λ32h″∘∇uN(τ)=λ1(uNttt(x,0),uNt(x,0))L2(Ω)+λ1α(uNtt(x,0),uNt(x,0))L2(Ω)+λ1ϱ2‖∇uN(x,0)‖2L2(Ω)+λ1β2‖uNt(x,0)‖2L2(Ω)+(λ1γ2+λ2δ2)‖∇uNt(x,0)‖2L2(Ω)+λ1∫τ0(uNttt,uNtt)L2(Ω)dt+(λ1α−λ2β)∫τ0‖utt(x,t)‖2L2(Ω)dt+(λ2ϱ−λ1δ)∫τ0‖∇uNt(x,t)‖2L2(Ω)dt+(λ2−λ3α)∫τ0‖uNttt(x,t)‖2L2(Ω)dt+λ2(uNttt(x,0),uNtt(x,0))L2(Ω)+(λ2α2−λ3β2)‖uNtt(x,0)‖2L2(Ω)λ32‖uNttt(x,0)‖2L2(Ω)+λ2ϱ(∇uN(x,0),∇uNt(x,0))L2(Ω)+(λ3δ−λ2γ)∫τ0‖∇uNtt(x,t)‖2L2(Ω)dt+λ3ϱ(∇uN(x,0),∇uNtt(x,0))L2(Ω)+λ3ϱ∫τ0(∇uNt,∇uNtt)L2(Ω)dt+λ3δ(∇uNt(x,0),∇uNtt(x,0))L2(Ω)−λ3γ2‖∇uNtt(x,0)‖2L2(Ω)+λ1ϱ∫∂ΩuN(x,τ)∫τ0∫ΩuN(ξ,t)dξdtdsx−λ1ϱ∫∂Ω∫τ0uN(x,t)∫ΩuN(ξ,t)dξdtdsx+(λ1δ−λ2ϱ)∫∂Ω∫τ0uNt(x,t)∫ΩuN(ξ,t)dξdtdsx−λ1δ∫∂Ω∫τ0uNt(x,t)∫ΩuN(ξ,0)dξdtdsx+(λ1γ−λ2δ)∫τ0∫∂ΩuNt(x,t)(∫ΩuNt(ξ,t)dξ)dsxdt−λ1γ∫τ0∫∂ΩuNt(x,t)(∫ΩuNt(ξ,0)dξ)dsxdt+λ2ϱ∫∂ΩuNτ(x,τ)∫τ0∫ΩuN(ξ,t)dξdtdsx+λ2δ∫∂ΩuNτ(x,τ)∫ΩuN(ξ,τ)dξdsx−λ2δ∫∂ΩuNτ(x,τ)∫ΩuN(ξ,0)dξdsx+λ2γ∫∂ΩuNτ(x,τ)∫ΩuNτ(ξ,τ)dξdsx−λ2γ∫∂ΩuNτ(x,τ)∫ΩuNt(ξ,0)dξdsx−λ2γ∫∂Ω∫τ0uNt(x,t)∫ΩuNtt(ξ,t)dξdtdsx+λ3ϱ∫∂ΩuNττ(x,τ)∫τ0∫ΩuN(ξ,t)dξdtdsx−λ3ϱ∫∂Ω∫τ0uNtt(x,t)∫ΩuN(ξ,t)dξdtdsx+λ3δ∫∂ΩuNττ(x,τ)∫ΩuN(ξ,τ)dξdsx−λ3δ∫∂ΩuNττ(x,τ)∫ΩuN(ξ,0)dξdsx−λ3δ∫∂Ω∫τ0uNtt(x,t)∫ΩuNt(ξ,t)dξdtds+λ3γ∫∂ΩuNττ(x,τ)∫ΩuNτ(ξ,τ)dξdsx−λ3γ∫∂ΩuNττ(x,τ)∫ΩuNt(ξ,0)dξdsx−λ3γ∫∂Ω∫τ0uNtt(x,t)∫ΩuNtt(ξ,t)dξdtds+λ1(F,uNt)L2(Qτ)+λ2(F,uNtt)L2(Qτ)+λ3(F,uNttt)L2(Qτ)+λ12∫τ0h′∘∇uN(t)dt−λ12h(t)‖∇uN(x,t)‖2L2(Ω)dt−λ1∫∂ΩuN(x,τ)∫τ0∫ΩH(t)uN(ξ,t)dξdtdsx+λ1∫τ0∫∂ΩuN(x,t)∫ΩH(t)uN(ξ,t)dξdsxdt+λ1∫∂ΩuN(x,τ)(∫τ0∫Ω[∫t0h(t−σ)(uN(ξ,t)−uN(ξ,σ))dσ]dξ)dsxdt−λ1∫τ0∫∂ΩuN(x,t)∫Ω[∫t0h(t−σ)(uN(ξ,t)−uN(ξ,σ))dσ]dξdsxdt−λ3∫∂ΩuNττ(x,τ)∫τ0∫ΩH(t)uN(ξ,t)dξdtdsx−λ32∫τ0(h″−h‴)∘∇uN(t)dt+λ3h(0)∫τ0‖∇uNt(x,t)‖2L2(Ω)dt+λ3∫τ0∫∂ΩuNtt(x,t)∫ΩH(t)uN(ξ,t)dξdsxdt+λ3∫∂ΩuNττ(x,τ)(∫τ0∫Ω[∫t0h(t−σ)(uN(ξ,t)−uN(ξ,σ))dσ]dξ)dsxdt−λ3∫τ0∫∂ΩuNtt(x,t)∫Ω[∫t0h(t−σ)(uN(ξ,t)−uN(ξ,σ))dσ]dξdsxdt−λ2∫∂ΩuNτ(x,τ)∫τ0∫ΩH(t)uN(ξ,t)dξdtdsx+λ2∫τ0∫∂ΩuNt(x,t)∫ΩH(t)uN(ξ,t)dξdsxdt+λ2∫∂ΩuNτ(x,τ)(∫τ0∫Ω[∫t0h(t−σ)(uN(ξ,t)−uN(ξ,σ))dσ]dξ)dsxdt−λ2∫τ0∫∂ΩuNt(x,t)∫Ω[∫t0h(t−σ)(uN(ξ,t)−uN(ξ,σ))dσ]dξdsxdt−λ22∫τ0h″∘∇uN(t)dt+λ22∫τ0h′(t)‖∇uN(x,t)‖2L2(Ω)dt. | (3.31) |
We can estimate all the terms in the RHS of (3.31) as follows
λ1ϱ∫∂ΩuN(x,τ)∫τ0∫ΩuN(ξ,t)dξdtdsx≤λ1ϱ2ε1(ε‖∇uN(x,τ)‖2L2(Ω)+l(ε)‖uN(x,τ)‖2L2(Ω))+λ1ϱ2ε1T|Ω||∂Ω|∫τ0‖uN(x,t)‖2L2(Ω)dt, | (3.32) |
−λ1ϱ∫∂Ω∫τ0uN(x,t)∫ΩuN(ξ,t)dξdtdsx≤λ1ϱ2ε∫τ0‖∇uN(x,t)‖2L2(Ω)dt+λ1ϱ2(l(ε)+|Ω||∂Ω|)∫τ0‖uN(x,t)‖2L2(Ω)dt, | (3.33) |
(λ1δ−λ2ϱ)∫∂Ω∫τ0uNt(x,t)∫ΩuN(ξ,t)dξdtdsx≤(λ1δ+λ2ϱ)2(ε∫τ0‖∇uNt(x,t)‖2L2(Ω)dt+l(ε)∫τ0‖uNt(x,t)‖2L2(Ω)dt)+(λ1δ+λ2ϱ)2|Ω||∂Ω|∫τ0‖uN(x,t)‖2L2(Ω)dt, | (3.34) |
−λ1δ∫∂Ω∫τ0uNt(x,t)∫ΩuN(ξ,0)dξdtdsx≤λ1δ2(ε∫τ0‖∇uNt(x,t)‖2L2(Ω)dt+l(ε)∫τ0‖uNt(x,t)‖2L2(Ω)dt)+λ1δ2|Ω||∂Ω|T‖uN(x,0)‖2L2(Ω), | (3.35) |
