# Solvability for a New Class of Moore-Gibson-Thompson Equation with Viscoelastic Memory, Source Terms, and Integral Condition

Authors: Salah Mahmoud Boulaaras; Abdelbaki Choucha; Djamel Ouchenane; Asma Alharbi; Mohamed Abdalla; Bahri Belkacem Cherif
Journal: Journal of Function Spaces
 (2021)

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## Abstract

This paper deals with the existence and uniqueness of solutions of a new class of Moore-Gibson-Thompson equation with respect to the nonlocal mixed boundary value problem, source term, and nonnegative memory kernel. Galerkin’s method was the main used tool for proving our result. This work is a generalization of recent homogenous work.


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## Body

## 1. Introduction

In this contribution, we are interested to study the existence and uniqueness of solutions of the following problem(1)Lux,t=auttt+βutt−c2Δu−bΔut−∫0tht−σΔuσdσ=Fx,t,ux,0=u0x,utx,0=u1x,uttx,0=u2x,∂u∂η=∫0t∫Ωξ,τdξdτ,x∈∂Ω.Here,a and β are physical parameters, and c is the speed of sound. The convolution term ∫0tht−sΔusds reflects the memory effect of materials due to vicoelasticity, F is a given function, and h is the relaxation function satisfying(H1)h∈C1ℝ+,ℝ+ is a nonincreasing function satisfying
(2)h0>0,1−h0=l>0,where h0=G∞=∫0∞hsds>0, Gt=∫0thsds, and h′′>0.(H2)∃ζ>0 satisfying
(3)h′t≤−ζht,t≥0.(H3)(4)β−a>0The phenomena resulting from sound waves (diffraction, interference, reflection) in terms of modeling are very important. As the existence of the third derivative is very important, especially in the field of thermodynamics (EIT), the study of these models is considered the beginning of an in-depth understanding of both convergent and good behavior. From the results extracted, the equation of MGT resulted in nonlinear acoustics, for much depth, see ([1–7]) and especially [8] where equation of MGT appeared for the first time. Also, nonlinear problems of great importance can be considered [9], where Galerkin’s method was applied in solving them, for more depth ([2, 3, 10–13]). Recently, in [14], the authors studied the equation of MGT with memory. Likewise, in [1], the authors used Galerkin’s method to demonstrate the ability to solve a mixed problem of MGT equation in the absence of viscous elasticity and memory. Based on work [9] and the works we mentioned earlier, we want to prove the existence and uniqueness of a weak solution to the problem (1).We divide this paper into the following: in the second part, we put some definitions and appropriate spaces. Then, we apply Galerkin’s method to prove the existence, and in the fourth part, we demonstrate the uniqueness.


## 2. Preliminaries

We will define the spaces:VQT and WQT by
(5)VQT=u∈W21QT:ut∈W21QT,u,∇u∈Lh2QT,WQT=u∈VQT:ux,T=0,Lh2QT=u∈VQT:∫0Th∘utdt<∞,where
(6)h∘ut=∫Ω∫0tht−σut−uσ2dσdx.Consider the equation(7)auttt,vL2QT+βutt,vL2QT−c2Δu,vL2QT−bΔut,vL2QT−Δw,vL2QT=F,vL2QT,where
(8)wx,t=∫0tht−σux,σdσ,and .,.L2QT stands for the inner product in L2QT, u is supposed to be a solution of (1) and v∈WQT. Evaluation of the inner product in [9] gives
(9)−autt,vtL2QT−βut,vtL2QT+c2∇u,∇vL2QT+b∇ut,∇vL2QT+∇w,∇vL2QT=F,vL2QT+c2∫0T∫∂Ωv∫0t∫Ωvξ,τdξdτdsxdt+b∫0T∫∂Ωv∫Ωuξ,tdξdsxdt−b∫0T∫∂Ωv∫Ωu0ξdξdsxdt+au2x,vx,0L2Ω+βu1x,vx,0L2Ω+∫0T∫∂Ωv∫0t∫Ωwξ,τdξdτdsxdt.We give two useful inequalities:(i)
Gronwall inequality. Let the nonnegative integrable functionsφt,ϕt on the interval I with the nondecreasing function ht. If ∀t∈I, we have
(10)ϕt≤φt+c∫0tϕsds,where c>0, hence,
(11)ϕt≤φtexpct.(ii)
Trace inequality (see [15]). If Φ∈W12Ω where Ω is a bounded domain in ℝn with smooth boundary ∂Ω, then for any ε>0,
(12)ΦL2∂Ω2≤ε∇ΦL2Ω2+lεΦL2Ω2,where lε>0.Definition 1.
We call a generalized solution to the problem (1) for each function u∈VQT that fulfills the equation (9) for each v∈WQT.


