# The Galerkin Method for Fourth-Order Equation of the Moore–Gibson–Thompson Type with Integral Condition

Authors: Ahlem Mesloub; Abderrahmane Zaraï; Fatiha Mesloub; Bahri-Belkacem Cherif; Mohamed Abdalla
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

In this manuscript, we consider the fourth order of the Moore–Gibson–Thompson equation by using Galerkin’s method to prove the solvability of the given nonlocal problem.


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## Body

## 1. Introduction

Research on the nonlinear propagation of sound in a situation of high amplitude waves has shown literature on physically well-founded partial differential models (see, e.g., [1–23]). This still very active field of research is carried by a wide range of applications such as the medical and industrial use of high-intensity ultrasound in lithotripsy, thermotherapy, ultrasound cleaning, and sonochemistry. The classical models of nonlinear acoustics are Kuznetsov’s equation, the Westervelt equation, and the KZK (Kokhlov-Zabolotskaya-Kuznetsov) equation. For a mathematical existence and uniqueness analysis of several types of initial boundary value problems for these nonlinear second order in time PDEs, we refer to [24–44]. Focusing on the study of the propagation of acoustic waves, it should be noted that the MGT equation is one of the equations of nonlinear acoustics describing acoustic wave propagation in gases and liquids. The behavior of acoustic waves depends strongly on the medium property related to dispersion, dissipation, and nonlinear effects. It arises from modeling high-frequency ultrasound (HFU) waves (see [10, 12, 34]). The derivation of the equation, based on continuum and fluid mechanics, takes into account viscosity and heat conductivity as well as effect of the radiation of heat on the propagation of sound. The original derivation dates back to [44]. This model is realized through the third-order hyperbolic equation:
(1)τuttt+utt−c2Δu−bΔut=0.The unknown functionu=ux,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 [44], Chen and Palmieri studied the blow-up result for the semilinear Moore–Gibson–Thompson equation with nonlinearity of derivative type in the conservative case defined as follows:
(2)βuttt+utt−Δu−βΔut=utp,x∈ℝn,t>0.This paper is related to the following works (see [16, 39]). Now, when we talk about the MGT equation with memory term, we have Lasieka and Wang in [17] who studied the exponential decay of the energy of the temporally third-order (Moore–Gibson–Thompson) equation with a memory term as follows:
(3)τuttt+αutt−c2Au−bAut−∫0tgt−sAwsds=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 ∫0tgt−sAwsds reflects the memory effects of materials due to viscoelasticity. In [18], Lasieka and Wang studied the general decay of solution of same problem above. The Moore–Gibson–Thompson equation with a nonlocal condition is a new posed problem. Existence and uniqueness of the generalized solution are established by using the Galerkin method. These problems can be encountered in many scientific domains and many engineering models (see previous works [5, 25–32, 35, 36, 40, 41]). Mesloub and Mesloub in [33] have applied the Galerkin method to a higher dimension mixed with nonlocal problem for a Boussinesq equation, while Boulaaras et al. investigated the Moore–Gibson–Thompson equation with the integral condition in [4]. Motivated by these outcomes, we improve the existence and uniqueness by the Galerkin method of the fourth-order equation of the Moore–Gibson–Thompson type with integral condition; this problem was cited by the work of Dell’Oro and Pata in [9].We define the problem as follows:(4)utttt+αuttt+βutt−ρΔu−δΔut−γΔutt=0,ux,0=u0x,utx,0=u1x,uttx,0=u2x,utttx,0=u3x∂u∂η=∫0t∫Ωuξ,τdξdτ,x∈∂Ω.The aim of this manuscript is to consider the following nonlocal mixed boundary value problem for the Moore–Gibson–Thompson (MGT) equation for allx;t∈QT=0,T, where Ω⊂ℝn is a bounded domain with sufficiently smooth boundary ∂Ω. solution of the posed problem.We divide this paper into the following: In “Preliminaries,” some definitions and appropriate spaces have been given. Then in “Solvability of the Problem,” we use Galerkin’s method to prove the existence, and in “Uniqueness of Solution,” we demonstrate the uniqueness.


## 2. Preliminaries

LetVQT and WQT be the set spaces defined, respectively, by
(5)VQT=u∈W21QT:ut∈W21QT:utt∈W21QT,WQT=u∈VQT:ux,T=0.Consider the equation(6)utttt,vL2QT+αuttt,vL2QT+βutt,vL2QT−ϱΔu,vL2QT−δΔut,vL2QTγΔutt,vL2QT=0,where .,.L2QT depend on the inner product in L2QT, u is supposed to be a solution of (1), and v∈WQT. Upon using (6) and (1), we find
(7)−uttt,vtL2QT−αutt,vtL2QT−βut,vtL2QT+ϱ∇u,∇vL2QT+δ∇ut,∇vL2QT−γ∇ut,∇vtL2QT=ϱ∫0T∫∂Ωv∫0t∫Ωuξ,τdξdτdsxdt+δ∫0T∫∂Ωv∫Ωuξ,tdξdsxdt−δ∫0T∫∂Ωv∫Ωu0ξdξdsxdt−γ∫0T∫∂Ωvt∫0t∫Ωuτξ,τdξdτdsxdt+u3x,vx,0L2Ω+αu2x,vx,0L2Ω+βu1x,vx,0L2Ω−γΔu1,vx,0L2Ω.Now, we give two useful inequalities:(i)
Gronwall inequality: if for anyt∈I, we have(8)yt≤ht+c∫0tysds,where ht and yt are two nonnegative integrable functions on the interval I with ht nondecreasing and c is constant, then
(9)yt≤htexpct(ii)
Trace inequality: whenw∈W12Ω, we have(10)wL2∂Ω2≤ε∇wL2Ω2+lεwL2Ω2,where Ω is a bounded domain in ℝn with smooth boundary ∂Ω, and lε is a positive constant.Definition 1.
If a functionu∈VQT satisfies Equation (3), each v∈WQT is called a generalized solution of problem (1).


## 3. Solvability of the Problem

Here, by using Galerkin’s method, we give the existence of problem (1).Theorem 2.
Ifu0∈,u1∈ and u2∈, u3∈, then there is at least one generalized solution in VQT to problem (1).Proof.
LetZkxk≥1 be a fundamental system in W21Ω, such that Zk,ZlL2Ω=δk,l. Now, we will find an approximate solution of the problem (1) in the form
(11)uNx,t=∑k=1NCktZkx,where the constants Ckt are defined by the conditions
(12)Ckt=uNx,t,ZkxL2Ω,k=1,⋯,N,and can be determined from the relations
(13)uttttN,ZlxL2Ω+αutttN,ZlxL2Ω+βuttN,ZlxL2Ω+ϱ∇uN,∇ZlxL2Ω+δ∇utN,∇ZlxL2Ω+γ∇uttN,∇ZlxL2Ω=ϱ∫∂ΩZlx∫0t∫ΩuNξ,τdξdτdsx+δ∫∂ΩZlx∫0t∫ΩuτNξ,τdξdτdsx+γ∫∂ΩZlx∫0t∫ΩuττNξ,τdξdτdsx.
Invoking to (11) in (6) gives for l=1,⋯,N.
(14)∫Ω∑k=1NCk′′′′tZkxZlx+αCk′′′tZkxZlx+βCk′′tZkxZlx+ϱCkt∇Zkx.∇Zlx+δCk′t∇Zkx.∇Zlx+γCk′′t∇Zk.∇Zldx=ϱ∑k=1N∫0tCkτ∫∂ΩZlx∫ΩZkξdξdsxdτ+δ∑k=1N∫0tCk′τ∫∂ΩZlx∫ΩZkξdξdsxdτ+γ∑k=1N∫0tCk′′τ∫∂ΩZlx∫ΩZkξdξdsxdτ.