λ2ϱ∫∂ΩuNτ(x,τ)∫τ0∫ΩuN(ξ,t)dξdtdsx≤λ2ϱ2(εε2‖∇uNτ(x,τ)‖2L2(Ω)+l(ε)ε2‖uNτ(x,τ)‖2L2(Ω))+λ2ϱ2ε2|Ω||∂Ω|T∫τ0‖uN(x,t)‖2L2(Ω)dt, | (3.36) |
λ2δ∫∂ΩuNτ(x,τ)∫ΩuN(ξ,τ)dξdsx≤λ2δ2ε3(ε‖∇uNτ(x,τ)‖2L2(Ω)+l(ε)‖uNτ(x,τ)‖2L2(Ω))+λ2δ2ε3|Ω||∂Ω|‖uN(x,τ)‖2L2(Ω), | (3.37) |
−λ2δ∫∂ΩuNτ(x,τ)∫ΩuN(ξ,0)dξdsx≤λ2δ2ε4(ε‖∇uNτ(x,τ)‖2L2(Ω)+l(ε)‖uNτ(x,τ)‖2L2(Ω))+λ2δ2ε4|Ω||∂Ω|‖uN(x,0)‖2L2(Ω), | (3.38) |
(λ1γ−λ2δ)∫∂Ω∫τ0uNt(x,t)∫ΩuNt(ξ,t)dξdtdsx≤(λ1γ+λ2δ)2ε∫τ0‖∇uNt(x,t)‖2L2(Ω)dt+(λ1γ+λ2δ)2(l(ε)+|Ω||∂Ω|)∫τ0‖uNt(x,t)‖2L2(Ω)dt, | (3.39) |
−λ1γ∫τ0∫∂ΩuNt(x,t)(∫ΩuNt(ξ,0)dξ)dsxdt≤λ1γ2(ε∫τ0‖∇uNt(x,t)‖2L2(Ω)dt+l(ε)∫τ0‖uNt(x,t)‖2L2(Ω)dt)+λ1γ2|Ω||∂Ω|T‖uNt(x,0)‖2L2(Ω), | (3.40) |
λ2γ∫∂ΩuNτ(x,τ)∫ΩuNτ(ξ,τ)dξdsx≤λ2γ2ε5(ε‖∇uNτ(x,τ)‖2L2(Ω)+l(ε)‖uNτ(x,τ)‖2L2(Ω))+λ2γ2ε5|Ω||∂Ω|‖uNτ(x,τ)‖2L2(Ω), | (3.41) |
−λ2γ∫∂ΩuNτ(x,τ)∫ΩuNt(ξ,0)dξdsx≤λ2γ2ε6(ε‖∇uNτ(x,τ)‖2L2(Ω)+l(ε)‖uNτ(x,τ)‖2L2(Ω))+λ2γ2ε6|Ω||∂Ω|‖uNt(x,0)‖2L2(Ω), | (3.42) |
−λ2γ∫∂Ω∫τ0uNt(x,t)∫ΩuNtt(ξ,t)dξdtdsx≤λ2γ2ε∫τ0‖∇uNt(x,t)‖2L2(Ω)dt+λ2γ2l(ε)∫τ0‖uNt(x,t)‖2L2(Ω)dt+λ2γ2|Ω||∂Ω|∫τ0‖uNtt(x,t)‖2L2(Ω)dt, | (3.43) |
λ3ϱ∫∂ΩuNττ(x,τ)∫τ0∫ΩuN(ξ,t)dξdtdsx≤λ3ϱ2(εε7‖∇uNττ(x,τ)‖2L2(Ω)+l(ε)ε7‖uNττ(x,τ)‖2L2(Ω))+λ3ϱ2ε7|Ω||∂Ω|T∫τ0‖uN(x,t)‖2L2(Ω)dt, | (3.44) |
and
−λ3ϱ∫∂Ω∫τ0uNtt(x,t)∫ΩuN(ξ,t)dξdtdsx≤λ3ϱ2(ε∫τ0‖∇uNtt(x,t)‖2L2(Ω)dt+l(ε)∫τ0‖uNtt(x,t)‖2L2(Ω)dt)+λ3ϱ2|Ω||∂Ω|∫τ0‖uN(x,t)‖2L2(Ω)dt, | (3.45) |
λ3δ∫∂ΩuNττ(x,τ)∫ΩuN(ξ,τ)dξdsx≤λ3δ2ε8(ε‖∇uNττ(x,τ)‖2L2(Ω)+l(ε)‖uNττ(x,τ)‖2L2(Ω))+λ3δ2ε8|Ω||∂Ω|‖uN(x,τ)‖2L2(Ω), | (3.46) |
−λ3δ∫∂ΩuNττ(x,τ)∫ΩuN(ξ,0)dξdsx≤λ3δ2ε9(ε‖∇uNττ(x,τ)‖2L2(Ω)+l(ε)‖uNττ(x,τ)‖2L2(Ω))+λ3δ2ε9|Ω||∂Ω|‖uN(x,0)‖2L2(Ω), | (3.47) |
−λ3δ∫∂Ω∫τ0uNtt(x,t)∫ΩuNt(ξ,t)dξdtdsx≤λ3δ2(ε∫τ0‖∇uNtt(x,t)‖2L2(Ω)dt+l(ε)∫τ0‖uNtt(x,t)‖2L2(Ω)dt)+λ3δ2|Ω||∂Ω|∫τ0‖uNt(x,t)‖2L2(Ω)dt, | (3.48) |
λ3γ∫∂ΩuNττ(x,τ)∫ΩuNτ(ξ,τ)dξdsx≤λ3γ2ε10(ε‖∇uNττ(x,τ)‖2L2(Ω)+l(ε)‖uNττ(x,τ)‖2L2(Ω))+λ3γ2ε10|Ω||∂Ω|‖uNτ(x,τ)‖2L2(Ω), | (3.49) |
−λ3γ∫∂ΩuNττ(x,τ)∫ΩuNt(ξ,0)dξdsx≤λ3γ2ε11(ε‖∇uNττ(x,τ)‖2L2(Ω)+l(ε)‖uNττ(x,τ)‖2L2(Ω))+λ3γ2ε11|Ω||∂Ω|‖uNt(x,0)‖2L2(Ω), | (3.50) |
−λ3γ∫∂Ω∫τ0uNtt(x,t)∫ΩuNtt(ξ,t)dξdtdsx≤λ3γ2ε∫τ0‖∇uNtt(x,t)‖2L2(Ω)dt+λ3γ2(l(ε)+|Ω||∂Ω|)∫τ0‖uNtt(x,t)‖2L2(Ω)dt, | (3.51) |
−λ12‖uNτττ(x,τ)‖2L2(Ω)−λ12‖uNτ(x,τ)‖2L2(Ω)≤λ1(uNτττ(x,τ),uNτ(x,τ))L2(Ω), | (3.52) |
−λ22‖uNτττ(x,τ)‖2L2(Ω)−λ22‖uNττ(x,τ)‖2L2(Ω)≤λ2(uNτττ(x,τ),uNττ(x,τ))L2(Ω), | (3.53) |
−λ1α2‖uNττ(x,τ)‖2L2(Ω)−λ1α2‖uNτ(x,τ)‖2L2(Ω)≤λ1α(uNττ(x,τ),uNτ(x,τ))L2(Ω), | (3.54) |
−λ2ϱε122‖∇uN(x,τ)‖2L2(Ω)−λ2ϱ2ε12‖∇uNτ(x,τ)‖2L2(Ω)≤λ2ϱ(∇uN(x,τ),∇uNτ(x,τ))L2(Ω), | (3.55) |