## 3. Solvability of the Problem

In this section, we use Galerkin’s method to prove the existence of a generalized solution of our problem.Theorem 2.
Ifu0∈W21Ω, u1∈W21Ω, u2∈L2Ω, and F∈L2QT, 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,ZlΩ=δk,l.
First, we will give an approximate solution of the problem (1) in the form
(13)uNx,t=∑k=1NCktZkx,where Ckt are constants given by the conditions, for k=1,⋯,N,(14)Lux,t,ZlxL2Ω=Fx,t,ZlxL2Ωand can be determined from the relations
(15)autttN,ZlxL2Ω+βuttN,ZlxL2Ω+c2∇uN,∇ZlxL2Ω+b∇utN,∇ZlxL2Ω+∇wN,∇ZlxL2Ω=Fx,t,ZlxL2Ω+c2∫∂ΩZlx∫0t∫ΩuNξ,τdξdτdsx+b∫∂ΩZlx∫0t∫ΩuNξ,τdξdτdsx+∫∂ΩZlx∫0t∫ΩwNξ,τdξdτdsx,substitution of (13) into (15), and we find for l=1,⋯,N.
(16)∫Ω∑k=1NaCk‴tZkxZlxβCk‴tZkxZlx+c2Ckt∇Zkx.∇Zlx+bCk′t∇Zkx.∇Zlx+∫0tht−σCkσdσ∇Zkx.∇Zlxdx=Fx,t,ZlxL2Ω+c2∑k=1N∫0tCkτ∫∂ΩZlx∫ΩZkξdξdsxdτ+b∑k=1N∫0tCk′τ∫∂ΩZlx∫ΩZkξdξdsxdτ+∑k=1N∫0t∫0τhτ−σCkσdσ∫∂ΩZlx∫ΩZkξdξdsxdτ
From (15) it follows that
(17)∑k=1NaCk‴tZkx,ZlxL2Ω+βCk″tZkx,ZlxL2Ω+c2Ckt∇Zkx,∇ZlxL2Ω+bCk′t∇Zkx,∇ZlxL2Ω+∫0tht−σCkσdσ∇Zkx,∇ZlxL2Ω=Fx,t,ZlxL2Ω+c2∑k=1N∫0tCkτ∫∂ΩZlx∫ΩZkξdξdsxdτ+b∑k=1N∫0tCk′τ∫∂ΩZlx∫ΩZkξdξdsxdτ+∑k=1N∫0t∫0τhτ−σCkσdσ∫∂ΩZlx∫ΩZkξdξdsxdτ
Let(18)Zk,ZlL2Ω=δkl=1,k=l0,k≠l,∇Zk,∇ZlL2Ω=γkl,∫∂ΩZlx∫ΩZkξdξds=χkl,Fx,t,ZlxL2Ω=Flt.
Then, (17) can be written as
(19)∑k=1NaδklCk‴t+βδklCk″t+c2Cktγkl+∫0tht−σCkσdσγkl+bCk′tγkl−∫0tc2Ckτχkl−∫0τhτ−σCkσdσχkldτ=Flt.
By differentiating (two times) with respect tot, it gives
(20)∑k=1NaδklCk‴t+βδklCk‴t+c2Ck″tγkl+h0Ck′tγkl+bCk″tγkl−c2Cktχkl−bCk″tχkl−h0Cktχkl=Fl″t.(21)∑k=1NaδklCk‴0+βδklCk″.
We find a system of differential equations of fifth order with respect tot, constant coefficients, and the initial conditions (21). Hence, we obtain a Cauchy problem of linear differential equations with smooth coefficients that is uniquely solvable. Thus, ∀n, ∃uN (x) satisfying (15).
Now, we prove thatuN is sequence bounded. To do this, we multiply each equation of (15) by the appropriate Ck′t summing over k from 1 to N. Hence, by integration the result equality with respect to t from 0 to τ, and τ≤T, it gives
(22)autttN,utNL2Qτ+βutttN,utNL2Qτ+c2∇uN,∇utNL2Qτ+b∇utN,∇utNL2Qτ+∇wN,∇utNL2Qτ=F,utNL2Qτ+c2∫0τ∫0τutNx,t∫0t∫Ωξ,ηdξdηdsxdt+b∫0τ∫∂ΩutNx,t∫0t∫ΩutNξ,ηdξdηdsxdt+∫0τ∫∂ΩutNx,t∫0t∫ΩwNξ,ηdξdηdsxdt.
Evaluation of the terms on the LHS of (22) gives
(23)autttN,utNL2Qτ=auττN,uτNL2Ω−auttNx,0,utNx,0L2Ω−a∫0τuttx,tL2Ω2dt,(24)βuttN,utNL2Qτ=β2uτNx,τL2Ω2−β2utNx,0L2Ω2,(25)c2∇uN,∇utNL2Qτ=c2c∇uNx,τL2Ω2−c2c∇uNx,0L2Ω2,(26)b∇utN,∇utNL2Qτ=b∫0τ∇utNx,tL2Ω2dt,(27)∇wN,∇utNL2Qτ=12h∘∇uNτ−12Gτ∇uNx,τL2Ω2−12∫0τh′Ntdt+12∫0τht∇uNx,τL2Ω2dt,(28)c2∫0τ∫∂ΩutN∫0t∫ΩuNξ,ηdξdηdsxdt=c2∫∂ΩuNx,τ∫0τ∫ΩuNξ,tdξdtdsx−c2∫∂Ω∫0τuNx,t∫ΩuNξ,tdξdtdsx,So,