From (7), it follows that
(15)∑k=1NCk′′′′tZkx,ZlxL2Ω+αCk′′′tZkx,ZlxL2Ω+βCk′′tZkx,ZlxL2Ω+ϱCkt∇Zk,∇ZlL2Ω+δCk′t∇Zkx,∇ZlxL2Ω+γCk′′t∇Zkx,∇ZlxL2Ω=ϱ∑k=1N∫0tCkτ∫∂ΩZlx∫ΩZkξdξdsxdτ+δ∑k=1N∫0tCk′τ∫∂ΩZlx∫ΩZkξdξdsxdτ+γ∑k=1N∫0tCk′′τ∫∂ΩZlx∫ΩZkξdξdsdτ,l=1,⋯,N.
Let(16)Zk,ZlL2Ω=δkl=1,k=l0,k≠l,∇Zk,∇ZlL2Ω=γkl,∫∂ΩZlx∫ΩZkξdξds=χkl.
Then (8) can be written as
(17)∑k=1NCk′′′′tδkl+αCk′′′tδkl+Ck′′tβδkl+γγkl+δCk′tγkl+ϱCktγkl−∫0tϱCkτχkl+δCk′τχkl+γCk′′τχkl=0.
A differentiation with respect tot yields
(18)∑k=1NCk′′′′′tδkl+αCk′′′′tδkl+Ck′′′tβδkl+γγkl+Ck′′tδγkl−γχkl+Ck′tϱγkl−δχklγkl−ϱCktχkl=0,∑k=1NCk′′′′0δkl+αCk′′′0δkl+Ck′′0βδkl+γγkl+δCk′0γkl+ϱCk0γkl=0Ck0=Zk,u0L2Ω,Ck′0=Zk,u1xL2Ω,Ck′′t=Zk,u2xL2Ω,Ck′′′t=Zk,u3xL2Ω.
Thus, for everyn, there exists a function uNx satisfying (6). Now, we will demonstrate that the sequence uN is bounded. To do this, we multiply each equation of (6) 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 τ≤T, which yields
(19)uttttN,utNL2Qτ+αutttN,utNL2Qτ+βuttN,utNL2Qτ+ϱ∇uN,∇utNL2Qτ+δ∇utN,∇utNL2Qτ+γ∇uttN,∇utNL2Qτ=ϱ∫0τ∫∂ΩutNx,t∫0t∫ΩuNξ,ηdξdηdsxdt+δ∫0τ∫∂ΩutNx,t∫0t∫ΩutNξ,ηdξdηdsxdt+γ∫0τ∫∂ΩutNx,t∫0t∫ΩuttNξ,ηdξdηdsxdt.
After simplification of theLHS of (19), we observe that
(20)uttttN,utNL2Qτ=−∫0τutttN,uttNL2Ωdt+uτττNx,τ,uτNx,τL2Ω−utttNx,0,utNx,0L2Ω,(21)αutttN,utNL2Qτ=αuττNx,τ,uτNx,τL2Ω−uttNx,0,utNx,0L2Ω−α∫0τuttx,tL2Ω2dt,(22)βuttN,utNL2Qτ=β2uτNx,τL2Ω2−β2utNx,0L2Ω2,(23)ϱ∇uN,∇utNL2Qτ=ϱ2∇uNx,τL2Ω2−ϱ2∇uNx,0L2Ω2,(24)δ∇utN,∇utNL2Qτ=δ∫0τ∇utNx,tL2Ω2dt,(25)γ∇uttN,∇utNL2Qτ=γ2∇uτNx,τL2Ω2−γ2∇utNx,0L2Ω2,(26)ϱ∫0τ∫∂ΩutN∫0t∫ΩuNξ,ηdξdηdsxdt=ϱ∫∂ΩuNx,τ∫0τ∫ΩuNξ,tdξdtdsx−ϱ∫∂Ω∫0τuNx,t∫ΩuNξ,tdξdtdsx,(27)δ∫0τ∫∂ΩutN∫0t∫ΩutNξ,ηdξdηdsxdt=δ∫∂Ω∫0τutNx,t∫ΩuNξ,tdξdtdsx−δ∫∂Ω∫0τutNx,t∫ΩuNξ,0dξdtdsx,(28)γ∫0τ∫∂ΩutNx,t∫0t∫ΩuttNξ,ηdξdηdsxdt=γ∫0τ∫∂ΩutNx,t∫ΩutNξ,tdξdsxdt−γ∫0τ∫∂ΩutNx,t∫ΩutNξ,0dξdsxdt.
Taking into account the equalities (20) and (21) in (12), we obtain
(29)uτττNx,τ,uτNx,τL2Ω+αuττNx,τ,uτNx,τL2Ω+β2uτNx,τL2Ω2+ϱ2∇uNx,τL2Ω2+γ2∇uτNx,τL2Ω2=utttNx,0,utNx,0L2Ω+αuttNx,0,utNx,0L2Ω+β2utNx,0L2Ω2+ϱ2∇uNx,0L2Ω2+γ2∇utNx,0L2Ω2+∫0τutttN,uttNL2Ωdt+α∫0τuttx,tL2Ω2dt−δ∫0τ∇utNx,tL2Ω2dt+ϱ∫∂ΩuNx,τ∫0τ∫ΩuNξ,tdξdtdsx−ϱ∫∂Ω∫0τuNx,t∫ΩuNξ,tdξdtdsx+δ∫∂Ω∫0τutNx,t∫ΩuNξ,tdξdtdsx−δ∫∂Ω∫0τutNx,t∫ΩuNξ,0dξdtdsx+γ∫0τ∫∂ΩutNx,t∫ΩutNξ,tdξdsxdt−γ∫0τ∫∂ΩutNx,t∫ΩutNξ,0dξdsxdt.
Now, multiplying each equation of (6) by the appropriate Ck′′t, we add them up from 1 to N and then integrate with respect to t from 0 to τ, with τ≤T, and we obtain
(30)uttttN,uttNL2Qτ+αutttN,uttNL2Qτ+βuttN,uttNL2Qτ+ϱ∇uN,∇uttNL2Qτ+δ∇utN,∇uttNL2Qτ+γ∇uttN,∇uttNL2Qτ=ϱ∫0τ∫∂ΩuttNx,t∫0t∫ΩuNξ,ηdξdηdsxdt+δ∫0τ∫∂ΩuttNx,t∫0t∫ΩutNξ,ηdξdηdsxdt+γ∫0τ∫∂ΩuttNx,t∫0t∫ΩuttNξ,ηdξdηdsxdt.
