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

Laboratory of mathematics, Informatics and Systems (LAMIS), Department of Mathematics and Computer Science, Larbi Tebessi University, 12002 Tebessa, Algeria Preparatory Institute for Engineering Studies in Sfax, Sfax, Tunisia Department of Mathematics, College of Sciences and Arts, Qassim University, Ar Rass, Saudi Arabia Mathematics Department, College of Science, King Khalid University, Abha, Saudi Arabia Mathematics Department, Faculty of Science, South Valley University, Qena 83523, Egypt


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., ). 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 . 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 vis-cosity 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 thirdorder hyperbolic equation: where τ, α, b, c 2 are physical parameters and A is a positive self-adjoint operator on a Hilbert space H: The convolution term Ð t 0 gðt − sÞAwðsÞds 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: The aim of this manuscript is to consider the following nonlocal mixed boundary value problem for the Moore-Gibson-Thompson (MGT) equation for all ðx ; tÞ ∈ Q T = ð0, TÞ, where Ω ⊂ ℝ n 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.

Preliminaries
Let VðQ T Þ and WðQ T Þ be the set spaces defined, respectively, by Consider the equation where ð:, :Þ L 2 ðQ T Þ depend on the inner product in L 2 ðQ T Þ, u is supposed to be a solution of (1), and v ∈ WðQ T Þ. Upon using (6) and (1), we find Now, we give two useful inequalities: (i) Gronwall inequality: if for any t ∈ I, we have where hðtÞ and yðtÞ are two nonnegative integrable functions on the interval I with hðtÞ nondecreasing and c is constant, then (ii) Trace inequality: when w ∈ W 2 1 ðΩÞ, we have where Ω is a bounded domain in ℝ n with smooth boundary ∂Ω, and lðεÞ is a positive constant.

Solvability of the Problem
Here, by using Galerkin's method, we give the existence of problem (1).
Proof. Let fZ k ðxÞg k≥1 be a fundamental system in W 1 2 ðΩÞ, such that ðZ k , Z l Þ L 2 ðΩÞ = δ k,l . Now, we will find an approximate solution of the problem (1) in the form where the constants C k ðtÞ are defined by the conditions 2 Advances in Mathematical Physics and can be determined from the relations Invoking to (11) in (6) gives for l = 1, ⋯, N.
From (7), it follows that Let Then (8) can be written as A differentiation with respect to t yields Thus, for every n, there exists a function u N ðxÞ satisfying (6). Now, we will demonstrate that the sequence u N is bounded. To do this, we multiply each equation of (6) by the appropriate C k ′ ðtÞ 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 After simplification of the LHS of (19), we observe that

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Taking into account the equalities (20) and (21) in (12), we obtain Now, multiplying each equation of (6) by the appropriate C k ′ ′ ðtÞ, 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 With the same reasoning in (12), we find Advances in Mathematical Physics Upon using (31) and (32) into (23), we have Now, multiplying each equation of (6) by the appropriate C k ′ ′ ′ðtÞ, 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 With the same reasoning in (12), we find A substitution of equalities (42) and (43) in (34) gives Multiplying (22) by λ 1 , (33) by λ 2 , and (44) by λ 3 , we get We can estimate all the terms in the right-hand side of (45) as follows: Advances in Mathematical Physics

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Combining inequalities (46)-(79) and equality (45) and making use of the following inequality:

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where we have where 9 Advances in Mathematical Physics Choosing ε 7 , ε 8 , ε 9 , ε 10 , ε 11 , ε 113 , and ε 14 sufficiently large the relation (80) reduces to where Applying the Gronwall inequality to (60) and then integrating from 0 to τ, it appears that We deduce from (84) that Therefore, the sequence fu N g N≥1 is 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; we have to show that uðx, tÞ is a generalized solution of (4). 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 (3) holds, we multiply each relation in (15) by a function p l ðtÞ ∈ W 1 2 ð0, TÞ, p l ðtÞ = 0, then add up the obtained equalities ranging from l = 1 to l = N, and integrate over t on ð0, TÞ. If we let η N = ∑ N k=1 p k ðtÞZ k ðxÞ, then we have 10 Advances in Mathematical Physics for all η N of the form ∑ N k=1 p l ðtÞZ k ðxÞ. Since therefore, we have Thus, the limit function u satisfies (3) for every η N = ∑ N k=1 p l ðtÞZ k ðxÞ. We denote by ℚ N the totality of all functions of the form η N = ∑ N k=1 p l ðtÞZ k ðxÞ, with p l ðtÞ ∈ W 1 2 ð0, TÞ, p l ðt Þ = 0: But ∪ N l=1 ℚ N is dense in WðQ T Þ, and then relation (3) holds for all u∈WðQ T Þ. Thus, we have shown that the limit function uðx, tÞ is a generalized solution of problem (4) in VðQ T Þ.

Uniqueness of Solution
Theorem 3. The problem (4) cannot have more than one generalized solution in VðQ T Þ.