(29)b∫0τ∫∂ΩutN∫0t∫ΩutNξ,ηdξdηdsxdt=b∫∂Ω∫0τutNx,t∫ΩuNξ,tdξdtdsx−b∫∂Ω∫0τutNx,t∫ΩuNξ,0dξdtdsx.Thus,
(30)∫0τ∫∂ΩutN∫0t∫ΩwNξ,ηdξdηdsxdt=∫∂ΩuNx,τ∫0τ∫ΩuNξ,tdξdtdsx−∫∂Ω∫0τuNx,t∫ΩwNξ,tdξdtdsx.
Taking into account the equalities (23)–(30) in (22), we end up with
(31)auττN,uτNL2Ω+β2uτNx,τL2Ω2+c22∇uNx,τL2Ω212h∘∇uNτ−12Gτ∇uNx,τL2Ω2=auttNx,0,utNx,0L2Ω+β2uτNx,0L2Ω2+c22∇uNx,0L2Ω2+a∫0τuttx,tL2Ω2dt+b∫0τ∇utNx,tL2Ω2dt+12∫0τh′Ntdt−12∫0τht∇uNx,tL2Ω2dt+c2∫∂ΩuNx,τ∫0τ∫ΩuNξ,tdξdtdsx−c2∫∂Ω∫0τuNx,t∫Ωξ,tdξdtdsx+b∫∂Ω∫0τutNx,t∫ΩuNξ,tdξdtdsx−b∫∂Ω∫0τutNx,t∫ΩuNξ,0dξdtdsx+∫∂ΩuNx,τ∫0τ∫Ωξ,tdξdtdsx−∫∂Ω∫0τuNx,t∫ΩwNξ,tdξdtdsx+F,utNL2Qτ
Now, multiplying the equations of (15) by Ck″t, collect them from 1 to N and integrating the result with respect to t from 0 to τ, and τ≤T, we find
(32)autttN,uttNL2Qτ+βuttN,uttNL2Qτ+c2∇uN,∇uttNL2Qτ+b∇utN,∇uttNL2Qτ+∇wN,∇uttNL2Qτ=F,uttNL2Qτ+c2∫0τ∫∂ΩuttNx,t∫0t∫ΩuNξ,ηdξdηdsxdt+b∫0τ∫∂ΩuttNx,t∫0t∫ΩuηNξ,ηdξdηdsxdt−∫0τ∫∂ΩuttNx,t∫0t∫ΩwNξ,ηdξdηdsxdt
With the same reasoning in (22), we find
(33)autttN,uttNL2Qτ=a2uττNx,τL2Ω2−a2uttNx,0L2Ω2,(34)βuttN,uttNL2Qτ=β∫0τuttNx,tL2Ω2dt,(35)c2∇uN,∇uttNL2Qτ=c2∇uNx,τ,∇uτNx,τL2Qτ−c2∇uNx,0,∇uτNx,0L2Ω−c2∫0τ∇utNx,tL2Ω2dt,(36)b∇utN,∇uttNL2Qτ=b2∇uτNx,τL2Ω2−b2∇utNx,0L2Ω2,(37)∇wN,∇uttNL2Qτ=12h′Nτ∇uNx,τL2Ω2−2∇wNτ,∇uτNL2Ω+12∫0τh″Ntdt−12∫0τh′t∇uNx,tL2Ω2dt,(38)c2∫0τ∫∂ΩuttN∫0τ∫ΩuNξ,ηdξdηdsxdt=c2∫∂ΩuτNx,τ∫0τ∫ΩuNξ,tdξtdsx−c2∫∂Ω∫0τutNx,t∫ΩuNξ,tdξdtdsx,(39)b∫0τ∫∂ΩuttNx,t∫0t∫ΩuηNξ,ηdξdηdsxdt=b∫∂ΩuτNx,τ∫ΩuNξ,τdξdsx−b∫∂ΩuτNx,τ∫ΩuNξ,0dξdsx−b∫∂Ω∫0τutNx,t∫ΩutNξ,tdξdtds.(40)∫0τ∫∂ΩuttN∫0t∫ΩwNξ,ηdξdηdsxdt=∫∂ΩuτNx,τ∫0τ∫ΩwNξ,tdξdtdsx−∫∂Ω∫0τutNx,t∫ΩwNξ,tdξdtdsx.
A substitution of equalities (33)–(40) in (22) gives
(41)a2uττNx,τL2Ω2+b2∇uτNx,τL2Ω2+c2∇uNx,τ,∇uτNx,τL2Ω−12h′Nτ+hτ∇uNx,τL2Ω2−2∇wNτ,∇uτNL2Ω=a2uttNx,0L2Ω2+c2∇uNx,0,∇utNx,0L2Ω+b2∇utNx,0L2Ω2−β∫0τuttNx,tL2Ω2dt+c2∫0τ∇utx,tL2Ω2−12∫0τh″Ntdt+12∫0τh′t∇uNx,tL2Ω2dt+c2∫∂ΩuτNx,τ∫0τ∫ΩuNξ,tdξdtdsx−c2∫∂Ω∫0τutNx,t∫ΩuNξ,tdξdtdsx+b∫∂ΩuτNx,τ∫ΩuNξ,τdξdsx−b∫∂ΩuτNx,τ∫ΩuNξ,0dξdsx−b∫∂Ω∫0τutNx,t∫ΩutNξ,tdξdtdsx∫∂ΩuτNx,t∫0τ∫ΩwNξ,tdξdtdsx−∫∂Ω∫0τutNx,t∫ΩwNξ,tdξdtdsx+F,uttNL2Qτ.
Multiplying (32) by λ and using (41), we get