−λ2ϱε132‖∇uN(x,τ)‖2L2(Ω)−λ2ϱ2ε13‖∇uNττ(x,τ)‖2L2(Ω)≤λ3ϱ(∇uN(x,τ),∇uNττ(x,τ))L2(Ω), | (3.56) |
−λ3δε142‖∇uNτ(x,τ)‖2L2(Ω)−λ3δ2ε14‖∇uNττ(x,τ)‖2L2(Ω)≤λ3δ(∇uNτ(x,τ),∇uNττ(x,τ))L2(Ω), | (3.57) |
λ1(uNttt(x,0),uNt(x,0))L2(Ω)≤λ12‖uNttt(x,0)‖2L2(Ω)+λ12‖uNt(x,0)‖2L2(Ω) | (3.58) |
λ1α(uNtt(x,0),uNt(x,0))L2(Ω)≤λ1α2‖uNtt(x,0)‖2L2(Ω)+λ1α2‖uNt(x,0)‖2L2(Ω), | (3.59) |
λ2(uNttt(x,0),uNtt(x,0))L2(Ω)≤λ22‖uNttt(x,0)‖2L2(Ω)+λ22‖uNtt(x,0)‖2L2(Ω), | (3.60) |
λ2ϱ(∇uN(x,0),∇uNt(x,0))L2(Ω)≤λ22ϱ‖∇uN(x,0)‖2L2(Ω)+λ22ϱ‖∇uNt(x,0)‖2L2(Ω), | (3.61) |
λ3ϱ(∇uN(x,0),∇uNtt(x,0))L2(Ω)≤λ32ϱ‖∇uN(x,0)‖2L2(Ω)+λ32ϱ‖∇uNtt(x,0)‖2L2(Ω), | (3.62) |
λ3δ(∇uNt(x,0),∇uNtt(x,0))L2(Ω)≤λ32δ‖∇uNt(x,0)‖2L2(Ω)+λ32δ‖∇uNtt(x,0)‖2L2(Ω), | (3.63) |
λ1∫τ0(uNttt,uNtt)L2(Ω)dt≤λ12∫τ0‖uNttt(x,t)‖2L2(Ω)dt+λ12∫τ0‖uNtt(x,t)‖2L2(Ω)dt, | (3.64) |
λ3ϱ∫τ0(∇uNt,∇uNtt)L2(Ω)dt≤λ3ϱ2∫τ0‖∇uNt(x,t)‖2L2(Ω)dt+λ3ϱ2∫τ0‖uNtt(x,t)‖2L2(Ω)dt, | (3.65) |
λ1∫∂ΩuN(x,τ)∫τ0∫ΩH(t)uN(ξ,t)dξdtdsx≤λ1h02(ε‖∇uN(x,τ)‖2L2(Ω)+l(ε)‖uN(x,τ)‖2L2(Ω))+λ1h02T|Ω||∂Ω|∫τ0‖uN(x,t)‖2L2(Ω)dt, | (3.66) |
−λ1∫∂Ω∫τ0uN(x,t)∫ΩH(t)uN(ξ,t)dξdtdsx≤λ1h02ε∫τ0‖∇uN(x,t)‖2L2(Ω)dt+λ1h02(l(ε)+|Ω||∂Ω|)∫τ0‖uN(x,t)‖2L2(Ω)dt, | (3.67) |
λ1∫∂ΩuN(x,τ)∫τ0∫Ω∫t0h(t−σ)(uN(ξ,t)−uN(ξ,σ))dσdξdtdsx≤λ1h02(ε‖∇uN(x,τ)‖2L2(Ω)+l(ε)‖uN(x,τ)‖2L2(Ω))+λ12T|Ω||∂Ω|∫τ0h∘uN(t)dt, | (3.68) |
−λ1∫∂Ω∫τ0uN(x,t)∫Ω∫t0h(t−σ)(uN(ξ,t)−uN(ξ,σ))dσdξdtdsx≤λ1h02ε∫τ0‖∇uN(x,t)‖2L2(Ω)dt+λ1h02l(ε)∫τ0‖uN(x,t)‖2L2(Ω)dt+λ12|Ω||∂Ω|∫τ0h∘uN(t)dt, | (3.69) |
λ2∫∂ΩuNτ(x,τ)∫τ0∫ΩH(t)uN(ξ,t)dξdtdsx≤λ2h02(ε‖∇uNτ(x,τ)‖2L2(Ω)+l(ε)‖uNτ(x,τ)‖2L2(Ω))+λ2h02T|Ω||∂Ω|∫τ0‖uN(x,t)‖2L2(Ω)dt, | (3.70) |
−λ2∫∂Ω∫τ0uNτ(x,t)∫ΩH(t)uN(ξ,t)dξdtdsx≤λ2h02ε∫τ0‖∇uNt(x,t)‖2L2(Ω)dt+λ2h02l(ε)∫τ0‖uNt(x,t)‖2L2(Ω)dt+λ2h02|Ω||∂Ω|∫τ0‖uN(x,t)‖2L2(Ω)dt, | (3.71) |
λ2∫∂ΩuNτ(x,τ)∫τ0∫Ω∫t0h(t−σ)(uN(ξ,t)−uN(ξ,σ))dσdξdtdsx≤λ2h02(ε‖∇uNτ(x,τ)‖2L2(Ω)+l(ε)‖uNτ(x,τ)‖2L2(Ω))+λ22T|Ω||∂Ω|∫τ0h∘uN(t)dt, | (3.72) |
−λ2∫∂Ω∫τ0uNt(x,t)∫Ω∫t0h(t−σ)(uN(ξ,t)−uN(ξ,σ))dσdξdtdsx≤λ2h02ε∫τ0‖∇uNt(x,t)‖2L2(Ω)dt+λ22h0l(ε)∫τ0‖uNt(x,t)‖2L2(Ω)dt+λ22|Ω||∂Ω|∫τ0h∘uN(t)dt, | (3.73) |
λ3∫∂ΩuNττ(x,τ)∫τ0∫ΩH(t)uN(ξ,t)dξdtdsx≤λ3h02ε18(ε‖∇uNττ(x,τ)‖2L2(Ω)+l(ε)‖uNττ(x,τ)‖2L2(Ω))+λ3h02ε18T|Ω||∂Ω|∫τ0‖uN(x,t)‖2L2(Ω)dt, | (3.74) |
−λ3∫∂Ω∫τ0uNττ(x,t)∫ΩH(t)uN(ξ,t)dξdtdsx≤λ3h02ε∫τ0‖∇uNtt(x,t)‖2L2(Ω)dt+λ3h02l(ε)∫τ0‖uNtt(x,t)‖2L2(Ω)dt+λ3h02|Ω||∂Ω|∫τ0‖uN(x,t)‖2L2(Ω)dt, | (3.75) |
λ3∫∂ΩuNττ(x,τ)∫τ0∫Ω∫t0h(t−σ)(uN(ξ,t)−uN(ξ,σ))dσdξdtdsx≤λ3h02ε15(ε‖∇uNττ(x,τ)‖2L2(Ω)+l(ε)‖uNττ(x,τ)‖2L2(Ω))+λ32ε15T|Ω||∂Ω|∫τ0h∘uN(t)dt, | (3.76) |
−λ3∫∂Ω∫τ0uNtt(x,t)∫Ω∫t0h(t−σ)(uN(ξ,t)−uN(ξ,σ))dσdξdtdsx≤λ3h02ε∫τ0‖∇uNtt(x,t)‖2L2(Ω)dt+λ32h0l(ε)∫τ0‖uNtt(x,t)‖2L2(Ω)dt+λ32|Ω||∂Ω|∫τ0h∘uN(t)dt, | (3.77) |