With the same reasoning in (12), we find
(31)uttttN,uttNL2Qτ=−∫0τutttNx,tL2Ω2dt+uτττNx,τ,uττNx,τL2Ω−utttNx,0,uttNx,0L2Ω,(32)αutttN,uttNL2Qτ=α2uττNx,τL2Ω2−α2uttNx,0L2Ω2,(33)βuttN,uttNL2Qτ=β∫0τuttNx,tL2Ω2dt,(34)ϱ∇uN,∇uttNL2Qτ=ϱ∇uNx,τ,∇uτNx,τL2Qτ−ϱ∇uNx,0,∇utNx,0L2Ω−ϱ∫0τ∇utNx,tL2Ω2dt,(35)δ∇utN,∇uttNL2Qτ=δ2∇uτNx,τL2Ω2−δ2∇utNx,0L2Ω2,(36)γ∇uttN,∇uttNL2Qτ=γ∫0τ∇uttNx,tL2Ω2dt,(37)ϱ∫0τ∫∂ΩuttN∫0t∫ΩuNξ,ηdξdηdsxdt=ϱ∫∂ΩuτNx,τ∫0τ∫ΩuNξ,tdξdtdsx−ϱ∫∂Ω∫0τutNx,t∫ΩuNξ,tdξdtdsx,(38)δ∫0τ∫∂ΩuttNx,t∫0t∫ΩutNξ,ηdξdηdsxdt=δ∫∂ΩuτNx,τ∫ΩuNξ,τdξdsx−δ∫∂ΩuτNx,τ∫ΩuNξ,0dξdsx−δ∫∂Ω∫0τutNx,t∫ΩutNξ,tdξdtds,(39)γ∫0τ∫∂ΩuttNx,t∫0t∫ΩuttNξ,ηdξdηdsxdt=γ∫∂ΩuτNx,τ∫ΩuτNξ,τdξdsx−γ∫∂ΩuτNx,τ∫ΩutNξ,0dξdsx−γ∫∂Ω∫0τutNx,t∫ΩuttNξ,tdξdtds,
Upon using (31) and (32) into (23), we have
(40)uτττNx,τ,uττNx,τL2Ω+α2uττNx,τL2Ω2+δ2∇uτNx,τL2Ω2+ϱ∇uNx,τ,∇uτNx,τL2Ω=∫0τutttNx,tL2Ω2dt+utttNx,0,uttNx,0L2Ω+α2uttNx,0L2Ω2−β∫0τuttNx,tL2Ω2dt+ϱ∇uNx,0,∇utNx,0L2Ω+ϱ∫0τ∇utx,tL2Ω2dt+δ2∇utNx,0L2Ω2−γ∫0τ∇uttNx,tL2Ω2dt+ϱ∫∂ΩuτNx,τ∫0τ∫ΩuNξ,tdξdtdsx−ϱ∫∂Ω∫0τutNx,t∫ΩuNξ,tdξdtdsx+δ∫∂ΩuτNx,τ∫ΩuNξ,τdξdsx−δ∫∂ΩuτNx,τ∫ΩuNξ,0dξdsx−δ∫∂Ω∫0τutNx,t∫ΩutNξ,tdξdtdsx+γ∫∂ΩuτNx,τ∫ΩuτNξ,τdξdsx−γ∫∂ΩuτNx,τ∫ΩutNξ,0dξdsx−γ∫∂Ω∫0τutNx,t∫ΩuttNξ,tdξdtdsx.
Now, multiplying each equation of (6) by the appropriate Ck′′′t, we add them up from 1 to N and then integrate with respect to t from 0 to τ, with τ≤T, and we obtain
(41)uttttN,utttNL2Qτ+αutttN,utttNL2Qτ+βuttN,utttNL2Qτ+ϱ∇uN,∇utttNL2Qτ+δ∇utN,∇utttNL2Qτ+γ∇uttN,∇utttNL2Qτ=ϱ∫0τ∫∂ΩutttNx,t∫0t∫ΩuNξ,ηdξdηdsxdt+δ∫0τ∫∂ΩutttNx,t∫0t∫ΩutNξ,ηdξdηdsxdt+γ∫0τ∫∂ΩutttNx,t∫0t∫ΩuttNξ,ηdξdηdsxdt.
With the same reasoning in (12), we find
(42)uttttN,utttNL2Qτ=12uτττNx,τL2Ω2−12utttNx,0L2Ω2,(43)αutttN,utttNL2Qτ=α∫0τutttNx,tL2Ω2,(44)βuttN,utttNL2Qτ=β2uττNx,τL2Ω2−β2uttNx,0L2Ω2,(45)ϱ∇uN,∇utttNL2Qτ=ϱ∇uNx,τ,∇uττNx,τL2Ω−ϱ∇uNx,0,∇uttNx,0L2Ω−ϱ∫0τ∇utN,∇uttNL2Ωdt,(46)δ∇utN,∇utttNL2Qτ=−δ∫0τ∇uttNx,tL2Ω2dt+δ∇uτNx,τ,∇uττNx,τL2Ω−δ∇utNx,0,∇uttNx,0L2Ω,(47)γ∇uttN,∇utttNL2Qτ=γ2∇uττNx,τL2Ω2−γ2∇uttNx,0L2Ω2,(48)ϱ∫0τ∫∂ΩutttN∫0t∫ΩuNξ,ηdξdηdsxdt=ϱ∫∂ΩuττNx,τ∫0τ∫ΩuNξ,tdξdtdsx−ϱ∫∂Ω∫0τuttNx,t∫ΩuNξ,tdξdtdsx,(49)δ∫0τ∫∂ΩutttNx,t∫0t∫ΩutNξ,ηdξdηdsxdt=δ∫∂ΩuττNx,τ∫ΩuNξ,τdξdsx−δ∫∂ΩuττNx,τ∫ΩuNξ,0dξdsx−δ∫∂Ω∫0τuttNx,t∫ΩutNξ,tdξdtds,(50)γ∫0τ∫∂ΩutttNx,t∫0t∫ΩuttNξ,ηdξdηdsxdt=γ∫∂ΩuττNx,τ∫ΩuτNξ,τdξdsx−γ∫∂ΩuττNx,τ∫ΩutNξ,0dξdsx−γ∫∂Ω∫0τuttNx,t∫ΩuttNξ,tdξdtds,
A substitution of equalities (42) and (43) in (34) gives
(51)12uτττNx,τL2Ω2+β2uττNx,τL2Ω2+ϱ∇uNx,τ,∇uττNx,τL2Ω+δ∇uτNx,τ,∇uττNx,τL2Ω+γ2∇uττNx,τL2Ω2=12utttNx,0L2Ω2−α∫0τutttNx,tL2Ω2−β2uttNx,0L2Ω2+ϱ∇uNx,0,∇uttNx,0L2Ω+ϱ∫0τ∇utN,∇uttNL2Ωdt+δ∫0τ∇uttNx,tL2Ω2dt+δ∇utNx,0,∇uttNx,0L2Ω−γ2∇uttNx,0L2Ω2+ϱ∫∂ΩuττNx,τ∫0τ∫ΩuNξ,tdξdtdsx−ϱ∫∂Ω∫0τuttNx,t∫ΩuNξ,tdξdtdsx+δ∫∂ΩuττNx,τ∫ΩuNξ,τdξdsx−δ∫∂ΩuττNx,τ∫ΩuNξ,0dξdsx−δ∫∂Ω∫0τuttNx,t∫ΩutNξ,tdξdtdsγ+∫∂ΩuττNx,τ∫ΩuτNξ,τdξdsx−γ∫∂ΩuττNx,τ∫ΩutNξ,0dξdsx−γ∫∂Ω∫0τuttNx,t∫ΩuttNξ,tdξdtds.