(42)λauττN,uτNL2Ω+λβ2uτNx,τL2Ω2+λc22∇uNx,τL2Ω2λ2h∘∇uNτ−λ2Gτ∇uNx,τL2Ω2a2uττNx,τL2Ω2+b2∇uτNx,τL2Ω2+c2∇uNx,τ,∇uτNx,τL2Ω−12h′Nτ+hτ∇uNx,τL2Ω2−2∇wNτ,∇uτNL2Ω=λF,utNL2Qτ+F,uttNL2Qτ+λauttNx,0,utNx,0L2Ω+λa−β∫0τuttx,tL2Ω2dt+c2−λb∫0τ∇utNx,tL2Ω2dt+λ2∫0τh′Ntdt−λ2∫0τht∇uNx,tL2Ω2dt+λc2∫∂ΩuNx,τ∫0τ∫ΩuNξ,tdξdtdsx+λc22∇uNx,0L2Ω2−λc2∫∂Ω∫0τuNx,t∫ΩuNξ,tdξdtdsx+λβ2utNx,0L2Ω2+λb−c2∫∂Ω∫0τutNx,t∫ΩuNξ,tdξdtdsx−λb∫∂Ω∫0τutNx,t∫ΩuNξ,0dξdtdsx+λ∫∂ΩuNx,τ∫0τ∫ΩwNξ,tdξdtdsx−λ∫∂Ω∫0τuNx,t∫ΩwNξ,tdξdtdsx+a2uttNx,0L2Ω2+c2∇uNx,0,∇utNx,0L2Ω+b2∇utNx,0L2Ω2−12∫0τh″Ntdt+12∫0τh′t∇uNx,tL2Ω2dt+c2∫∂ΩuτNx,τ∫0τ∫ΩuNξ,tdξdtdsx+b∫∂ΩuτNx,τ∫ΩuNx,τdξdsx−b∫∂ΩuτNx,τ∫ΩuNξ,0dξdsx−b∫∂Ω∫0τutNξ,tdξdtdsx∫∂ΩuτNx,τ∫0τ∫ΩwNξ,tdξdtdsx−∫∂Ω∫0τuτNx,t∫ΩwNξ,tdξdtdsx.where 0<λ<1.
With the help of Cauchy and the trace inequalities, we can estimate all the terms in the RHS of (42) that gives
(43)c2λ∫∂ΩuNx,τ∫0τ∫ΩuNξ,tdξdtdsx≤c2λ2ε1ε∇uNx,τL2Ω2+lεuNx,τL2Ω2+c2λ2ε1TΩ∂Ω∫0τuNx,tL2Ω2dt,(44)−c2λ∫∂Ω∫0τuNx,t∫ΩuNξ,tdξdtdsx≤c2λ2ε∫0τ∇uNx,tL2Ω2dt+c2λ2lε+Ω∂Ω∫0τuNx,tL2Ω2dt,(45)bλ−c2∫∂Ω∫0τutNx,t∫ΩuNξ,τdξdtdsx≤bλ−c22ε∫0τ∇utNx,tL2Ω2dt+lε∫0τ∇utNx,tL2Ω2dt+bλ−c22Ω∂Ω∫0τuNx,tL2Ω2dt,(46)−bλ∫∂Ω∫0τutNx,t∫ΩuNξ,0dξdtdsx≤bλ2ε∫0τ∇utNx,tL2Ω2dt+lε∫0τutNx,tL2Ω2dt+bλ2Ω∂ΩTuNx,0L2Ω2,(47)c2∫∂ΩuτNx,τ∫0τ∫ΩuNξ,tdξdtdsx≤c22εε2∇uτNx,τL2Ω2+lεε2uτNx,τL2Ω2+c22ε2Ω∂ΩT∫0τuNx,tL2Ω2dt,(48)+b∫∂ΩuτNx,τ∫ΩuNξ,τdξdsx≤b2ε3ε∇uτNx,τL2Ω2+lεuτNx,τL2Ω2+b2ε3Ω∂ΩuNx,τL2Ω2,(49)−b∫∂ΩuτNx,τ∫ΩuNξ,0dξdsx≤b2ε4ε∇uτNx,τL2Ω2+lεuτNx,τL2Ω2+b2ε4Ω∂ΩuNx,τL2Ω2,(50)−b∫∂ΩuτNx,t∫ΩuτNξ,τdξdtdsx≤b2∫0τ∇uτNx,τL2Ω2dt+b2lεΩ∂Ω∫0τuτNx,tL2Ω2dt,(51)−λa2uττNx,τL2Ω2−λa2uττNx,τL2Ω2≤λauττNx,τ,uτNx,τL2Ω,(52)−c2ε72∇uNx,τL2Ω2−c22ε7∇uτNx,τL2Ω2≤c2uNx,τ,∇uτNx,τL2Ω,(53)λauttNx,0,uτNx,0≤λa2uttNx,0L2Ω2+λa2uτNx,0L2Ω2,(54)c2∇uNx,0,uτNx,0L2Ω≤c22∇uNx,0L2Ω2+c22∇uτNx,0L2Ω2.(55)−Gτ21ε8+1ε9∇uNx,τL2Ω2−ε82h∘∇uNτ−ε92Gτ∇uNx,τL2Ω2≤2∇wNx,τ,∇uτNx,τL2Ω,(56)∫∂ΩuτNx,τ∫0τ∫ΩwNξ,tdξdtdsx≤12ε5+12ε5′ε∇uτNx,τL2Ω2+lεuτNx,τL2Ω2+Ω∂ΩTε52∫0τh∘uNtdt+ε5′2h0∫0τuNx,tL2Ω2,(57)λ∫∂ΩuNx,τ∫0τ∫ΩwNξ,tdξdtdsx≤λ12ε6+12ε6′ε∇uNx,τL2Ω2+lεuNx,τL2Ω2+λΩ∂ΩTε62∫0τh∘uNtdt+ε6′2h0∫0τuNx,τL2Ω2,(58)∫∂Ω∫0τutNx,τ∫ΩwNξ,tdξdtdsx≤ε∫0τ∇utNx,tL2Ω2dt+lε∫0τ∇utNx,tL2Ω2dt+12Ω∂ΩT∫0τh∘uNtdt+h0∫0τuNx,tL2Ω2dt,(59)λ∫∂Ω∫0τuNx,t∫ΩuNξ,tdξdtdsx≤λε∫0τ∇uNx,tL2Ω2dt+lε∫0τuNx,tL2Ω2dt+λ2Ω∂ΩT∫0τh∘uNtdt+h0∫0τuNx,tL2Ω2dt,(60)λF,uτNL2Qτ≤λ2∫0τFx,tL2Ω2dt+12∫0τuttx,tL2Ω2dt.
Combining inequalities (45)-(60) and equality (44) and make use of the following inequality
(61)m1uNx,τL2Ω2≤m1uNx,tL2Ω2+m1uτNx,tL2Ω2+m1uNx,0L2Ω2,m2utNx,τL2Ω2≤m2utNx,tL2Ω2+m2uttNx,tL2Ω2+m2utNx,0L2Ω2,m3∇uNx,τL2Ω2≤m3∇uNx,tL2Qτ2+m3∇utNx,tL2Qτ2+m3∇uNx,0L2Ω2,h∘uNτ≤12h∘uNtdt+h02utNx,tL2Qτ2−∫0τh′Ntdt,h∘∇uNτ≤12∫0τh∘∇uNtdt+h02∇utNx,tL2Qτ2−∫0τh′Ntdt,where
(62)m1≔c2λε1lε+b2ε3Ω∂Ω+λ12ε6+12ε6′lε,m2≔lεc22ε2+b2ε2+b2ε4+12ε5+12ε5′,m3≔c2ε72+h0+c2λ2ε1+c2ε92+λ12ε6+12ε6′ε,and we have
