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

Salah Mahmoud Boulaaras , Abdelbaki Choucha, Djamel Ouchenane, Asma Alharbi, Mohamed Abdalla , and Bahri Belkacem Cherif 1,7 Department of Mathematics, College of Sciences and Arts, ArRas, Qassim University, Saudi Arabia Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran 1, Oran, 31000 Oran, Algeria Laboratory of Operator Theory and PDEs: Foundations and Applications, Department of Mathematics, Faculty of Exact Sciences, University of El Oued, Algeria Laboratory of Pure and Applied Mathematics, Amar Teledji Laghouat University, Algeria Mathematics Department, College of Science, King Khalid University, Abha 61413, Saudi Arabia Mathematics Department, Faculty of Science, South Valley University, Qena 83523, Egypt Preparatory Institute for Engineering Studies in Sfax, Tunisia


Introduction
In this contribution, we are interested to study the existence and uniqueness of solutions of the following problem Here, a and β are physical parameters, and c is the speed of sound. The convolution term Ð t 0 hðt − sÞΔuðsÞds reflects the memory effect of materials due to vicoelasticity, F is a given function, and h is the relaxation function satisfying (H1) h ∈ C 1 ðℝ+,ℝ + Þ is a nonincreasing function satisfying where h 0 = Gð∞Þ = Ð ∞ 0 hðsÞds > 0, GðtÞ = Ð t 0 hðsÞds, and h ′ ′ > 0.
(H2) ∃ζ > 0 satisfying The 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][2][3][4][5][6][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][11][12][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.

Preliminaries
We will define the spaces: VðQ T Þ and WðQ T Þ by where Consider the equation where w x, t ð Þ= and ð:, :Þ L2ðQTÞ stands for the inner product in L 2 ðQ T Þ, u is supposed to be a solution of (1) and v ∈ WðQ T Þ. Evaluation of the inner product in [9] gives We give two useful inequalities: (i) Gronwall inequality. Let the nonnegative integrable functions φðtÞ, ϕðtÞ on the interval I with the nondecreasing function hðtÞ. If ∀t ∈ I, we have where c > 0, hence, (ii) Trace inequality (see [15]). If Φ ∈ W 2 1 ðΩÞ where Ω is a bounded domain in ℝ n with smooth boundary ∂Ω, then for any ε > 0, where lðεÞ > 0.
We call a generalized solution to the problem (1) for each function u ∈ VðQ T Þ that fulfills the equation (9) for each v ∈ WðQ T Þ.

Solvability of the Problem
First, we will give an approximate solution of the problem (1) in the form where C k ðtÞ are constants given by the conditions, for k = 1, ⋯, N, and can be determined from the relations substitution of (13) into (15), and we find for l = 1, ⋯, N.
From (15) it follows that Let Then, (17) can be written as By differentiating (two times) with respect to t, it gives We find a system of differential equations of fifth order with respect to t, 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, ∃u N (x) satisfying (15). Now, we prove that u N is sequence bounded. To do this, we multiply each equation of (15) by the appropriate C k ′ ðtÞ 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

Journal of Function Spaces
Evaluation of the terms on the LHS of (22) gives Thus, Taking into account the equalities (23)-(30) in (22), we end up with Now, multiplying the equations of (15) by C k ″ðtÞ, collect them from 1 to N and integrating the result with respect to t from 0 to τ, and τ ≤ T, we find With the same reasoning in (22), we find Journal of Function Spaces A substitution of equalities (33)-(40) in (22) gives Multiplying (32) by λ and using (41), we get 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 Journal of Function Spaces Combining inequalities (45)-(60) and equality (44) and make use of the following inequality where

Journal of Function Spaces and we have
where Choosing ε 2 , ε 3 , ε 4 , ε 5 , ε 5 ′ , ε 7 , ε 8 and ε 9 sufficiently large By using (2)-(4), the relation (64) reduces to Journal of Function Spaces where Using the inequality of Gronwall to (67) and integrating the result from 0 to τ that gives where We deduce from (69) that Hence, fu N g N≥1 is sequence bounded in VðQ T Þ, and we can extract from it a subsequence for which we use the same notation which converges weakly in VðQ T Þ to a limit function uðx, tÞ, and we have to show that uðx, tÞ is a generalized solution of (1). Since u N ðx, tÞ ⟶ uðx, tÞ in L 2 ðQ T Þ and u N ðx, 0Þ ⟶ ζðxÞ in L 2 ðΩÞ, then uðx, 0Þ = ζðxÞ. Now to prove that (15) holds, we multiply each of the relations (15) by a function p l ðtÞ ∈ W 1 2 ð0, TÞ, p l ðTÞ = 0. Hence, collect them the obtained equalities ranging from l = 1 to l = N and integrating the result over t on ð0, TÞ. If we let η N = ∑ N k=1 p l ðtÞZ k ðxÞ, then we have for all η N of the form ∑ N k=1 p l ðtÞZ k ðxÞ and α > 0: Since Thus, the limit function u satisfies (15) for every η N = ∑ N k=1 p l ðtÞZ k ðxÞ.
We define the totality of all functions of the form η N = ∑ N k=1 p l ðtÞZ k ðxÞ by ℚ N , with p l ðtÞ ∈ W 1 2 ð0, TÞ, p l ðTÞ = 0. But ∪ N l=1 ℚ N is dense in WðQ T Þ, hence the relation (15) holds ∀u ∈ WðQ T Þ. Then, we have shown that the limit function uðx, tÞ is a generalized solution of problem (1) in VðQ T Þ. Proof. Suppose that ∃ u 1 ,u 2 ∈ VðQ T Þ two different generalized solutions for the problem (1). Hence, the difference U = u 1 − u 2 solves

Uniqueness of the Problem
9 Journal of Function Spaces and (9) gives where It is obvious that v ∈ WðQ T Þ and v t ðx, tÞ = −Uðx, tÞ for all t ∈ ½0, τ. By integration by parts in the LHS of (75) that yields Plugging (78)-(82) into (75), we obtain then v k k 2 Applying the inequality of the trace, the RHS of (83) gives Journal of Function Spaces Combining the relations (86)-(83) and (87)-(88), we get Next, multiplying (74) by U tt and integrating the result over Q τ = Ω × ð0, τÞ, we find An integration by parts in (91) yields Substitution (91)-(95) into (90), we get the equality The RHS of (96) can be bounded as follows

Journal of Function Spaces
So, combining inequalities (97)-(102), we obtain And using the inequalities