−λ3H(τ)(∇uNττ,∇uN)2L2(Ω)≥−λ3h02ε16‖∇uNττ(x,τ)‖2L2(Ω)−λ3h0ε162‖∇uN(x,τ)‖2L2(Ω), | (3.78) |
−λ3h(τ)(∇uNτ,∇uN)2L2(Ω)≥−λ3h(0)2‖∇uNτ(x,τ)‖2L2(Ω)−λ3h(0)2‖∇uN(x,τ)‖2L2(Ω), | (3.79) |
λ3∫Ω∇uNττ[∫τ0h(τ−σ)(∇uN(τ)−∇uN(σ)dσ]dx≥−λ3h02ε17‖∇uNττ(x,τ)‖2L2(Ω)−λ3ε172h∘∇uN(τ), | (3.80) |
−λ3∫Ω∇uNτ[∫τ0h′(τ−σ)(∇uN(τ)−∇uN(σ)dσ]dx≥−λ3h02‖∇uNτ(x,τ)‖2L2(Ω)+λ32h′∘∇uN(τ), | (3.81) |
λ2∫Ω∇uNτ[∫τ0h(τ−σ)∇uN(σ)dσ]dx≥−λ22h∘∇uN(τ)−λ2(h0+1)2‖∇uNτ(x,τ)‖2L2(Ω)−λ2h02‖∇uN(x,τ)‖2L2(Ω), | (3.82) |
λ1(F,uNt)L2(Qτ)≤λ12∫τ0‖F(x,t)‖2L2(Ω)dt+λ12∫τ0‖ut(x,t)‖2L2(Ω)dtλ2(F,uNtt)L2(Qτ)≤λ22∫τ0‖F(x,t)‖2L2(Ω)dt+λ22∫τ0‖utt(x,t)‖2L2(Ω)dtλ3(F,uNttt)L2(Qτ)≤λ32∫τ0‖F(x,t)‖2L2(Ω)dt+λ32∫τ0‖uttt(x,t)‖2L2(Ω)dt. | (3.83) |
Substituting (3.32)-(3.83) into (3.31) and make use of the following inequality
m1‖uN(x,τ)‖2L2(Ω)≤m1‖uN(x,t)‖2L2(Qτ)+m1‖uNt(x,t)‖2L2(Qτ)+m1‖uN(x,0)‖2L2(Ω)m2‖uNτ(x,τ)‖2L2(Ω)≤m2‖uNt(x,t)‖2L2(Qτ)+m2‖uNtt(x,t)‖2L2(Qτ)+m2‖uNt(x,0)‖2L2(Ω)m3‖uNττ(x,τ)‖2L2(Ω)≤m3‖uNtt(x,t)‖2L2(Qτ)+m3‖uNttt(x,t)‖2L2(Qτ)+m3‖uNtt(x,0)‖2L2(Ω)m4‖∇uN(x,τ)‖2L2(Ω)≤m4‖∇uN(x,t)‖2L2(Qτ)+m4‖∇uNt(x,t)‖2L2(Qτ)+m4‖∇uN(x,0)‖2L2(Ω)m5‖∇uNτ(x,τ)‖2L2(Ω)≤m5‖∇uNt(x,t)‖2L2(Qτ)+m5‖∇uNtt(x,t)‖2L2(Qτ)+m5‖∇uNt(x,0)‖2L2(Ω)m6h∘uN(τ)≤m6‖uNt(x,t)‖2L2(Qτ)+m6∫τ0h∘uN(t)dtm7h∘∇uN(τ)≤m7‖∇uNt(x,t)‖2L2(Qτ)+m7∫τ0h∘∇uN(t)dt−m8h′∘∇uN(τ)≤m8‖∇uNt(x,t)‖2L2(Qτ)−m8∫τ0h′∘∇uN(t)dt, |
where
m1=λ1ϱε1l(ε)+λ2δ2ε3|Ω||∂Ω|+λ3δ2ε8|Ω||∂Ω|+λ1h0l(ε),m2=λ2ϱ2l(ε)ε2+λ2δ2l(ε)ε3+λ2δ2l(ε)ε4+λ2γ2(l(ε)ε5+ε5|Ω||∂Ω|)+λ2γ2l(ε)ε6+λ3γ2ε10|Ω||∂Ω|+λ1(1+α)2+λ2h0l(ε),m3=λ3ϱ2l(ε)ε7+λ3δ2l(ε)ε8+λ3δ2l(ε)ε9+λ3γ2l(ε)ε10+λ3γ2l(ε)ε11+λ22+λ1α2+λ3h02ε18l(ε)+λ32ε15l(ε),m4=λ1h02ε1ε+λ2ϱ2ε12+λ2ϱ2ε13+λ1h0ε+λ32+λ3h02ε16+λ3h(0)2+λ2h02+λ1ϱ2ε1ε,m5=λ2ϱ2εε2+λ2δ2εε3+λ2δ2εε4+λ2γ2εε5+λ2γ2εε6+λ2ϱ2ε12+λ3δε142+λ2h0ε+λ3(h0+h(0))2+λ2(h0+1)2,m7=λ2ε172+λ22,m8=λ32,m6=1, |
we have
λ1ϱ2ε1l(ε)‖uN(x,τ)‖2L2(Ω)+λ1β2‖uNτ(x,τ)‖2L2(Ω)+(λ2α2+λ3β2)‖uNττ(x,τ)‖2L2(Ω)+{λ32−λ12−λ22}‖uNτττ(x,τ)‖2L2(Ω)+λ1ϱ2‖∇uN(x,τ)‖2L2(Ω)+{λ1γ2+λ2δ2}‖∇uNτ(x,τ)‖2L2(Ω)+h∘uN(τ)+λ12h∘∇uN(τ)−λ22h′∘∇uN(τ)+{λ3γ2−λ3ϱ2εε7−λ3δ2εε8−λ3δ2εε9−λ3γ2εε10−λ3γ2εε11−λ2ϱ2ε13−λ3δ2ε14−λ3h02εε16−λ3h02εε17−λ3h02ε18−λ3h02ε15}‖∇uNττ(x,τ)‖2L2(Ω)≤γ7‖uN(x,0)‖2L2(Ω)+{λ22+λ1α2+(λ2α2−λ3β2)+m3}‖uNtt(x,0)‖2L2(Ω)+{λ12+λ22+λ32}‖uNttt(x,0)‖2L2(Ω)+{λ1ϱ2+λ2ϱ2+λ3ϱ2+m4}‖∇uN(x,0)‖2L2(Ω)+γ8‖uNt(x,0)‖2L2(Ω)+{λ2ϱ2+λ3δ2+λ1γ2+λ2δ2+m5}‖∇uNt(x,0)‖2L2(Ω)+{λ3ϱ2+3λ3δ2−λ3γ2−λ2γ}‖∇uNtt(x,0)‖2L2(Ω)+(γ1+m1)∫τ0‖uN(x,t)‖2L2(Ω)dt+(γ2+m1+m2)∫τ0‖uNt(x,t)‖2L2(Ω)dt+{λ12+λ2−λ3α+m3}∫τ0‖uNttt(x,t)‖2L2(Ω)dt−m8∫τ0h′∘∇uN(t)dt+{γ6+m4}∫τ0‖∇uN(x,t)‖2L2(Ω)dt+(γ3+m2+m3)∫τ0‖uNtt(x,t)‖2L2(Ω)dt+(γ4+m4+m5+m7+m8)∫τ0‖∇uNt(x,t)‖2L2(Ω)dt+(γ5+m5)∫τ0‖∇uNtt(x,t)‖2L2(Ω)dt+∫τ0h∘uN(t)dt+m7∫τ0h∘∇uN(t)dt+λ1+λ2+λ32∫τ0‖F(x,t)‖2L2(Ω)dt, | (3.84) |