(63)c2λ2ε1lεuNx,τL2Ω2+λ2β−autNx,τL2Ω2+λ2c2−Gτ∇uNx,τL2Ω2+b2−c2ε2ε2−bε2ε3−bε2ε4−c22ε7−ε2ε5−ε2ε5′−Gτ21ε8+1ε9∇uτNx,τL2Ω2+a2−λa2uττNx,τL2Ω2+λ+12h∘∇uNτ+h∘uNτ≤γ1∫0τuNx,tL2Ω2dt+γ2∫0τutNx,tL2Ω2dt+c2λ2ε+λε+m3⏟α1∫0τ∇uNx,tL2Ω2dt+γ3∫0τ∇uτNx,tL2Ω2dt+λ+3ξ2⏟α2∫0τh∘∇uNτ+γ4∫0τh∘uNτ+λa+β+m22⏟α3∫0τuttNx,tL2Ω2dt+bλ2Ω∂ΩT+b2ε4Ω∂Ω+m1⏟α4uNx,0L2Ω2+λa2+λβ2+m2⏟α5utNx,0L2Ω2+λ+12∫0τFx,tL2Ω2dt+λ2c2+c22+12h0+m3⏟α6∇uNx,0L2Ω2+b2+c22⏟α7∇utNx,0L2Ω2+λa2+a2⏟α7∇uttNx,0L2Ω2,where
(64)γ1≔c2λ2ε1T∂ΩΩ+c2λ2lε+∂ΩΩ+bλ+c22∂ΩΩ+c22ε2+c22ε5′h0+λ+12∂ΩΩT+λlε+m1,γ2≔bλ+c22lε+bλ2lε+b2lε+∂ΩΩ+lε+h02ξ+λ2+m2,(65)γ3≔c2+λb+bλ+c22ε+bλ+12ε+λε+h0λ2+h02ξ+m3,γ4≔ε52+λ+12∂ΩΩT+3ξ2.
Choosingε2,ε3,ε4,ε5,ε5′,ε7,ε8 and ε9 sufficiently large
(66)α0=b2−c2ε2ε2−bε2ε3−bε2ε4−c22ε7−ε2ε5−ε2ε5′−Gτ21ε8+1ε9>0.By using (2)-(4), the relation (64) reduces to
(67)uNx,τL2Ω2+∇uNx,τL2Ω2+uτNx,τL2Ω2+∇uτNx,τL2Ω2+uττNx,τL2Ω2+h∘uNτ+h∘∇uNτ≤D∫0τuNx,tL2Ω2+∇uNx,tL2Ω2+utNx,tL2Ω2+∇utNx,tL2Ω2+uttNx,tL2Ω2+h∘uNt+h∘∇uNt+FL2Ω2dt+DuNx,0W21Ω2+utNx,0W21Ω2+uttNx,0L2Ω2,≤D∫0τuNx,tL2Ω2+uttNx,tL2Ω2+h∘uNt+h∘∇uNt+FL2Ω2dt+DuNx,0W21Ω2+utNx,0W21Ω2+uttNx,0L2Ω2+h∘uN0+h∘∇uN0,where
(68)D≔maxαi,i=1⋯8minc2λ/2ε1lε,λ/2β−a,λ/2c2−Gτ,a/21−λ,λ+1/2,1,α0.
Using the inequality of Gronwall to (67) and integrating the result from 0 to τ that gives
(69)uNx,tW21Qτ2+utNx,tW21Qτ2+uNx,th,W21Qτ2+uttNx,tL2Qτ2≤DeDTu0xW21Ω2+u1xW21Ω2+u2xL2Ω2+FL2Ω2.where
(70)ux,th,W21Qτ2≔∫0τh∘utdt+∫0τh∘∇utdt,ux,0h,W21Qτ2=0.
We deduce from (69) that
(71)uNx,tw21Qτ2+utNx,tw21Qτ2+uNx,th,W21Qτ2+uttNx,tL2Qτ2≤A.
Hence,uNN≥1 is sequence 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, and we have to show that ux,t is a generalized solution of (1). Since uNx,t⟶ux,t in L2QT and uNx,0⟶ζx in L2Ω, then ux,0=ζx.
Now to prove that (15) holds, we multiply each of the relations (15) by a function plt∈W210,T, plT=0. Hence, collect them the obtained equalities ranging from l=1 to l=N and integrating the result over t on 0,T. If we let ηN=∑k=1NpltZkx, then we have
(72)auttN,ηtNL2QT+βutN,ηtNL2QT−c2∇uN,∇ηNL2QT−b∇utN,∇ηNL2QT−∇wN,∇ηNL2QT=auttNx,0,ηN0L2Ω+βutNx,0,ηN0L2Ω+c2∫∂Ω∫0TηNx,t∫0t∫ΩuNξ,τdξdτdtdsx+b∫∂Ω∫0TηNx,t∫ΩuNξ,τdξdtdsx−b∫∂Ω∫0TηNx,t∫ΩuNξ,0dξdtdsx+∫∂Ω∫0TηNx,t∫0t∫ΩwNξ,τdξdτdtdsx+F,ηtNL2QT,for all ηN of the form ∑k=1NpltZkx and α>0.
Since(73)∫0t∫ΩuNξ,τ−uξ,τdξdτ≤TΩuN−uL2QT,uN−uL2QT→0asN→∞.Thus, the limit function u satisfies (15) for every ηN=∑k=1NpltZkx.
We define the totality of all functions of the formηN=∑k=1NpltZkx by ℚN, with plt∈W210,T, plT=0.
But∪l=1NℚN is dense in WQT, hence the relation (15) holds ∀u∈WQT. Then, we have shown that the limit function ux,t is a generalized solution of problem (1) in VQT.