Multiplying (22) by λ1, (33) by λ2, and (44) by λ3, we get
(52)λ1uτττNx,τ,uτNx,τL2Ω+λ1αuττNx,τ,uτNx,τL2Ω+λ1β2uτNx,τL2Ω2+λ1ϱ2∇uNx,τL2Ω2+λ1γ2+λ2δ2∇uτNx,τL2Ω2+λ2uτττNx,τ,uττNx,τL2Ω+λ2α2+λ3β2uττNx,τL2Ω2+λ2ϱ∇uNx,τ,∇uτNx,τL2Ω+λ32uτττNx,τL2Ω2+λ3ϱ∇uNx,τ,∇uττNx,τL2Ω+λ3δ∇uτNx,τ,∇uττNx,τL2Ω+λ3γ2∇uττNx,τL2Ω2=λ1utttNx,0,utNx,0L2Ω+λ1αuttNx,0,utNx,0L2Ω+λ1β2utNx,0L2Ω2+λ1ϱ2∇uNx,0L2Ω2+λ1γ2+λ2δ2∇utNx,0L2Ω2+λ1∫0τutttN,uttNL2Ωdt+λ1α−λ2β∫0τuttx,tL2Ω2dt+λ2ϱ−λ1δ∫0τ∇utNx,tL2Ω2dt+λ2−λ3α∫0τutttNx,tL2Ω2dt+λ2utttNx,0,uttNx,0L2Ω+λ2α2−λ3β2uttNx,0L2Ω2+λ2ϱ∇uNx,0,∇utNx,0L2Ω+λ3δ−λ2γ∫0τ∇uttNx,tL2Ω2dt+λ32utttNx,0L2Ω2+λ3ϱ∇uNx,0,∇uttNx,0L2Ω+λ3ϱ∫0τ∇utN,∇uttNL2Ωdt,+λ3δ∇utNx,0,∇uttNx,0L2Ω−λ3γ2∇uttNx,0L2Ω2+λ1ϱ∫∂ΩuNx,τ∫0τ∫ΩuNξ,tdξdtdsx−λ1ϱ∫∂Ω∫0τuNx,t∫ΩuNξ,tdξdtdsx+λ1δ−λ2ϱ∫∂Ω∫0τutNx,t∫ΩuNξ,tdξdtdsx−λ1δ∫∂Ω∫0τutNx,t∫ΩuNξ,0dξdtdsx+λ1γ−λ2δ∫0τ∫∂ΩutNx,t∫ΩutNξ,tdξdsxdt−λ1γ∫0τ∫∂ΩutNx,t∫ΩutNξ,0dξdsxdt.+λ2ϱ∫∂ΩuτNx,τ∫0τ∫ΩuNξ,tdξdtdsx+λ2δ∫∂ΩuτNx,τ∫ΩuNξ,τdξdsx−λ2δ∫∂ΩuτNx,τ∫ΩuNξ,0dξdsx+λ2γ∫∂ΩuτNx,τ∫ΩuτNξ,τdξdsx−λ2γ∫∂ΩuτNx,τ∫ΩutNξ,0dξdsx−λ2γ∫∂Ω∫0τutNx,t∫ΩuttNξ,tdξdtdsx+λ3ϱ∫∂ΩuττNx,τ∫0τ∫ΩuNξ,tdξdtdsx−λ3ϱ∫∂Ω∫0τuttNx,t∫ΩuNξ,tdξdtdsx+λ3δ∫∂ΩuττNx,τ∫ΩuNξ,τdξdsx−λ3δ∫∂ΩuττNx,τ∫ΩuNξ,0dξdsx−λ3δ∫∂Ω∫0τuttNx,t∫ΩutNξ,tdξdtds+λ3γ∫∂ΩuττNx,τ∫ΩuτNξ,τdξdsx−λ3γ∫∂ΩuττNx,τ∫ΩutNξ,0dξdsx−λ3γ∫∂Ω∫0τuttNx,t∫ΩuttNξ,tdξdtds.
We can estimate all the terms in the right-hand side of (45) as follows:
(53)λ1ϱ∫∂ΩuNx,τ∫0τ∫ΩuNξ,tdξdtdsx≤λ1ϱ2ε1ε∇uNx,τL2Ω2+lεuNx,τL2Ω2+λ1ϱ2ε1TΩ∂Ω∫0τuNx,tL2Ω2dt,−λ1ϱ∫∂Ω∫0τuNx,t∫ΩuNξ,tdξdtdsx≤λ1ϱ2ε∫0τ∇uNx,tL2Ω2dt+λ1ϱ2lε+Ω∂Ω∫0τuNx,tL2Ω2dt,λ1δ−λ2ϱ∫∂Ω∫0τutNx,t∫ΩuNξ,tdξdtdsx≤λ1δ+λ2ϱ2ε∫0τ∇utNx,tL2Ω2dt+lε∫0τutNx,tL2Ω2dt+λ1δ+λ2ϱ2Ω∂Ω∫0τuNx,tL2Ω2dt,−λ1δ∫∂Ω∫0τutNx,t∫ΩuNξ,0dξdtdsx≤λ1δ2ε∫0τ∇utNx,tL2Ω2dt+lε∫0τutNx,tL2Ω2dt+λ1δ2Ω∂ΩTuNx,0L2Ω2,λ2ϱ∫∂ΩuτNx,τ∫0τ∫ΩuNξ,tdξdtdsx≤λ2ϱ2εε2∇uτNx,τL2Ω2+lεε2uτNx,τL2Ω2+λ2ϱ2ε2Ω∂ΩT∫0τuNx,tL2Ω2dt,λ2δ∫∂ΩuτNx,τ∫ΩuNξ,τdξdsx≤λ2δ2ε3ε∇uτNx,τL2Ω2+lεuτNx,τL2Ω2+λ2δ2ε3Ω∂ΩuNx,τL2Ω2,−λ2δ∫∂ΩuτNx,τ∫ΩuNξ,0dξdsx≤λ2δ2ε4ε∇uτNx,τL2Ω2+lεuτNx,τL2Ω2+λ2δ2ε4Ω∂ΩuNx,0L2Ω2,λ1γ−λ2δ∫∂Ω∫0τutNx,t∫ΩutNξ,tdξdtdsx≤λ1γ+λ2δ2ε∫0τ∇utNx,tL2Ω2dt+λ1γ+λ2δ2lε+Ω∂Ω∫0τutNx,tL2Ω2dt,(54)−λ1γ∫0τ∫∂ΩutNx,t∫ΩutNξ,0dξdsxdt≤λ1γ2ε∫0τ∇utNx,tL2Ω2dt+lε∫0τutNx,tL2Ω2dt+λ1γ2Ω∂ΩTutNx,0L2Ω2,λ2γ∫∂ΩuτNx,τ∫ΩuτNξ,τdξdsx≤λ2γ2ε5ε∇uτNx,τL2Ω2+lεuτNx,τL2Ω2+λ2γ2ε5Ω∂ΩuτNx,τL2Ω2,−λ2γ∫∂ΩuτNx,τ∫ΩutNξ,0dξdsx≤λ2γ2ε6ε∇uτNx,τL2Ω2+lεuτNx,τL2Ω2+λ2γ2ε6Ω∂ΩutNx,0L2Ω2,−λ2γ∫∂Ω∫0τutNx,t∫ΩuttNξ,tdξdtdsx≤λ2γ2ε∫0τ∇utNx,tL2Ω2dt+λ2γ2lε∫0τutNx,tL2Ω2dt+λ2γ2Ω∂Ω∫0τuttNx,tL2Ω2dt,λ3ϱ∫∂ΩuττNx,τ∫0τ∫ΩuNξ,tdξdtdsx≤λ3ϱ2εε7∇uττNx,τL2Ω2+lεε7uττNx,τL2Ω2+λ3ϱ2ε7Ω∂ΩT∫0τuNx,tL2Ω2dt,−λ3ϱ∫∂Ω∫0τuttNx,t∫ΩuNξ,tdξdtdsx≤λ3ϱ2ε∫0τ∇uttNx,tL2Ω2dt+lε∫0τuttNx,tL2Ω2dt+λ3ϱ2Ω∂Ω∫0τuNx,tL2Ω2dt,λ3δ∫∂ΩuττNx,τ∫ΩuNξ,τdξdsx≤λ3δ2ε8ε∇uττNx,τL2Ω2+lεuττNx,τL2Ω2+λ3δ2ε8Ω∂ΩuNx,τL2Ω2,(55)−λ3δ∫∂ΩuττNx,τ∫ΩuNξ,0dξdsx≤λ3δ2ε9ε∇uττNx,τL2Ω2+lεuττNx,τL2Ω2+λ3δ2ε9Ω∂ΩuNx,0L2Ω2,(56)−λ3δ∫∂Ω∫0τuttNx,t∫ΩutNξ,tdξdtdsx≤λ3δ2ε∫0τ∇uttNx,tL2Ω2dt+lε∫0τuttNx,tL2Ω2dt+λ3δ2Ω∂Ω∫0τutNx,tL2Ω2dt,λ3γ∫∂ΩuττNx,τ∫ΩuτNξ,τdξdsx≤λ3γ2ε10ε∇uττNx,τL2Ω2+lεuττNx,τL2Ω2+λ3γ2ε10Ω∂ΩuτNx,τL2Ω2,(57)−λ3γ∫∂ΩuττNx,τ∫ΩutNξ,0dξdsx≤λ3γ2ε11ε∇uττNx,τL2Ω2+lεuττNx,τL2Ω2+λ3γ2ε11Ω∂ΩutNx,0L2Ω2,−λ3γ∫∂Ω∫0τuttNx,t∫ΩuttNξ,tdξdtdsx≤λ3γ2ε∫0τ∇uttNx,tL2Ω2dt+λ3γ2lε+Ω∂Ω∫0τuttNx,tL2Ω2dt,−λ12uτττNx,τL2Ω2−λ12uτNx,τL2Ω2≤λ1uτττNx,τ,uτNx,τL2Ω,(58)−λ22uτττNx,τL2Ω2−λ22uττNx,τL2Ω2≤λ2uτττNx,τ,uττNx,τL2Ω,−λ1α2uττNx,τL2Ω2−λ1α2uτNx,τL2Ω2≤λ1αuττNx,τ,uτNx,τL2Ω,−λ2ϱε122∇uNx,τL2Ω2−λ2ϱ2ε12∇uτNx,τL2Ω2≤λ2ϱ∇uNx,τ,∇uτNx,τL2Ω,−λ2ϱε132∇uNx,τL2Ω2−λ2ϱ2ε13∇uττNx,τL2Ω2≤λ3ϱ∇uNx,τ,∇uττNx,τL2Ω,−λ3δε142∇uτNx,τL2Ω2−λ3δε142∇uττNx,τL2Ω2≤λ3δ∇uτNx,τ,∇uττNx,τL2Ω,λ1utttNx,0,utNx,0L2Ω≤λ12utttNx,0L2Ω2+λ12utNx,0L2Ω2,λ1αuttNx,0,utNx,0L2Ω≤λ1α2uttNx,0L2Ω2+λ1α2utNx,0L2Ω2,λ2utttNx,0,uttNx,0L2Ω≤λ22utttNx,0L2Ω2+λ22uttNx,0L2Ω2,λ2ϱ∇uNx,0,∇utNx,0L2Ω≤λ22ϱ∇uNx,0L2Ω2+λ22ϱ∇utNx,0L2Ω2,λ3ϱ∇uNx,0,∇uttNx,0L2Ω≤λ32ϱ∇uNx,0L2Ω2+λ32ϱ∇uttNx,0L2Ω2,λ3δ∇utNx,0,∇uttNx,0L2Ω≤λ32δ∇utNx,0L2Ω2+λ32δ∇uttNx,0L2Ω2,(59)λ1∫0τutttN,uttNL2Ωdt≤λ12∫0τutttNx,tL2Ω2dt+λ12∫0τuttNx,tL2Ω2dt,(60)λ3ϱ∫0τ∇utN,∇uttNL2Ωdt≤λ3ϱ2∫0τ∇utNx,tL2Ω2dt+λ3ϱ2∫0τuttNx,tL2Ω2dt.
Combining inequalities (46)–(79) and equality (45) and making use of the following inequality:
(61)m1uNx,τL2Ω2≤m1uNx,tL2Qτ2+m1utNx,tL2Qτ2+m1uNx,0L2Ω2,m2uτNx,τL2Ω2≤m2utNx,tL2Qτ2+m2uttNx,tL2Qτ2+m2utNx,0L2Ω2,m3uττNx,τL2Ω2≤m3uttNx,tL2Qτ2+m3utttNx,tL2Qτ2+m3uttNx,0L2Ω2,m4∇uNx,τL2Ω2≤m4∇uNx,tL2Qτ2+m4∇utNx,tL2Qτ2+m4∇uNx,0L2Ω2,m5∇uτNx,τL2Ω2≤m5∇utNx,tL2Qτ2+m5∇uttNx,tL2Qτ2+m5∇utNx,0L2Ω2,where
(62)m1=λ2δ2ε3Ω∂Ω+λ3δ2ε8Ω∂Ω,m2=λ2ϱ2lεε2+λ2δ2lεε3+λ2δ2lεε4+λ2γ2lεε5+ε5Ω∂Ω+λ2γ2lεε6+λ3γ2ε10Ω∂Ω+λ12,m3=λ3ϱ2lεε7+λ3δ2lεε8+λ3δ2lεε9+λ3γ2lεε10+λ3γ2lεε11+λ22+λ1α2,m4=λ1ϱ2ε1ε+λ2ϱ2ε12+λ2ϱ2ε13,m5=λ2ϱ2εε2+λ2δ2εε3+λ2δ2εε4+λ2γ2εε5+λ2γ2εε6+λ2ϱ2ε12+λ3δε142,
we have(63)λ1ϱ2ε1lεuNx,τL2Ω2+λ1β2uτNx,τL2Ω2+λ2α2+λ3β2uττNx,τL2Ω2+λ32−λ12−λ22uτττNx,τL2Ω2+λ1ϱ2∇uNx,τL2Ω2+λ1γ2+λ2δ2∇uτNx,τL2Ω2+−λ3ϱ2εε7−λ3δ2εε8−λ3δ2εε9−λ3γ2εε10−λ3γ2εε11−λ2ϱ2ε13−λ3δ2ε14+λ3γ2∇uττNx,τL2Ω2≤λ1δ2Ω∂ΩT+λ2δ2ε4Ω∂Ω+λ3δ2ε9Ω∂Ω+m1uNx,0L2Ω2λ1γ2Ω∂ΩT+λ2γ2ε6Ω∂Ω+λ3γ2ε11Ω∂Ω+λ12+λ1α2+λ1β2+m2utNx,0L2Ω2+λ22+λ1α2+λ2α2−λ3β2+m3uttNx,0L2Ω2+λ12+λ22+λ32utttNx,0L2Ω2+λ1ϱ2+λ2ϱ2+λ3ϱ2+m4∇uNx,0L2Ω2+λ2ϱ2+λ3δ2+λ1γ2+λ2δ2+m5∇utNx,0L2Ω2,+λ3ϱ2+λ3δ2−λ3γ2∇uttNx,0L2Ω2+γ1+m1∫0τuNx,tL2Ω2dt+γ2+m1+m2∫0τutNx,tL2Ω2dt+γ3+m2+m3∫0τuttNx,tL2Ω2dt+λ12+λ2−λ3α+m3∫0τutttNx,tL2Ω2dt+λ1ϱ2ε+m4∫0τ∇uNx,tL2Ω2dt+γ4+m4+m5∫0τ∇utNx,tL2Ω2dt+γ5+m5∫0τ∇uttNx,tL2Ω2dt,
where(64)γ1=λ1ϱ2ε1TΩ∂Ω+λ1ϱ2lε+Ω∂Ω+λ1δ+λ2ϱ2Ω∂Ω+λ2ϱ2ε2TΩ∂Ω+λ3ϱ2ε7TΩ∂Ω+λ3ϱ2Ω∂Ω,γ2=λ1δ+λ2ϱ2lε+λ1δ2lε+λ1γ+λ2δ2lε+Ω∂Ω+λ1γ2lε+λ2γ2lε+λ3δ2Ω∂Ω,γ3=λ2γ2Ω∂Ω+λ3ϱ2lε+λ3δ2lε+λ3γ2lε+Ω∂Ω+λ12+λ1α−λ2β,γ4=λ1δ+λ2ϱ2ε+λ1δ2ε+λ1γ+λ2δ2ε+λ1γ2ε+λ2γ2ε+λ3ϱ2+λ2ρ−λ1δ,γ5=λ3δ2ε+λ3γ2ε+λ3ϱ2+λ3δ−λ2γ.
