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Existence and uniqueness for Moore-Gibson-Thompson equation with, source terms, viscoelastic memory and integral condition

  • 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+uttc2ΔubΔut=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+uttΔuβΔut=|ut|p,xRn,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+αuttc2AubAutt0g(ts)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(ts)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+βuttϱΔuδΔutγΔutt+t0h(tσ)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)uη=t0Ωu(ξ,τ)dξdτ,   xΩ. (1.1)

    The convolution term t0h(ts)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) hC1(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),t0. (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 ΩRn 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)={uW12(QT):ut,uttW12(QT),u,uL2h(QT)},

    and

    W(QT)={uV(QT):u(x,T)=0}.L2h(QT)={uV(QT):T0hu(t)dt<},

    where

    hu(t)=Ωt0h(tσ)(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(tσ)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 vW(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)+ϱT0Ωv(t0Ωu(ξ,τ)dξdτ)dsxdt+δT0ΩvΩu(ξ,t)dξdsxdtδT0ΩvΩu0(ξ)dξdsxdtγT0Ωvt(t0Ωuτ(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)T0Ωv(t0Ωw(ξ,τ)dξdτ)dsxdt. (2.2)

    Now, we give two useful inequalities:

    ● Gronwall inequality: If for any tI, we have

    y(t)h(t)+ct0y(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 wW21(Ω), we have

    w2L2(Ω)εw2L2(Ω)+l(ε)w2L2(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 uV(QT) satisfies Eq (2.1) for each vW(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,u2W12(Ω), u3L2(Ω) and FL2(QT), then there is at least one generalized solution in V(QT) to problem (1.1).

    Proof. Let {Zk(x)}k1 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)=Nk=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)+ϱΩZl(x)(t0ΩuN(ξ,τ)dξdτ)dsx+δΩZl(x)(t0ΩuNτ(ξ,τ)dξdτ)dsx+γΩZl(x)(t0ΩuNττ(ξ,τ)dξdτ)dsxΩZl(x)(t0ΩwN(ξ,τ)dξdτ)dsx, (3.3)

    Invoking to (3.1) in (3.3) gives for  l=1,...,N.

    ΩNk=1{Ck(t)Zk(x)Zl(x)+αCk(t)Zk(x)Zl(x)+βCk(t)Zk(x)Zl(x)+ϱCk(t)Zk(x).Zl(x)+δCk(t)Zk(x).Zl(x)+γCk(t)Zk.Zl(t0h(tσ)Ck(σ)dσ)Zk(x).Zl(x)}dx=(F(x,t),Zl(x))L2(Ω)+ϱNk=1t0Ck(τ)(ΩZl(x)ΩZk(ξ)dξdsx)dτ+δNk=1t0Ck(τ)(ΩZl(x)ΩZk(ξ)dξdsx)dτ+γNk=1t0Ck(τ)(ΩZl(x)ΩZk(ξ)dξdsx)dτNk=1t0τ0h(τσ)Ck(σ)(ΩZl(x)ΩZk(ξ)dξdsx)dσdτ. (3.4)

    From (3.4) it follows that

    Nk=1Ck(t)(Zk(x),Zl(x))L2(Ω)+αCk(t)(Zk(x),Zl(x))L2(Ω)+βCk(t)(Zk(x),Zl(x))L2(Ω)+ϱCk(t)(Zk,Zl)L2(Ω)+δCk(t)(Zk(x),Zl(x))L2(Ω)+γCk(t)(Zk(x),Zl(x))L2(Ω)(t0h(tσ)Ck(σ)dσ)(Zk,Zl)L2(Ω)}dx=(F(x,t),Zl(x))L2(Ω)+ϱNk=1t0Ck(τ)(ΩZl(x)ΩZk(ξ)dξdsx)dτ+δNk=1t0Ck(τ)(ΩZl(x)ΩZk(ξ)dξdsx)dτ+γNk=1t0(Ck(τ)ΩZl(x)ΩZk(ξ)dξds)dτNk=1t0τ0h(τσ)Ck(σ)(ΩZl(x)ΩZk(ξ)dξdsx)dσdτ,    l=1,...,N. (3.5)

    Let

    (Zk,Zl)L2(Ω)=δkl={1,   k=l0,   kl
    (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

    Nk=1Ck(t)δkl+αCk(t)δkl+Ck(t)(βδkl+γγkl)+δCk(t)γkl+ϱCk(t)γklt0(ϱCk(τ)χkl+δCk(τ)χkl+γCk(τ)χklh(tτ)Ck(τ)γkl)t0τ0h(τσ)Ck(σ)dσχkldσdτ=Fl(t). (3.6)

    A differentiation with respect to t (two times), yields

    Nk=1C′′′′′′k(t)δkl+αC′′′′′k(t)δkl+Ck(t)(βδkl+γγkl)+Ck(t)(δγklγχkl)+Ck(t)(ϱγklδχkl)(ϱ+h(0))Ck(t)χkl+h(0)Ck(t)χkl=Fl(t), (3.7)
    {Nk=1[Ck(0)δkl+αCk(0)δkl+Ck(0)(βδkl+γγkl)+δCk(0)γkl+ϱCk(0)γkl]=Fl(0)Ck(0)=(Zk,u0)L2(Ω), Ck(0)=(Zk,u1(x))L2(Ω),Ck(0)=(Zk,u2(x))L2(Ω),Ck(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 Ck(t) summing over k from 1 to N then integrating the resultant equality with respect to t from 0 to τ, with τT, 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)+ϱτ0ΩuNt(x,t)(t0ΩuN(ξ,η)dξdη)dsxdt+δτ0ΩuNt(x,t)(t0ΩuNt(ξ,η)dξdη)dsxdt+γτ0ΩuNt(x,t)(t0ΩuNtt(ξ,η)dξdη)dsxdtτ0ΩuNt(x,t)(t0ΩwN(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)ατ0utt(x,t)2L2(Ω)dt,β(uNtt,uNt)L2(Qτ)=β2uNτ(x,τ)2L2(Ω)β2uNt(x,0)2L2(Ω),ϱ(uN,uNt)L2(Qτ)=ϱ2uN(x,τ)2L2(Ω)ϱ2uN(x,0)2L2(Ω),δ(uNt,uNt)L2(Qτ)=δτ0uNt(x,t)2L2(Ω)dt,γ(uNtt,uNt)L2(Qτ)=γ2uNτ(x,τ)2L2(Ω)γ2uNt(x,0)2L2(Ω),(wN,uNt)L2(Qτ)=12huN(τ)12H(τ)uN(x,τ)2L2(Ω)12τ0huN(t)dt+12h(t)uN(x,t)2L2(Ω)dt, (3.10)
    ϱτ0ΩuNt(t0ΩuN(ξ,η)dξdη)dsxdt=ϱΩuN(x,τ)τ0ΩuN(ξ,t)dξdtdsxϱΩτ0uN(x,t)ΩuN(ξ,t)dξdtdsx, (3.11)
    δτ0ΩuNt(t0ΩuNt(ξ,η)dξdη)dsxdt=δΩτ0uNt(x,t)ΩuN(ξ,t)dξdtdsxδΩτ0uNt(x,t)ΩuN(ξ,0)dξdtdsx, (3.12)
    γτ0ΩuNt(x,t)(t0ΩuNtt(ξ,η)dξdη)dsxdt=γτ0ΩuNt(x,t)(ΩuNt(ξ,t)dξ)dsxdtγτ0ΩuNt(x,t)(ΩuNt(ξ,0)dξ)dsxdt. (3.13)
    τ0ΩuNt(t0ΩwN(ξ,η)dξdη)dsxdt=τ0ΩuNt(t0ΩH(η)uN(ξ,η)dξdη)dsxdt+τ0ΩuNt(t0Ω[η0h(ησ)(uN(ξ,η)uN(ξ,σ))dσ]dξdη)dsxdt=ΩuN(x,τ)τ0ΩH(t)uN(ξ,t)dξdtdsx+τ0ΩuN(x,t)ΩH(t)uN(ξ,t)dξdsxdt+ΩuN(x,τ)(τ0Ω[t0h(tσ)(uN(ξ,t)uN(ξ,σ))dσ]dξ)dsxdtτ0ΩuN(x,t)Ω[t0h(tσ)(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)+β2uNτ(x,τ)2L2(Ω)+ϱ2uN(x,τ)2L2(Ω)+γ2uNτ(x,τ)2L2(Ω)+12huN(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)+ϱ2uN(x,0)2L2(Ω)+γ2uNt(x,0)2L2(Ω)+τ0(uNttt,uNtt)L2(Ω)dt+ατ0utt(x,t)2L2(Ω)dtδτ0uNt(x,t)2L2(Ω)dt+β2uNt(x,0)2L2(Ω)+ϱΩuN(x,τ)τ0ΩuN(ξ,t)dξdtdsx+(F,uNt)L2(Qτ)ϱΩτ0uN(x,t)ΩuN(ξ,t)dξdtdsx+δΩτ0uNt(x,t)ΩuN(ξ,t)dξdtdsxδΩτ0uNt(x,t)ΩuN(ξ,0)dξdtdsx+γτ0ΩuNt(x,t)(ΩuNt(ξ,t)dξ)dsxdtγτ0ΩuNt(x,t)(ΩuNt(ξ,0)dξ)dsxdt12τ0huN(t)dt+12h(t)uN(x,t)2L2(Ω)dtΩuN(x,τ)τ0ΩH(t)uN(ξ,t)dξdtdsx+τ0ΩuN(x,t)ΩH(t)uN(ξ,t)dξdsxdt+ΩuN(x,τ)(τ0Ω[t0h(tσ)(uN(ξ,t)uN(ξ,σ))dσ]dξ)dsxdtτ0ΩuN(x,t)Ω[t0h(tσ)(uN(ξ,t)uN(ξ,σ))dσ]dξdsxdt. (3.15)