## 4. Uniqueness of the Problem

Theorem 3.
The problem (1) cannot have more than one generalized solution in VQT.Proof.
Suppose that∃u1,u2∈VQT two different generalized solutions for the problem (1). Hence, the difference U=u1−u2 solves
(74)aUttt+βUtt−c2ΔU−bΔUt−∫0tht−sΔUsds=0,Ux,0=Utx,0=Uttx,0=0,∂U∂η=∫0t∫ΩUξ,τdξdτ,x∈∂Ω,and (9) gives
(75)aUtt,vtL2QT+βUt,vtL2QT+c2∇U,∇vL2QT+b∇Ut,∇vL2QT+∇Wt,∇vL2QT=−c2∫0T∫∂Ωv∫0t∫ΩUξ,τdξdτdsxdt−b∫0T∫∂Ωv∫ΩUξ,τdξdsxdt−∫0T∫∂Ωvt∫ΩWξ,τdξdsxdt.where
(76)Wx,t=∫0tht−σUx,σdσ.
Let the function(77)vx,t=∫tτUx,sds0≤t≤τ,0,τ≤t≤T.
It is obvious thatv∈WQT and vtx,t=−Ux,t for all t∈0,τ. By integration by parts in the LHS of (75) that yields
(78)−aUtt,vtL2QT=aUτx,τ,Ux,τL2Ω−a∫0τUtx,tL2Ω2dt,(79)−βUt,vtL2QT=β2Ux,τL2Ω2,(80)c2∇U,∇vL2QT=c22∇vx,0L2Ω2,(81)∇W,∇vL2QT≤h0∫0τ∇vx,tL2Ω2dt+h02∫0τ∇Ux,tL2Ω2dt+h02∫0τh0∇Utdt,(82)b∇Ut,∇vL2QT=b∫0τ∇vtx,tL2Ω2dt.
Plugging (78)–(82) into (75), we obtain
(83)aUτx,τ,Ux,τL2Ω+β2Ux,τL2Ω2+c22∇vx,0L2Ω2=a∫0τUτx,tL2Ω2dt−b∫0τvtx,tL2Ω2dth0∫0τ∇vx,tL2Ω2dt+h02∫0τ∇Ux,tL2Ω2dt+h02h0∇Utdt−∫0T∫∂Ωv∫0t∫ΩWξ,τdξdτdsxdt−c2∫0T∫∂Ωv∫0t∫ΩUξ,τdξdτdsxdt−b∫0T∫∂Ωv∫ΩUξ,τdξdsxdt.
Now since(84)v2x,t=∫tτUx,sds2≤τ∫0τU2x,sds,then
(85)vL2QT2≤τ2UL2QT2≤T2UL2QT2.
Applying the inequality of the trace, the RHS of (83) gives
(86)c2∫0T∫∂Ωv∫0t∫ΩUξ,τdξdτdsxdt≤c22T2lε+Ω∂Ω∫0τUx,tL2Ω2dt+c22ε∫0τ∇vx,tL2Ω2dt,(87)b∫0T∫∂Ωv∫ΩUξ,tdξdsxdt≤b2T2lε+Ω∂Ω∫0τUx,tL2Ω2dt+b2ε∫0τ∇vx,tL2Ω2dt,(88)∫0T∫∂Ωv∫0t∫ΩWξ,τdξdτdsxdt≤12T2lε+Ω∂Ω∫0τh∘Utdt+12T2h∘lε+Ω∂Ω∫0τUx,tL2Ω2dt+ε∫0τ∇vx,tL2Ω2dt.
Combining the relations (86)–(83) and (87)–(88), we get
(89)aUτx,τ,Ux,τL2Ω+β2Ux,τL2Ω2+c22∇vx,0L2Ω2≤c22T2lε+Ω∂Ω+b+h02T2lε+Ω∂Ω∫0τUx,tL2Ω2dt+c2+b2+1ε+h0∫0τ∇vx,tL2Ω2dt+a∫0τUtx,tL2Ω2dt+h02∫0τ∇Ux,tL2Ω2dt+h02∫0τh∘∇Utdt+12T2lε+Ω∂Ω∫0τh∘Utdt.
Next, multiplying (74) by Utt and integrating the result over Qτ=Ω×0,τ, we find
(90)aUttt,UttL2QT+βUtt,UttL2QT−c2ΔU,UttL2QT−bΔUt,UttL2QT−ΔW,UttL2QT=0.
An integration by parts in (91) yields
(91)aUttt,UttL2QT=a2Uττx,τL2Ω2,(92)βUtt,UttL2QT=β∫0τUttx,tL2Ω2dt,(93)−c2ΔU,UttL2QT=c2∇Uτx,τ,∇Ux,τL2Ω−c2∫0τ∇Utx,tL2Ω2dt−c2∫∂ΩUτx,τ∫0τ∫ΩUξ,ηdξdηdsx+c2∫∂Ω∫0τUtx,t∫ΩUξ,tdξdtdsx,(94)−bΔUt,UttL2QT=b2∇Utx,τL2Ω2−b∫∂ΩUtx,τ∫ΩUξ,τdξdsx+b∫0τ∫∂ΩUtx,t∫ΩUtξ,tdξdsxdt,(95)∇W,∇UttL2QT=−12h′∘∇Uτ−12hτ∇Ux,τL2QT2+2∇Wτ,∇UτL2Ω−12∫0τh′t∇Ux,tL2Ω2dt+12∫0τh″∘∇Utdt+∫∂ΩUτx,τ∫0τ∫ΩWξ,σdξdσdsx−∫0τ∫∂ΩUtx,t∫ΩWξ,tdξdsxdt.