Choosingε7,ε8,ε9,ε10,ε11,ε113, and ε14 sufficiently large
(65)λ3ϱ2εε7+λ3δ2εε8+λ3δ2εε9+λ3γ2εε10+λ3γ2εε11+λ3δ2ε14+λ2ϱ2ε13<λ3γ2,the relation (80) reduces to
(66)uNx,τL2Ω2+∇uNx,τL2Ω2+uτNx,τL2Ω2+∇uτNx,τL2Ω2+uττNx,τL2Ω2+∇uττNx,τL2Ω2+uτττNx,τL2Ω2≤D∫0τuNx,tL2Ω2+∇uNx,tL2Ω2+utNx,tL2Ω2+∇utNx,tL2Ω2+uttNx,tL2Ω2+∇uttNx,tL2Ω2+utttNx,tL2Ω2dt+DuNx,0W21Ω2+utNx,0W21Ω2+uttNx,0W21Ω2+utttNx,0L2Ω2,where
(67)D≔maxλ1δ/2Ω∂ΩT+λ2δ/2ε4Ω∂Ω+λ3δ/2ε9Ω∂Ω+m1,λ1γ/2Ω∂ΩT+λ2γ/2ε6Ω∂Ω+λ3γ/2ε11Ω∂Ω+λ1/2+λ1α/2+λ1β/2+m2,λ2/2+λ1α/2+λ2α/2−λ3β/2+m3,λ1/2+λ2/2+λ3/2λ1ϱ/2+λ2ϱ/2+λ3ϱ/2+m4,λ2ϱ/2+λ3δ/2+λ1γ/2+λ2δ/2+m5,λ3ϱ/2+λ3δ/2−λ3γ/2,γ1+m1,γ2+m1+m2,γ3+m2+m3,λ1/2+λ2−λ3α+m3,λ1ϱ/2ε+m4,γ4+m4+m5,γ5+m5minλ1ϱ/2ε1lε,λ1β/2,λ2α/2+λ3β/2,λ3/2−λ1/2−λ2/2,λ1ϱ/2,λ1γ/2+λ2δ/2,,−λ3ϱ/2ε/ε7−λ3δ/2ε/ε8−λ3δ/2ε/ε9−λ3γ/2ε/ε10−λ3γ/2ε/ε11−λ2ϱ/2ε13−λ3δ/2ε14−λ3γ/2.
Applying the Gronwall inequality to (60) and then integrating from 0 to τ, it appears that
(68)uNx,tW21Qτ2+utNx,tW21Qτ2+uttNx,tW21Qτ2≤DeDTu0xW21Ω2+u1xW21Ω2+u2xL2Ω2+u3xL2Ω2.
We deduce from (84) that
(69)uNx,tw21Qτ2+utNx,tw21Qτ2+uttNx,tw21Qτ2≤A.
Therefore, the sequenceuNN≥1 is bounded in VQT, and we can extract from it a subsequence for which we use the same notation which converges weakly in VQT to a limit function ux,t; we have to show that ux,t is a generalized solution of (4). Since uNx,t→ux,t in L2QT and uNx,0→ζx in L2Ω, then ux,0=ζx.
Now to prove that (3) holds, we multiply each relation in (15) by a function plt∈W210,T,plt=0, then add up the obtained equalities ranging from l=1 to l=N, and integrate over t on 0,T. If we let ηN=∑k=1NpktZkx, then we have
(70)−utttN,ηtNL2QT−αuttN,ηtNL2QT−βutN,ηtNL2QT+ϱ∇uN,∇ηNL2QT+δ∇utN,∇ηNL2QT−γ∇utN,∇ηtNL2QT=ϱ∫∂Ω∫0TηNx,t∫0t∫ΩuNξ,τdξdτdtdsx+δ∫∂Ω∫0TηNx,t∫ΩuNξ,tdξdtdsx−δ∫∂Ω∫0TηNx,t∫ΩuNξ,0dξdtdsx−γ∫0T∫∂ΩηtN∫ΩuNξ,tdξdsxdt+γ∫0T∫∂ΩηtN∫ΩuNξ,0dξdsxdt−γΔutNx,0,ηN0L2Ω+utttNx,0,ηN0L2Ω+αuttNx,0,ηN0L2Ω+βuttNx,0,ηN0L2Ω,for all ηN of the form ∑k=1NpltZkx.
Since(71)∫0t∫ΩuNξ,τ−uξ,τdξdτ≤TΩuN−uL2QT,∫0TηNx,t∫ΩutNξ,t−utξ,tdξdt≤Ω∫0TηNx,t2dt1/2utN−utL2QT,∫0TηNx,t∫ΩuNξ,0−uξ,0dξdt≤Ω∫0TηNx,t21/2uNx,0L2QT,uN−uL2QT→0,asN→∞,therefore, we have
(72)ϱ∫∂Ω∫0TηNx,t∫0t∫ΩuNξ,τdξdτdtdsx,→ϱ∫∂Ω∫0Tηx,t∫0t∫Ωuξ,τdξdτdtdsx,δ∫∂Ω∫0TηNx,t∫ΩuNξ,tdξdtdsx,→δ∫∂Ω∫0Tηx,t∫Ωuξ,tdξdtdsx,−δ∫∂Ω∫0TηNx,t∫ΩuNξ,0dξdtds,→−δ∫∂Ω∫0Tηx,t∫Ωuξ,0dξdtds,−γ∫0T∫∂ΩηtN∫ΩuNξ,tdξdsxdt,→−γ∫0T∫∂Ωηt∫Ωuξ,tdξdsxdt,γ∫0T∫∂ΩηtN∫ΩuNξ,0dξdsxdt,→γ∫0T∫∂Ωηt∫Ωuξ,0dξdsxdt.
Thus, the limit functionu satisfies (3) for every ηN=∑k=1NpltZkx. We denote by ℚN the totality of all functions of the form ηN=∑k=1NpltZkx, with plt∈W210,T, plt=0.
But∪l=1NℚN is dense in WQT, and then relation (3) holds for all u∈WQT. Thus, we have shown that the limit function ux,t is a generalized solution of problem (4) in VQT.


## 4. Uniqueness of Solution

Theorem 3.
The problem (4) cannot have more than one generalized solution in VQT.Proof.
Suppose that there exist two different generalized solutionsu1∈VQT and u2∈VQT for the problem (1). Then, the difference U=u1−u2 solves
(73)Utttt+αUttt+βUtt−ϱΔU−δΔUt−γΔUtt=0,Ux,0=Utx,0=Uttx,0=Utttx,0=0∂u∂η=∫0t∫Ωuξ,τdξdτ,x∈∂Ω,and (3) gives
(74)−Uttt,vtL2QT−αUtt,vtL2QT−βUt,vtL2QT+ϱ∇U,∇vL2QT+δ∇Ut,∇vL2QT−γ∇Ut,∇vtL2QT=ϱ∫0T∫∂Ωv∫0t∫Ωuξ,τdξdτdsxdt+δ∫0T∫∂Ωv∫Ωuξ,tdξdsxdt−γ∫0T∫∂Ωvt∫Ωuτξ,tdξdtdsxdt.
Consider the function(75)vx,t=∫tτUx,sds,0≤t≤τ,0,τ≤t≤T.
It is obvious thatv∈WQT and vtx,t=−Ux,t for all t∈0,τ. Integration by parts in the left hand side of (75) gives
(76)−Uttt,vtL2QT=Uττx,τ,Ux,τL2Ω−12Uτx,τL2Ω2,(77)−αUtt,vtL2QT=αUτx,τ,Ux,τL2Ω−α∫0τUtx,tL2Ω2dt,(78)−βUt,vtL2QT=β2Ux,τL2Ω2,(79)ϱ∇U,∇vL2QT=ϱ2∇vx,0L2Ω2,(80)δ∇Ut,∇vL2QT=δ∫0τ∇vtx,tL2Ω2dt,(81)−γ∇Ut,∇vtL2QT=γ2∇Ux,τL2Ω2,
Plugging (76)–(95) into (88), we obtain
(82)Uττx,τ,Ux,τL2Ω+αUτx,τ,Ux,τL2Ω+β2Ux,τL2Ω2+ϱ2∇vx,0L2Ω2+γ2∇Ux,τL2Ω2−12Uτx,τL2Ω2=α∫0τUtx,tL2Ω2dt−δ∫0τ∇vtx,tL2Ω2dt+ϱ∫0T∫∂Ωv∫0t∫ΩUξ,τdξdτdsxdt+δ∫0T∫∂Ωv∫ΩUξ,tdξdsxdt−γ∫0T∫∂Ωvt∫ΩUξ,tdξdsdt.