    Now, multiplying each equation of (3.3) by the appropriate Ck(t), add them up from 1 to N and them integrate with respect to t from 0 to τ, with τT, 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)+ϱτ0ΩuNtt(x,t)(t0ΩuN(ξ,η)dξdη)dsxdt+δτ0ΩuNtt(x,t)(t0ΩuNt(ξ,η)dξdη)dsxdt+γτ0ΩuNtt(x,t)(t0ΩuNtt(ξ,η)dξdη)dsxdtτ0ΩuNtt(x,t)(t0ΩwN(ξ,η)dξdη)dsxdt. (3.16)

    With the same reasoning in (3.9), we find

    (uNtttt,uNtt)L2(Qτ)=τ0uNttt(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)=α2uNττ(x,τ)2L2(Ω)α2uNtt(x,0)2L2(Ω),β(uNtt,uNtt)L2(Qτ)=βτ0uNtt(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)ϱτ0uNt(x,t)2L2(Ω)dt,δ(uNt,uNtt)L2(Qτ)=δ2uNτ(x,τ)2L2(Ω)δ2uNt(x,0)2L2(Ω),γ(uNtt,uNtt)L2(Qτ)=γτ0uNtt(x,t)2L2(Ω)dt(wN,uNtt)L2(Qτ)=12{huN(τ)+h(τ)uN(x,τ)2L2(Ω)2(wN(τ),uNτ)L2(Ω)}+12τ0huN(t)dt12τ0h(t)uN(x,t)2L2(Ω)dt, (3.17)
    ϱτ0ΩuNtt(t0ΩuN(ξ,η)dξdη)dsxdt=ϱΩuNτ(x,τ)τ0ΩuN(ξ,t)dξdtdsxϱΩτ0uNt(x,t)ΩuN(ξ,t)dξdtdsx, (3.18)
    δτ0ΩuNtt(x,t)(t0ΩuNt(ξ,η)dξdη)dsxdt=δΩuNτ(x,τ)ΩuN(ξ,τ)dξdsxδΩuNτ(x,τ)ΩuN(ξ,0)dξdsxδΩτ0uNt(x,t)ΩuNt(ξ,t)dξdtds, (3.19)
    γτ0ΩuNtt(x,t)(t0ΩuNtt(ξ,η)dξdη)dsxdt=γΩuNτ(x,τ)ΩuNτ(ξ,τ)dξdsxγΩuNτ(x,τ)ΩuNt(ξ,0)dξdsxγΩτ0uNt(x,t)ΩuNtt(ξ,t)dξdtds, (3.20)
    τ0ΩuNtt(t0ΩwN(ξ,η)dξdη)dsxdt=τ0ΩuNtt(t0ΩH(η)uN(ξ,η)dξdη)dsxdt+τ0ΩuNtt(t0Ω[η0h(ησ)(uN(ξ,η)uN(ξ,σ))dσ]dξdη)dsxdt=ΩuNτ(x,τ)τ0ΩH(t)uN(ξ,t)dξdtdsx+τ0ΩuNt(x,t)ΩH(t)uN(ξ,t)dξdsxdt+ΩuNτ(x,τ)(τ0Ω[t0h(tσ)(uN(ξ,t)uN(ξ,σ))dσ]dξ)dsxdtτ0ΩuNt(x,t)Ω[t0h(tσ)(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)+α2uNττ(x,τ)2L2(Ω)+δ2uNτ(x,τ)2L2(Ω)+ϱ(uN(x,τ),uNτ(x,τ))L2(Ω)+12h(τ)uN(x,τ)2L2(Ω)12huN(τ)+(wN(τ),uNτ)L2(Ω)=τ0uNttt(x,t)2L2(Ω)dt+(uNttt(x,0),uNtt(x,0))L2(Ω)+α2uNtt(x,0)2L2(Ω)βτ0uNtt(x,t)2L2(Ω)dt+ϱ(uN(x,0),uNt(x,0))L2(Ω)+ϱτ0ut(x,t)2L2(Ω)dt+δ2uNt(x,0)2L2(Ω)+(F,uNtt)L2(Qτ)γτ0uNtt(x,t)2L2(Ω)dt+ϱΩuNτ(x,τ)τ0ΩuN(ξ,t)dξdtdsxϱΩτ0uNt(x,t)ΩuN(ξ,t)dξdtdsx+δΩuNτ(x,τ)ΩuN(ξ,τ)dξdsxδΩuNτ(x,τ)ΩuN(ξ,0)dξdsxδΩτ0uNt(x,t)ΩuNt(ξ,t)dξdtdsx+γΩuNτ(x,τ)ΩuNτ(ξ,τ)dξdsxγΩuNτ(x,τ)ΩuNt(ξ,0)dξdsxγΩτ0uNt(x,t)ΩuNtt(ξ,t)dξdtdsxΩuNτ(x,τ)τ0ΩH(t)uN(ξ,t)dξdtdsx+τ0ΩuNt(x,t)ΩH(t)uN(ξ,t)dξdsxdt+ΩuNτ(x,τ)(τ0Ω[t0h(tσ)(uN(ξ,t)uN(ξ,σ))dσ]dξ)dsxdtτ0ΩuNt(x,t)Ω[t0h(tσ)(uN(ξ,t)uN(ξ,σ))dσ]dξdsxdt. (3.22)

    Now, multiplying each equation of (3.3) by the appropriate Ck(t), add them up from 1 to N and them integrate with respect to t from 0 to τ, with τT, 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)+ϱτ0ΩuNttt(x,t)(t0ΩuN(ξ,η)dξdη)dsxdt+δτ0ΩuNttt(x,t)(t0ΩuNt(ξ,η)dξdη)dsxdt+γτ0ΩuNttt(x,t)(t0ΩuNtt(ξ,η)dξdη)dsxdtτ0ΩuNttt(x,t)(t0ΩwN(ξ,η)dξdη)dsxdt. (3.23)

    With the same reasoning in (3.9), we find

    (uNtttt,uNttt)L2(Qτ)=12uNτττ(x,τ)2L2(Ω)12uNttt(x,0)2L2(Ω)α(uNttt,uNttt)L2(Qτ)=ατ0uNttt(x,t)2L2(Ω),β(uNtt,uNttt)L2(Qτ)=β2uNττ(x,τ)2L2(Ω)β2uNtt(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)ϱτ0(uNt,uNtt)L2(Ω)dt,δ(uNt,uNttt)L2(Qτ)=δτ0uNtt(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)=γ2uNττ(x,τ)2L2(Ω)γ2uNtt(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)12uN(x,τ)2L2(Ω)+ΩuNτττ0h(τσ)(uN(τ)uN(σ))dσdx+ΩuNττ0h(τσ)(uN(τ)uN(σ))dσdx+12huN(τ)+12τ0(hh)uN(t)dth(0)τ0uNt(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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