where
γ1=λ1ϱ2ε1T|Ω||∂Ω|+λ1ϱ2(l(ε)+|Ω||∂Ω|)+(λ1δ+λ2ϱ2)|Ω||∂Ω|+λ2ϱ2ε2T|Ω||∂Ω|+λ3ϱ2ε7T|Ω||∂Ω|+λ3ϱ2|Ω||∂Ω|++λ1h02l(ε)+[λ3h02ε18+(λ3+λ2+λ1)h02+(λ1+λ2)h0T2]|Ω||∂Ω|γ2=(λ1δ+λ2ϱ2)l(ε)+λ1δ2l(ε)+(λ1γ+λ2δ2)(l(ε)+|Ω||∂Ω|)+λ1γ2l(ε)+λ2γ2l(ε)+λ3δ2|Ω||∂Ω|+λ2h0l(ε), |
γ3=λ2γ2|Ω||∂Ω|+λ3ϱ2l(ε)+λ3δ2l(ε)+λ3γ2(l(ε)+|Ω||∂Ω|)+λ12+(λ1α−λ2β)+λ3ϱ2+λ3h0l(ε),γ4=(λ1δ+λ2ϱ2)ε+λ1δ2ε+(λ1γ+λ2δ2)ε+λ1γ2ε+λ2γ2ε+λ3ϱ2+(λ2ϱ−λ1δ)+h(0)λ3+λ3h0ε,γ5=λ3δ2ε+λ3γ2ε+λ3ϱ2+(λ3δ−λ2γ)+λ3h0ε,γ6=λ1ϱ2ε+λ1h0ε,γ7=λ1δ2|Ω||∂Ω|T+λ2δ2ε4|Ω||∂Ω|+λ3δ2ε9|Ω||∂Ω|+m1,γ8=λ1γ2|Ω||∂Ω|T+λ2γ2ε6|Ω||∂Ω|+λ3γ2ε11|Ω||∂Ω|+λ12+λ1α2+λ1β2+m2. |
Choosing ε7, ε8, ε9, ε10, ε11, ε13, ε14,ε15, ε16, ε17 and ε18 sufficiently large
β0:=λ3γ2−λ3ϱ2εε7−λ3δ2εε8−λ3δ2εε9−λ3γ2εε10−λ3γ2εε11−λ3δ2ε14−λ2ϱ2ε13−λ3h02εε16−λ32εε17−λ3h02ε18−λ32ε15>0, | (3.85) |
the relation (3.84) reduces to
{‖uN(x,τ)‖2L2(Ω)+‖∇uN(x,τ)‖2L2(Ω)+‖uNτ(x,τ)‖2L2(Ω)+‖∇uNτ(x,τ)‖2L2(Ω)+‖uNττ(x,τ)‖2L2(Ω)+‖∇uNττ(x,τ)‖2L2(Ω)+‖uNτττ(x,τ)‖2L2(Ω)+h∘∇uN(τ)+h∘uN(τ)−h′∘∇uN(τ)} | (3.86) |
≤D∫τ0{‖uN(x,t)‖2L2(Ω)+‖∇uN(x,t)‖2L2(Ω)+‖uNt(x,t)‖2L2(Ω)+‖∇uNt(x,t)‖2L2(Ω)+‖uNtt(x,t)‖2L2(Ω)+‖∇uNtt(x,t)‖2L2(Ω)+‖uNttt(x,t)‖2L2(Ω)+h∘∇uN(t)+h∘uN(t)−h′∘∇uN(t)+‖F‖2L2(Ω)}dt+D{‖uN(x,0)‖2W12(Ω)+‖uNt(x,0)‖2W12(Ω)+‖uNtt(x,0)‖2W12(Ω)+‖uNttt(x,0)‖2L2(Ω)+h∘∇uN(0)+h∘uN(0)−h′∘∇uN(0)}, | (3.87) |
where
D:=max{λ1δ2|Ω||∂Ω|T+λ2δ2ε4|Ω||∂Ω|+λ3δ2ε9|Ω||∂Ω|+m1,λ1γ2|Ω||∂Ω|T+λ2γ2ε6|Ω||∂Ω|+λ3γ2ε11|Ω||∂Ω|+λ12+λ1α2+λ1β2+m2,λ22+λ1α2+λ2α2−λ3β2+m3,λ1+λ2+λ32,λ1ϱ2+λ2ϱ2+λ3ϱ2+m4,λ2ϱ2+λ3δ2+λ1γ2+λ2δ2+m5,γ1+m1,γ2+m1+m2,γ3+m2+m3,λ12+λ2−λ3α+m3,λ3ϱ2+λ3δ2−λ3γ2,γ6+m4,γ4+m4+m5,γ5+m5,m7,m8,1}min{λ1ϱ2ε1l(ε),λ1β2,λ2α2+λ3β2,λ32−λ12−λ22,λ1ϱ2,λ1γ2+λ2δ2,1,λ12,λ22,β0}. | (3.88) |
Applying the Gronwall inequality to (3.87) and then integrate from 0 to τ appears that
‖uN(x,t)‖2W12(Qτ)+‖uNt(x,t)‖2W12(Qτ)+‖uNtt(x,t)‖2W12(Qτ)+‖uN(x,t)‖h≤DeDT{‖u0(x)‖2W12(Ω)+‖u1(x)‖2W12(Ω)+‖u2(x)‖2L2(Ω)+‖u3(x)‖2L2(Ω)+‖F‖2L2(Ω)}.. | (3.89) |
We deduce from (3.89) that
‖uN(x,t)‖2W12(Qτ)+‖uNt(x,t)‖2W12(Qτ)+‖uNtt(x,t)‖2W12(Qτ)+‖uN(x,t)‖h≤A, | (3.90) |
where
‖uN(x,t)‖h:=∫τ0(h∘∇uN(t)+h∘uN(t)−h′∘∇uN(t))dt. |
Therefore the sequence {uN}N≥1 is bounded in V(QT), and we can extract from it a subsequence for which we use the same notation which converges weakly in V(QT) to a limit function u(x,t) we have to show that u(x,t) is a generalized solution of (1.1). Since uN(x,t)→u(x,t) in L2(QT) and uN(x,0)→ζ(x) in L2(Ω), then u(x,0)=ζ(x).