Now, since(83)v2x,t=∫tτUx,sds2≤τ∫0τU2x,sds,then
(84)vL2Qτ2≤τ2UL2Qτ2≤T2UL2Qτ2.
Using the trace inequality, the right-hand side of (96) can be estimated as follows:
(85)ϱ∫0T∫∂Ωv∫0t∫ΩUξ,τdξdτdsxdt≤ϱ2T2lε+Ω∂Ω∫0τUx,tL2Ω2dt+ϱ2ε∫0τ∇vx,tL2Ω2dt,δ∫0T∫∂Ωv∫ΩUξ,tdξdsxdt≤δ2T2lε+Ω∂Ω∫0τUx,tL2Ω2dt+δ2ε∫0τ∇vx,tL2Ω2dt,−γ∫0T∫∂Ωvt∫ΩUξ,tdξdsdt=γ∫0T∫∂Ωv∫ΩUtξ,tdξdsdt=γ∫0τ∫∂Ωv∫ΩUtξ,tdξdsdt≤γΩ∂Ω2UtL2Qτ2+η22ε∇vL2Qτ2+γ2lεT2UL2Qτ2.
Combining the relations (98)-(101) and (96), we get
(86)Uττx,τ,Ux,τL2Ω+αUτx,τ,Ux,τL2Ω+β2Ux,τL2Ω2+ϱ2∇vx,0L2Ω2+γ2∇Ux,τL2Ω2−12Uτx,τL2Ω2≤ϱ2T2lε+Ω∂Ω+δ2T2lε+Ω∂Ω+γ2lεT2∫0τUx,tL2Ω2dt+α+γΩ∂Ω2∫0τUtx,tL2Ω2dt+12∫0τUttx,tL2Ω2dt+ϱ+δ+γ2ε∫0τ∇vx,tL2Ω2dt.
Next, multiplying the differential equation in (73) by Uttt and integrating over Qτ=Ω×0,τ, we obtain
(87)Utttt,UtttL2Qτ+αUttt,UtttL2Qτ+βUtt,UtttL2Qτ−ϱΔU,UtttL2Qτ−δΔUt,UtttL2Qτ−γΔUt,UtttL2Qτ=0.
An integration by parts in (102) yields
(88)Utttt,UtttL2Qτ=12Uτττx,τL2Ω2,(89)αUttt,UtttL2Qτ=α∫0τUtttx,tL2Ω2dt,(90)βUtt,UtttL2Qτ=β2Uττx,τL2Ω2,(91)−ϱΔU,UtttL2Qτ=ϱ∇Ux,τ,∇Uττx,τL2Ω−ϱ2∇Uτx,τL2Ω2−ϱ∫∂ΩUττx,τ∫0τ∫ΩUξ,ηdξdηdsx+ϱ∫∂Ω∫0τUttx,t∫ΩUξ,tdξdtdsx,(92)−δΔUt,UtttL2Qτ=δ∇Uτx,τ,∇Uττx,τL2Ω−δ∫0τ∇Uttx,L2Ω2dt−δ∫∂ΩUττx,τ∫ΩUξ,τdξdsx+δ∫0τ∫∂ΩUttx,t∫ΩUtξ,tdξdsxdt,(93)−γΔUtt,UtttL2Qτ=γ2∇Uττx,τL2Ω2−γ∫∂ΩUττx,τ∫ΩUτξ,τdξdsx+γ∫0τ∫∂ΩUttx,t∫ΩUttξ,tdξdsxdt.
Substituting (88)–(108) into (102), we get the equality
(94)12Uτττx,τL2Ω2+β2Uττx,τL2Ω2+ϱ∇Ux,τ,∇Uττx,τL2Ω+δ∇Uτx,τ,∇Uττx,τL2Ω+γ2∇Uττx,τL2Ω2−ϱ2∇Uτx,τL2Ω2=−α∫0τUtttx,tL2Ω2dt+δ∫0τ∇Uttx,L2Ω2dt+ϱ∫∂ΩUττx,τ∫0τ∫ΩUξ,ηdξdηdsx−ϱ∫∂Ω∫0τUttx,t∫ΩUξ,tdξdtdsx+δ∫∂ΩUττx,τ∫ΩUξ,τdξdsx−δ∫0τ∫∂ΩUttx,t∫ΩUtξ,tdξdsxdt+γ∫∂ΩUττx,τ∫ΩUτξ,τdξdsx−γ∫0τ∫∂ΩUttx,t∫ΩUttξ,tdξdsxdt.
The right-hand side of (109) can be bounded as follows:(95)ϱ∫∂ΩUττx,τ∫0τ∫ΩUξ,ηdξdηdsx≤ϱ2ε1′ε∇Uττx,τL2Ω2+lεUττx,τL2Ω2+ϱ2ε1′T∂ΩΩ∫0τUx,tL2Ω2dt,−ϱ∫∂Ω∫0τUttx,t∫ΩUξ,tdξdtdsx≤ϱ2∫0τε∇Uttx,tL2Ω2+lεUttx,tL2Ω2dt+ϱ2Ω∂Ω∫0τUx,tL2Ω2dt,δ∫∂ΩUττx,τ∫ΩUξ,τdξdsx≤δ2ε2′ε∇Uττx,τL2Ω2+lεUττx,τL2Ω2+δ2ε2′TΩ∂ΩUx,τL2Ω2,−δ∫0τ∫∂ΩUttx,t∫ΩUtξ,tdξdsxdt≤δ2ε∫0τ∇Uttx,tL2Ω2dt+δ2lε∫0τUttx,tL2Ω2dt+δ2TΩ∂Ω∫0τUtx,tL2Ω2dt,γ∫∂ΩUττx,τ∫ΩUτξ,τdξdsx≤γ2ε3′ε∇Uττx,τL2Ω2+lεUττx,τL2Ω2+γ2ε3′TΩ∂ΩUτx,τL2Ω2,−γ∫0τ∫∂ΩUttx,t∫ΩUttξ,tdξdsxdt≤γ2lε∫0τUttx,tL2Ω2dt+γ2ε∫0τ∇Uttx,tL2Ω2dt+γ2TΩ∂Ω∫0τUttx,tL2Ω2dt.
So, combining inequalities (110)–(115) and equality (109), we obtain(96)12Uτττx,τL2Ω2+β2−ϱ2ε1′lε−δ2ε2′lε−δ2εεUττx,τL2Ω2−γ2ε3′TΩ∂ΩUτx,τL2Ω2−δ2ε2′TΩ∂ΩUx,τL2Ω2+γ2−ϱ2ε1′ε−δ2ε2′ε−γ2ε3′ε∇Uττx,τL2Ω2−ϱ2∇Uτx,τL2Ω2+ϱ∇Ux,τ,∇Uττx,τL2Ω+δ∇Uτx,τ,∇Uττx,τL2Ω≤−α∫0τUtttx,tL2Ω2dt+ϱ2lε+δ2lε+γ2lε+γ2TΩ∂Ω∫0τUttx,tL2Ω2dt+ϱ2ε1′T∂ΩΩ+ϱ2Ω∂Ω∫0τUx,tL2Ω2dt+δ2TΩ∂Ω∫0τUtx,tL2Ω2dt+δ+ϱ2ε+δ2ε+γ2ε∫0τ∇Uttx,L2Ω2dt.