Substitution (91)–(95) into (90), we get the equality
(96)a2Uττx,τL2Ω2+c2∇Uτx,τ,∇Ux,τL2Ω+b2∇Uτx,τL2Ω2−12h′∘∇Uτ−12hτ∇Ux,τL2Ω2+2∇Wτ,∇UτL2Ω=−β∫0τUttx,tL2Ω2dt+c2∫0τ∇Utx,tL2Ω2+c2∫∂ΩUτx,τ∫0τ∫ΩUξ,tdξdtdsx−c2∫∂Ω∫0τUtx,t∫ΩUξ,tdξdtdsx+b∫∂ΩUτx,τ∫ΩUξ,τdξdsx−b∫0τ∫∂ΩUtx,t∫ΩWξ,tdξdsxdt+∫∂ΩUτx,τ∫0τ∫ΩWξ,σdξdσdsx−∫0τ∫∂ΩUtx,t∫ΩWξ,tdξdsxdt+12∫0τh′t∇Ux,tL2Ω2dt−12∫0τh″∘∇Utdt.
The RHS of (96) can be bounded as follows
(97)c2∫∂ΩUτx,τ∫0T∫ΩUξ,tdξdtdsx≤c22ε1′ε∇Uτξ,τL2Ω2+lεUτx,τL2Ω2+c22ε1′T∂ΩΩ∫0TUx,tL2Ω2dt,(98)−c2∫∂Ω∫0τUtx,t∫ΩUξ,τdξdtdsx≤c22∫0Tε∇Utx,tL2Ω2+lεUtx,tL2Ω2dt+c22Ω∂Ω∫0τUx,tL2Ω2dt,(99)b∫∂ΩUτx,τ∫ΩUξ,τdξdsx≤b2ε2′ε∇Uτxξ,τL2Ω2+lεUτx,τL2Ω2+b2ε2′TΩ∂ΩUx,τL2Ω2,(100)−b∫0τ∫∂ΩUtx,t∫ΩUξ,tdξdsxdt≤b2lε+TΩ∂Ω∫0τUtx,tL2Ω2dt+b2ε∫0τ∇Utx,tL2Ω2dt,(101)∫∂ΩUτx,τ∫0τ∫ΩWξ,τdξdtdsx≤12ε3+12ε4ε∇Uτx,τL2Ω2+lεUτx,τL2Ω2+12ε3Th0∂ΩΩ∫0τUx,tL2Ω2dt+12ε4T∂ΩΩ∫0τh∘Utdt,(102)−∫∂Ω∫0τUtx,t∫ΩWξ,tdξdtdsx≤∫0τε∇Utx,tL2Ω2+lεUtx,tL2Ω2dt+12h0Ω∂Ω∫0τUx,tL2Ω2dt+12Ω∂Ω∫0τh∘Utdt,
So, combining inequalities (97)–(102), we obtain
(103)a2Uττx,τL2Ω2−12h′∘∇Uτ−12hτ∇Ux,τL2Ω2+c2∇Uτx,τ,∇Ux,τL2Ω+2∇Wx,τ,∇Uτx,τL2Ω+b2−c22ε1′ε−b2ε2′ε−12ε3+12ε4ε∇Uτx,τL2Ω2−c2lε2ε1′+blε2ε2′+12ε3+12ε4lεUτx,τL2Ω2−b2ε2′TΩ∂ΩUx,τL2Ω2≤β∫0τUttx,tL2Ω2dt+12TΩε3+1∫0τh∘Utdt+12∫0τh′′∘Utdt−12∫0τh′tUτx,τL2Ω2dt+c2+c22ε+b2ε∫0τ∇Utx,tL2Ω2dt+c22ε1′TΩ∂Ω+12c2+h0+Th0ε4Ω∂Ω∫0τUx,tL2Ω2dt+b2lε+TΩ∂Ω+12c2+1lε∫0τUtx,tL2Ω2dt.
Adding side to side (89) and (103) that gives
(104)aUτx,τ,Ux,τL2Ω+β2−b2ε2′TΩ∂ΩUx,τL2Ω2c22∇vx,0L2Ω2+a2Uττx,τL2Ω2+c2∇Uτx,τ,∇Ux,τL2Ω+b2−c22ε1′ε−b2ε2′ε−12ε3+12ε4ε∇Uτx,τL2Ω2−c22ε1′+b2ε2′+12ε3+12ε4lεlεUτx,τL2Ω2−12h′∘∇Uτ+2∇Wx,τ,∇Uτx,τL2Ω≤γ4∫0τUx,tL2Ω2dt+a+b2lε+TΩ∂Ω+c22+1lε∫0τUtx,tL2Ω2dt+c2+b2+1ε+h0∫0τ∇vx,tL2Ω2dt+c2+c22ε+b2ε+h04∫0τ∇Utx,tL2Ω2dt212Ω∂Ωε3T+1+T2+12T2lε∫0τh∘Utdt−12∫0τh′′∘∇Utdt+h02∫0τh∘∇Utdt+h04+h02∫0τ∇Ux,tL2Ω2dt,where
(105)γ4≔c22T2lε+Ω∂Ω+b2T2lε+Ω∂Ω+12T2lε+Ω∂Ωh0+c22ε1ε1′TΩ∂Ω+c22Ω∂Ω+12h0Ω∂Ωε4T2+1.
Now, the last term on the RHS of (104), we give the function θ (x,t) by
(106)θx,t≔∫0tUx,sds.
Hence, we use (77), and we get
(107)vx,t=θx,τ−θx,t,∇vx,0=∇θx,τ,∇vL2Qτ2=∇θx,τ−∇θx,tL2Ω2≤2τ∇θx,τL2Ω2+∇θx,tL2Qτ2.