Now to prove that (2.1) holds, we multiply each of the relations (3.5) by a function pl(t)∈W12(0,T), pl(t)=0, then add up the obtained equalities ranging from l=1 to l=N, and integrate over t on (0,T). If we let ηN=N∑k=1pk(t)Zk(x), then we have
−(uNttt,ηNt)L2(QT)−α(uNtt,ηNt)L2(QT)−β(uNt,ηNt)L2(QT)+ϱ(∇uN,∇ηN)L2(QT)+δ(∇uNt,∇ηN)L2(QT)−γ(∇uNt,∇ηNt)L2(QT)−(∇wN,∇ηN)L2(QT)=ϱ∫∂Ω∫T0ηN(x,t)(∫t0∫ΩuN(ξ,τ)dξdτ)dtdsx+δ∫∂Ω∫T0ηN(x,t)∫ΩuN(ξ,t)dξdtdsx−δ∫∂Ω∫T0ηN(x,t)∫ΩuN(ξ,0)dξdtdsx−γ∫T0∫∂ΩηNt(∫ΩuN(ξ,t)dξ)dsxdt+γ∫T0∫∂ΩηNt(∫ΩuN(ξ,0)dξ)dsxdt−γ(ΔuNt(x,0),ηN(0))L2(Ω)−∫∂Ω∫T0ηN(x,t)(∫t0∫ΩwN(ξ,τ)dξdτ)dtdsx+(F,ηNt)L2(QT)+(uNttt(x,0),ηN(0))L2(Ω)+α(uNtt(x,0),ηN(0))L2(Ω)+β(uNtt(x,0),ηN(0))L2(Ω), | (3.91) |
for all ηN of the form N∑k=1pl(t)Zk(x).
Since
∫t0∫Ω((uN(ξ,τ)−u(ξ,τ))dξdτ≤√T|Ω|‖uN−u‖L2(QT), |
∫T0ηN(x,t)∫Ω(uNt(ξ,t)−ut(ξ,t))dξdt≤√|Ω|(∫T0(ηN(x,t))2dt)1/2‖uNt−ut‖L2(QT), |
∫T0ηN(x,t)∫Ω(u(N(ξ,0)−u(ξ,0))dξdt≤√|Ω|(∫T0(ηN(x,t))2dt)1/2‖uN(x,0)−u(x,0)‖L2(QT), |
and
‖uN−u‖L2(QT)→0, asN→∞, |
therefore we have
ϱ∫∂Ω∫T0ηN(x,t)∫t0∫ΩuN(ξ,τ)dξdτdtdsx→ϱ∫∂Ω∫T0η(x,t)∫t0∫Ωu(ξ,τ)dξdτdtdsx, |
δ∫∂Ω∫T0ηN(x,t)∫ΩuN(ξ,t)dξdtdsx→δ∫∂Ω∫T0η(x,t)∫Ωu(ξ,t)dξdtdsx, |
−δ∫∂Ω∫T0ηN(x,t)∫ΩuN(ξ,0)dξdtds→−δ∫∂Ω∫T0η(x,t)∫Ωu(ξ,0)dξdtds, |
−γ∫T0∫∂ΩηNt(∫ΩuN(ξ,t)dξ)dsxdt→−γ∫T0∫∂Ωηt(∫Ωu(ξ,t)dξ)dsxdt, |
γ∫T0∫∂ΩηNt(∫ΩuN(ξ,0)dξ)dsxdt→γ∫T0∫∂Ωηt(∫Ωu(ξ,0)dξ)dsxdt. |
∫∂Ω∫T0ηN(x,t)∫t0∫ΩwN(ξ,τ)dξdτdtdsx→ϱ∫∂Ω∫T0η(x,t)∫t0∫Ωw(ξ,τ)dξdτdtdsx. |
Thus, the limit function u satisfies (2.1) for every ηN=N∑k=1pl(t)Zk(x). We denote by QN the totality of all functions of the form ηN=N∑k=1pl(t)Zk(x), with pl(t)∈W12(0,T), pl(t)=0.
But ∪Nl=1QN is dense in W(QT), then relation (2.1) holds for all u ∈W(QT). Thus we have shown that the limit function u(x,t) is a generalized solution of problem (1.1) in V(QT).
Theorem 2. The problem (1.1) cannot have more than one generalized solution in V(QT).
Proof. Suppose that there exist two different generalized solutions u1∈V(QT) and u2∈V(QT) for the problem (1.1). Then, U=u1−u2 solves
{Utttt+αUttt+βUtt−ϱΔU−δΔUt−γΔUtt+∫t0h(t−σ)Δu(σ)dσ=0,U(x,0)=Ut(x,0)=Utt(x,0)=Uttt(x,0)=0∂u∂η=∫t0∫Ωu(ξ,τ)dξdτ, x∈∂Ω. | (4.1) |
and (2.1) gives
−(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)=ϱ∫T0∫∂Ωv(∫t0∫Ωu(ξ,τ)dξdτ)dsxdt+δ∫T0∫∂Ωv∫ΩU(ξ,t)dξdsxdt−γ∫T0∫∂Ωvt(∫ΩUτ(ξ,t)dξdt)dsxdt−∫T0∫∂Ωv(∫t0∫ΩW(ξ,τ)dξdτ)dsxdt, | (4.2) |
where
W(x,t):=∫t0h(t−σ)ΔU(σ)dσ. |
Consider the function
v(x,t)={∫τtU(x,s)ds,0≤t≤τ,0,τ≤t≤T. | (4.3) |
It is obvious that v∈W(QT) and vt(x,t)=−U(x,t) for all t∈[0,τ]. Integration by parts in the left hand side of (4.2) gives
−(Uttt,vt)L2(QT)=(Uττ(x,τ),U(x,τ))L2(Ω)−12‖Uτ(x,τ)‖2L2(Ω), | (4.4) |
−α(Utt,vt)L2(QT)=α(Uτ(x,τ),U(x,τ))L2(Ω)−α∫τ0‖Ut(x,t)‖2L2(Ω)dt, | (4.5) |
−β(Ut,vt)L2(QT)=β2‖U(x,τ)‖2L2(Ω), | (4.6) |
ϱ(∇U,∇v)L2(QT)=ϱ2‖∇v(x,0)‖2L2(Ω), | (4.7) |
δ(∇Ut,∇v)L2(QT)=δ∫τ0‖∇vt(x,t)‖2L2(Ω)dt, | (4.8) |
−γ(∇Ut,∇vt)L2(QT)=γ2‖∇U(x,τ)‖2L2(Ω), | (4.9) |