Adding side to side (101) and (116), we obtain
(97)β2−δ2ε2′TΩ∂ΩUx,τL2Ω2+−12−γ2ε3′TΩ∂ΩUτx,τL2Ω2+β2−ϱ2ε1′lε−lεδ2ε2′−γ2ε3′lεUττx,τL2Ω2+12Uτττx,τL2Ω2+Uττx,τ,Ux,τL2Ω+αUτx,τ,Ux,τL2Ω+ϱ2∇vx,0L2Ω2+ϱ∇Ux,τ,∇Uττx,τL2Ω+δ∇Uτx,τ,∇Uττx,τL2Ω+γ2∇Ux,τL2Ω2−ϱ2∇Uτx,τL2Ω2+γ2−ϱ2ε1′ε−δ2ε2′−γ2ε3′ε∇Uττx,τL2Ω2≤ϱ2ε1′T∂ΩΩ+ϱ2Ω∂Ω+ϱ2T2lε+Ω∂Ω+δ2T2lε+Ω∂Ω+γ2lεT2∫0τUx,tL2Ω2dt+α+γΩ∂Ω2+δ2TΩ∂Ω∫0τUtx,tL2Ω2dt+12+lεϱ2+δ2lε+γ2lε+γ2TΩ∂Ω∫0τUttx,tL2Ω2dt−α∫0τUtttx,tL2Ω2dt+δ2ε+γ2ε+εϱ2+δ∫0τ∇Uttx,tL2Ω2dt+ϱ+δ+γ2ε∫0τ∇vx,tL2Ω2dt.
Now, to deal with the last term on the right-hand side of (117), we define the functionθx,t by the relation
(98)θx,t≔∫0tUx,sds.
Hence, using (89), it follows that
(99)vx,t=θx,τ−θx,t,∇vx,0=∇θx,τ,∇vL2Qτ2=∇θx,τ−∇θx,tL2Ω2≤2τ∇θx,τL2Ω2+∇θx,tL2Qτ2.
And we make use of the following inequality:(100)−α2Uτx,τL2Ω2−α2Ux,τL2Ω2≤αUτx,τ,Ux,τL2Ω,−12Uττx,τL2Ω2−12Ux,τL2Ω2≤Uττx,τ,Ux,τL2Ω,−ϱ2ε4′∇Uττx,τL2Ω2−ϱ2ε4′∇Ux,τL2Ω2≤ϱ∇Ux,τ,∇Uττx,τL2Ω,−δ2ε5′∇Uττx,τL2Ω2−δ2ε5′∇Uτx,τL2Ω2≤δ∇Uτx,τ,∇Uττx,τL2Ω,m1Ux,τL2Ω2≤m1Ux,tL2Qτ2+m1Utx,tL2Qτ2,m2Uτx,τL2Ω2≤m2Utx,tL2Qτ2+m2Uttx,tL2Qτ2,m3Uττx,τL2Ω2≤m3Uttx,tL2Qτ2+m3Utttx,tL2Qτ2,m4∇Ux,τL2Ω2≤m4∇Ux,tL2Qτ2+m4∇Utx,tL2Qτ2,m5∇Uτx,τL2Ω2≤m5∇Utx,tL2Qτ2+m5∇Uttx,tL2Qτ2.
Let(101)m1≔12+δ2ε2′TΩ∂Ω+α2m2≔1+γ2ε3′TΩ∂Ω+α2m3≔ϱ2ε1′lε+lεδ2ε2′+γ2ε3′lε+12m4≔ϱ2ε4′m5≔1+ϱ2+δ2ε5′,choosing ε1′,ε2′, ε3′, ε4′, and ε5′ sufficiently large:
(102)ϱ2ε1′ε+δ2ε2′+γ2ε3′ε+ϱ2ε4′+δ2ε5′<γ2.
Sinceτ is arbitrary, we get that ϱ/2−τεϱ+δ+γ>0; thus, inequality (117) takes the form
(103)β2Ux,τL2Ω2+12Uτx,τL2Ω2+β2Uττx,τL2Ω2+12Uτττx,τL2Ω2+ϱ2−τεϱ+δ+γ∇θx,τL2Ω2+γ2∇Ux,τL2Ω2+∇Uτx,τL2Ω2+γ2−ϱ2ε1′ε−δ2ε2′−γ2ε3′ε−ϱ2ε4′−δ2ε5′∇Uττx,τL2Ω2≤γ1′+m1∫0τUx,tL2Ω2dt+γ2′+m1+m2∫0τUtx,tL2Ω2dt+γ3′+m2+m3∫0τUttx,tL2Ω2dt+m3−α∫0τUtttx,tL2Ω2dt+εϱ+δ+γ∫0τ∇θx,tL2Ω2dt+m4∫0τ∇Ux,tL2Ω2dt+m4+m5∫0τ∇Utx,tL2Ω2dt+δ2ε+γ2ε+εϱ2+δ+m5∫0τ∇Uttx,tL2Ω2dt,where
(104)γ1′≔ϱ2ε1′T∂ΩΩ+ϱ2Ω∂Ω+ϱ2T2lε+Ω∂Ω+δ2T2lε+Ω∂Ω+γ2lεT2γ2′≔α+γΩ∂Ω2+δ2TΩ∂Ωγ3′≔12+lεϱ2+δ2lε+γ2lε+γ2TΩ∂Ω.
We obtain(105)Ux,τL2Ω2+Uτx,τL2Ω2+Uττx,τL2Ω2+Uτττx,τL2Ω2+∇Ux,τL2Ω2+∇Uτx,τL2Ω2+∇Uττx,τL2Ω2+∇θx,τL2Ω2≤D∫0τUx,tL2Ω2+Utx,tL2Ω2+Uttx,tL2Ω2+Utttx,tL2Ω2+∇Ux,tL2Ω2+∇Utx,tL2Ω2+∇Uttx,tL2Ω2+∇θx,tL2Ω2dt,where
(106)D≔maxγ1′+m1,γ2′+m1+m24,γ3′+m2+m3,m3−α,m4,m4+m5,δ/2ε+γ/2ε+εϱ/2+δ+m5,εϱ+δ+γminβ/2,1/2,,γ/2,γ/2−ϱ/2ε1′ε−δ/2ε2′−γ/2ε3′ε−ϱ/2ε4′−δ/2ε5′,,ϱ/2−τεϱ+δ+γ.
Further, applying Gronwall’s lemma to (133), we deduce that(107)Ux,τL2Ω2+Uτx,τL2Ω2+Uττx,τL2Ω2+Uτττx,τL2Ω2+∇Ux,τL2Ω2+∇Uτx,τL2Ω2+∇Uττx,τL2Ω2+∇θx,τL2Ω2≤0,∀τ∈0,ϱ2ερ+δ+γ.
We proceed in the same way for the intervalsτ∈m−1ϱ/2εϱ+δ+γ,mϱ/2εϱ+δ+γ to cover the whole interval 0,T, and thus proving that Ux,τ=0, for all τ in 0,T. Thus, the uniqueness is proved.


## 5. Conclusion

In the study of the propagation of acoustic waves, it should be noted that the Moore–Gibson–Thompson equation is one of the equations of nonlinear acoustics describing acoustic wave propagation in gases and liquids. The behavior of acoustic waves depends strongly on the medium property related to dispersion, dissipation, and nonlinear effects. It arises from modeling high-frequency ultrasound (HFU) waves (see [10, 12, 34]). In this work, we have studied the solvability of the nonlocal mixed boundary value problem for the fourth order of the Moore–Gibson–Thompson equation. Galerkin’s method was the main used tool for proving the solvability of the given nonlocal problem. In the next work, we will try to use the same method with the Hall-MHD equations which are nonlinear partial differential equation that arises in hydrodynamics and some physical applications. It was subsequently applied to problems in the percolation of water in porous subsurface strata (see for example [45–48]) by using some famous algorithms (see [49–51])



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