And using the inequalities(108)m1′Ux,τL2Ω2≤m1′Ux,tL2Qτ2+m1Utx,tL2Qτ2,m2′Uτx,τL2Ω2≤m2′Utx,tL2Qτ2+m2Uttx,tL2Qτ2,m3′∇Ux,τL2Ω2≤m3′∇Ux,tL2Qτ2+m3∇Utx,tL2Qτ2,m4′h∘Uτ≤m4′1+ζ2∫0τh′∘Utdt+m4∇Utx,τL2Qτ2,−a2Uτx,τL2Ω2−a2Ux,τL2Ω2≤aUτx,τ,Ux,τL2Ω,−c22ε3′∇Uτx,τL2Ω2−c22ε3′∇Ux,τL2Ω2≤c2∇Uτx,τ,∇Ux,τL2Ω,−14ε9+h04ε8∇Uτx,τL2Ω2−h0ε8∇Ux,τL2Ω2−ε9h0∇Uτ≤2∇Wx,τ,∇Uτx,τL2Ω.
Let(109)m1′≔a2+b2ε2′TΩ∂Ω,m2′≔1+a2−c22ε3′−c22ε1′+b2ε2′+12ε3+12ε4lε,m3′≔c22ε3′+h0ε8,and we choose ε1′,ε2′,ε3′,ε8 and ε9 sufficiently large
(110)c22ε1′ε+b2ε2′ε+c22ε3′+14ε9+h04ε9ε<b2.
Asτ is arbitrary, and we get
(111)A≔c22−2τεc2+b/2+1+h0>0.Thus, inequality (104) takes the form
(112)β2Ux,τL2Ω2+Uτx,τL2Ω2+∇Ux,τL2Ω2+b2−c22ε1′ε−b2ε2′ε−c22ε3′+14ε9+h04ε8⏟γ6∇Uτx,τL2Ω2−12h′∘∇Uτ−h′∘Uτ+a2Uττx,τL2Ω2+A∇θx,τL2Ω2≤γ4+m1′∫0τUx,tL2Ω2dt+γ5∫0τUtx,tL2Ω2dt+m3′+h04∫0τ∇Ux,tL2Ω2dt+β+m2′∫0τUttx,tL2Ω2dt+c2+c22ε+b2ε+h04+h02+m3′⏟γ7∫0τ∇Utx,tL2Ω2dt+2εc2+b2+1+h0⏟γ8∫0τ∇θx,tL2Ω2dt−12∫0τh′′∘∇Utdt−ε9ζ+1∫0τh′∘∇Utdt−1ζ12Ω∂Ωε3T+1+T2+12T2lε+1⏟γ9∫0τh′∘Utdt,where
(113)γ5≔a+b2lε+TΩ∂Ω+c22+1lε+h02+m1′+m2′.
We get(114)Ux,τL2Ω2+Uτx,τL2Ω2+∇Ux,τL2Ω2−h′∘Uτ+∇Uτx,τL2Ω2+Uττx,τL2Ω2+∇θx,τL2Ω2−h′∘∇Uτ≤D′∫0τUx,tL2Ω2+Utx,tL2Ω2+∇Ux,tL2Ω2−h′∘Ut+∇Utx,tL2Ω2+Uttx,tL2Ω2+∇θx,tL2Ω2−h′∘∇Utdt,where
(115)D′≔maxβ/2,1,η2/2,γ6,a/2,Aminγ4+m1′,γ5,m3′+h0/4,γ7,β+m2′,ε9/ζ+1,γ8,γ9.
Hence, applying Gronwall’s lemma to (114) gives
(116)Ux,τL2Ω2+Uτx,τL2Ω2+∇Uτx,τL2Ω2−h′∘Uτ+Uττx,τL2Ω2+∇Ux,τL2Ω2+∇θx,τL2Ω2−h′∘∇Uτ≤0,for all τ∈0,c2/4εc2+b/2+1+h0.
For the intervals, we use the same method(117)τ∈m−1c24εc2+b/2+1+h0,mc24εc2+b/2+1+h0
to cover the whole interval0,T and thus proving that Ux,τ=0, ∀τ in 0,T. Hence, the uniqueness is proved.


## 5. Conclusion

The objective of this work is the study of solvability of the Moore-Gibson-Thompson equation with viscoelastic memory term and integral condition by using the Galerkin method. The MGT equation is a nonlinear partial differential equation that arises in hydrodynamics and some physical applications. Recent developments in numerical schemes for solving MGT have placed immense interest in nonlinear dispersive wave models. In the next work, we will try to use the same method with Boussinesque and 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 [6, 15–24], for example, [10, 11, 25, 26]) by using some famous algorithms (see [27–29]).



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