−(∇W,∇v)L2(QT)≤h0∫τ0‖∇v(x,t)‖2L2(Ω)dt+h02∫τ0‖∇U(x,t)‖2L2(Ω)dt+12∫τ0h∘∇U(t)dt. | (4.10) |
Plugging (4.4)-(4.10) into (4.2) we get
(Uττ(x,τ),U(x,τ))L2(Ω)+α(Uτ(x,τ),U(x,τ))L2(Ω)+β2‖U(x,τ)‖2L2(Ω)+ϱ2‖∇v(x,0)‖2L2(Ω)+γ2‖∇U(x,τ)‖2L2(Ω)−12‖Uτ(x,τ)‖2L2(Ω)≤α∫τ0‖Ut(x,t)‖2L2(Ω)dt−δ∫τ0‖∇vt(x,t)‖2L2(Ω)dt+h0∫τ0‖∇v(x,t)‖2L2(Ω)dt+h02∫τ0‖∇U(x,t)‖2L2(Ω)dt+12∫τ0h∘∇U(t)dt+ϱ∫T0∫∂Ωv(∫t0∫ΩU(ξ,τ)dξdτ)dsxdt+δ∫T0∫∂Ωv∫ΩU(ξ,t)dξdsxdt−γ∫T0∫∂Ωvt(∫ΩU(ξ,t)dξ)dsdt−∫T0∫∂Ωv(∫t0∫ΩW(ξ,τ)dξdτ)dsxdt. | (4.11) |
Now since
v2(x,t)=(∫τtU(x,s)ds)2≤τ∫τ0U2(x,s)ds, |
then
‖v‖2L2(Qτ)≤τ2‖U‖2L2(Qτ)≤T2‖U‖2L2(Qτ). | (4.12) |
Using the trace inequality, the RHS of (4.11) can be estimated as follows
ϱ∫T0∫∂Ωv(∫t0∫ΩU(ξ,τ)dξdτ)dsxdt≤ϱ2T2{l(ε)+|Ω||∂Ω|}∫τ0‖U(x,t)‖2L2(Ω)dt+ϱ2ε∫τ0‖∇v(x,t)‖2L2(Ω)dt, | (4.13) |
and
δ∫T0∫∂Ωv∫ΩU(ξ,t)dξdsxdt≤δ2{T2l(ε)+|Ω||∂Ω|}∫τ0‖U(x,t)‖2L2(Ω)dt+δ2ε∫τ0‖∇v(x,t)‖2L2(Ω)dt, | (4.14) |
and
−γ∫T0∫∂Ωvt(∫ΩU(ξ,t)dξ)dsdt=γ∫τ0∫∂Ωv(∫ΩUt(ξ,t)dξ)dsdt≤γ|Ω||∂Ω|2‖Ut‖2L2(Qτ)+γT22ε‖∇v‖2L2(Qτ)+γ2l(ε)T2‖U‖2L2(Qτ). | (4.15) |
−∫T0∫∂Ωv(∫t0∫ΩW(ξ,τ)dξdτ)dsxdt=−∫T0∫∂Ωv(∫t0∫ΩH(τ)U(ξ,τ)dξdτ)dsxdt+∫T0∫∂Ωv(∫t0∫Ω[∫τ0h(τ−σ)(U(ξ,τ)−U(ξ,σ))dσ]dξdτ)dsxdt≤h02T2{l(ε)+|Ω||∂Ω|}∫τ0‖U(x,t)‖2L2(Ω)dt+h02ε∫τ0‖∇v(x,t)‖2L2(Ω)dt+12l(ε)∫τ0‖U(x,t)‖2L2(Ω)dt+12|Ω||∂Ω|∫τ0h∘U(t)dt+12ε∫τ0‖∇v(x,t)‖2L2(Ω)dt. | (4.16) |
Combining the relations (4.13)-(4.16) and (4.11) we get
(Uττ(x,τ),U(x,τ))L2(Ω)+α(Uτ(x,τ),U(x,τ))L2(Ω)+β2‖U(x,τ)‖2L2(Ω)+ϱ2‖∇v(x,0)‖2L2(Ω)+γ2‖∇U(x,τ)‖2L2(Ω)−12‖Uτ(x,τ)‖2L2(Ω)≤{ϱ2T2(l(ε)+|Ω||∂Ω|)+δ2(T2l(ε)+|Ω||∂Ω|)+γ2l(ε)T2+h02T2(l(ε)+|Ω||∂Ω|)+12l(ε)}∫τ0‖U(x,t)‖2L2(Ω)dt+(α+γ|Ω||∂Ω|2)∫τ0‖Ut(x,t)‖2L2(Ω)dt+12∫τ0‖Utt(x,t)‖2L2(Ω)dt+{(ϱ+δ+γ+h02)ε+h0}∫τ0‖∇v(x,t)‖2L2(Ω)dt+h0∫τ0‖∇v(x,t)‖2L2(Ω)dt+h02∫τ0‖∇U(x,t)‖2L2(Ω)dt+12∫τ0h∘∇U(t)dt+12|Ω||∂Ω|∫τ0h∘U(t)dt. | (4.17) |
Next, multiplying the differential equation in (4.1) by Uttt and integrating over Qτ=Ω×(0,τ), we obtain
(Utttt,Uttt)L2(Qτ)+α(Uttt,Uttt)L2(Qτ)+β(Utt,Uttt)L2(Qτ)−ϱ(ΔU,Uttt)L2(Qτ)−δ(ΔUt,Uttt)L2(Qτ)−γ(ΔUt,Uttt)L2(Qτ)+(ΔW,Uttt)L2(Qτ)=0. | (4.18) |
An integration by parts in (4.18) yields
(Utttt,Uttt)L2(Qτ)=12‖Uτττ(x,τ)‖2L2(Ω), | (4.19) |
α(Uttt,Uttt)L2(Qτ)=α∫τ0‖Uttt(x,t)‖2L2(Ω)dt, | (4.20) |
β(Utt,Uttt)L2(Qτ)=β2‖Uττ(x,τ)‖2L2(Ω), | (4.21) |
−ϱ(ΔU,Uttt)L2(Qτ)=ϱ(∇U(x,τ),∇Uττ(x,τ))L2(Ω)−ϱ2‖∇Uτ(x,τ)‖2L2(Ω)−ϱ∫∂ΩUττ(x,τ)(∫τ0∫ΩU(ξ,η)dξdη)dsx+ϱ∫∂Ω∫τ0Utt(x,t)∫ΩU(ξ,t)dξdtdsx, | (4.22) |
−δ(ΔUt,Uttt)L2(Qτ)=δ(∇Uτ(x,τ),∇Uττ(x,τ))L2(Ω)−δ∫τ0‖∇Utt(x,)‖2L2(Ω)dt−δ∫∂ΩUττ(x,τ)∫ΩU(ξ,τ)dξdsx+δ∫τ0∫∂ΩUtt(x,t)∫ΩUt(ξ,t)dξdsxdt, | (4.23) |
−γ(ΔUtt,Uttt)L2(Qτ)=γ2‖∇Uττ(x,τ)‖2L2(Ω)−γ∫∂ΩUττ(x,τ)∫ΩUτ(ξ,τ)dξdsx+γ∫τ0∫∂ΩUtt(x,t)∫ΩUtt(ξ,t)dξdsxdt. | (4.24) |
(ΔW,Uttt)L2(Qτ)=−H(τ)(∇U(x,τ),∇Uττ(x,τ))L2(Ω)+∫Ω∇Uττ∫τ0h(τ−σ)(∇U(τ)−∇U(σ))dσdx−∫τ0(∇Utt,∫t0h′(t−σ)(∇U(t)−∇U(σ))dσ)L2(Ω)dt+∫τ0h(t)(∇Utt,∇U(t))L2(Ω)dt+∫∂ΩUττ(x,τ)(∫τ0∫ΩW(ξ,η)dξdη)dsx−∫∂Ω∫τ0Utt(x,t)∫ΩW(ξ,t)dξdtdsx, | (4.25) |
Substitution (4.19)-(4.25) into (4.18) we get the equality
12‖Uτττ(x,τ)‖2L2(Ω)+β2‖Uττ(x,τ)‖2L2(Ω)+ϱ(∇U(x,τ),∇Uττ(x,τ))L2(Ω)+δ(∇Uτ(x,τ),∇Uττ(x,τ))L2(Ω)+γ2‖∇Uττ(x,τ)‖2L2(Ω)−ϱ2‖∇Uτ(x,τ)‖2L2(Ω)−H(τ)(∇U(x,τ),∇Uττ(x,τ))L2(Ω)+∫Ω∇Uττ∫τ0h(τ−σ)(∇U(τ)−∇U(σ))dσdx=−α∫τ0‖Uttt(x,t)‖2L2(Ω)dt+δ∫τ0‖∇Utt(x,)‖2L2(Ω)dt+ϱ∫∂ΩUττ(x,τ)(∫τ0∫ΩU(ξ,η)dξdη)dsx−ϱ∫∂Ω∫τ0Utt(x,t)∫ΩU(ξ,t)dξdtdsx+δ∫∂ΩUττ(x,τ)∫ΩU(ξ,τ)dξdsx−δ∫τ0∫∂ΩUtt(x,t)∫ΩUt(ξ,t)dξdsxdt+γ∫∂ΩUττ(x,τ)∫ΩUτ(ξ,τ)dξdsx−γ∫τ0∫∂ΩUtt(x,t)∫ΩUtt(ξ,t)dξdsxdt−∫τ0(∇Utt,∫t0h′(t−σ)(∇U(t)−∇U(σ))dσ)L2(Ω)dt+∫τ0h(t)(∇Utt,∇U(t))L2(Ω)dt+∫∂ΩUττ(x,τ)(∫τ0∫ΩW(ξ,η)dξdη)dsx−∫∂Ω∫τ0Utt(x,t)∫ΩW(ξ,t)dξdtdsx. | (4.26) |
The right hand side of (4.26) can be bounded as follows
ϱ∫∂ΩUττ(x,τ)(∫τ0∫ΩU(ξ,η)dξdη)dsx≤ϱ2ε′1(ε‖∇Uττ(x,τ)‖2L2(Ω)+l(ε)‖Uττ(x,τ)‖2L2(Ω))+ϱ2ε′1T|∂Ω||Ω|∫τ0‖U(x,t)‖2L2(Ω)dt, | (4.27) |
−ϱ∫∂Ω∫τ0Utt(x,t)∫ΩU(ξ,t)dξdtdsx≤ϱ2∫τ0{ε‖∇Utt(x,t)‖2L2(Ω)+l(ε)‖Utt(x,t)‖2L2(Ω)}dt+ϱ2|Ω||∂Ω|∫τ0‖U(x,t)‖2L2(Ω)dt, | (4.28) |
δ∫∂ΩUττ(x,τ)∫ΩU(ξ,τ)dξdsx≤δ2ε′2(ε‖∇Uττ(x,τ)‖2L2(Ω)+l(ε)‖Uττ(x,τ)‖2L2(Ω))+δ2ε′2T|Ω||∂Ω|‖U(x,τ)‖2L2(Ω), | (4.29) |
−δ∫τ0∫∂ΩUtt(x,t)∫ΩUt(ξ,t)dξdsxdt≤δ2ε∫τ0‖∇Utt(x,t)‖2L2(Ω)dt+δ2l(ε)∫τ0‖Utt(x,t)‖2L2(Ω)dt+δ2T|Ω||∂Ω|∫τ0‖Ut(x,t)‖2L2(Ω)dt, | (4.30) |
γ∫∂ΩUττ(x,τ)∫ΩUτ(ξ,τ)dξdsx≤γ2ε′3(ε‖∇Uττ(x,τ)‖2L2(Ω)+l(ε)‖Uττ(x,τ)‖2L2(Ω))+γ2ε′3T|Ω||∂Ω|‖Uτ(x,τ)‖2L2(Ω), | (4.31) |
−γ∫τ0∫∂ΩUtt(x,t)∫ΩUtt(ξ,t)dξdsxdt≤γ2l(ε)∫τ0‖Utt(x,t)‖2L2(Ω)dt+γ2ε∫τ0‖∇Utt(x,t)‖2L2(Ω)dt+γ2T|Ω||∂Ω|∫τ0‖Utt(x,t)‖2L2(Ω)dt, | (4.32) |
∫∂ΩUττ(x,τ)(∫τ0∫ΩW(ξ,η)dξdη)dsx≤(h02ε′6+12ε′7)(ε‖∇Uττ(x,τ)‖2L2(Ω)+l(ε)‖Uττ(x,τ)‖2L2(Ω))+h02ε′6T|∂Ω||Ω|∫τ0‖U(x,t)‖2L2(Ω)dt+12ε′7|∂Ω||Ω|∫τ0h∘U(t)dt, | (4.33) |
−∫∂Ω∫τ0Utt(x,t)∫ΩW(ξ,t)dξdtdsx≤h0+12∫τ0{ε‖∇Utt(x,t)‖2L2(Ω)+l(ε)‖Utt(x,t)‖2L2(Ω)}dt+h02|Ω||∂Ω|∫τ0‖U(x,t)‖2L2(Ω)dt+12|∂Ω||Ω|∫τ0h∘U(t)dt. | (4.34) |
∫Ω∇Uττ∫τ0h(τ−σ)(∇U(τ)−∇U(σ))dσdx≥−12ε′8h0‖∇Uττ(x,τ)‖2L2(Ω)−12ε′8h∘∇U(τ), | (4.35) |
−H(τ)(∇U(x,τ),∇Uττ(x,τ))L2(Ω)≥−12ε′9h0‖∇Uττ(x,τ)‖2L2(Ω)−12ε′9h0‖∇U(x,τ)‖2L2(Ω), | (4.36) |
∫τ0h(t)∇Utt(x,t)∇U(x,t)dt≤h(0)2∫τ0‖∇Utt(x,τ)‖2L2(Ω)dt+h(0)2∫τ0‖∇U(x,τ)‖2L2(Ω)dt, | (4.37) |
∫τ0∇Utt∫t0h′(t−σ)(∇U(t)−∇U(σ))dσdx≤−h(t)−h(0)2∫τ0‖∇Utt(x,τ)‖2L2(Ω)dt−12∫τ0h′∘∇U(t)dt. | (4.38) |
So, combining inequalities (4.27)-(4.38) and equality (4.26) we obtain
12‖Uτττ(x,τ)‖2L2(Ω)+{β2−ϱ2ε′1l(ε)−δ2ε′2l(ε)−γ2ε′3l(ε)−(h02ε′6+12ε′7)l(ε)}‖Uττ(x,τ)‖2L2(Ω)−γ2ε′3T|Ω||∂Ω|‖Uτ(x,τ)‖2L2(Ω)−δ2ε′2T|Ω||∂Ω|‖U(x,τ)‖2L2(Ω)+{γ2−ϱ2ε′1ε−δ2ε′2ε−γ2ε′3ε+(h02ε′6+12ε′7)ε−(12ε′8+12ε′9)h0}‖∇Uττ(x,τ)‖2L2(Ω)−ε′82h∘∇U(τ)−ε′92h0‖∇U(x,τ)‖2L2(Ω)−ϱ2‖∇Uτ(x,τ)‖2L2(Ω)+ϱ(∇U(x,τ),∇Uττ(x,τ))L2(Ω)+δ(∇Uτ(x,τ),∇Uττ(x,τ))L2(Ω)≤−α∫τ0‖Uttt(x,t)‖2L2(Ω)dt+{ϱ2l(ε)+δ2l(ε)+γ2l(ε)+γ2T|Ω||∂Ω|+(h0+10)l(ε)}∫τ0‖Utt(x,t)‖2L2(Ω)dt+{ϱ2ε′1T|∂Ω||Ω|+(ϱ2+h02(1+Tε′6))|Ω||∂Ω|}∫τ0‖U(x,t)‖2L2(Ω)dt+δ2T|Ω||∂Ω|∫τ0‖Ut(x,t)‖2L2(Ω)dt+{δ+ϱ2ε+δ2ε+γ2ε+h0+12ε+3h(0)2}∫τ0‖∇Utt(x,)‖2L2(Ω)dt−12∫τ0h′∘∇U(t)dt−h(0)2∫τ0‖∇U(x,τ)‖2L2(Ω)dt+1+ε′72|∂Ω||Ω|∫τ0h′∘U(t)dt. | (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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