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.

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:

The unknown function denotes the scalar acoustic velocity, denotes the speed of sound, and denotes the thermal relaxation. Besides, the coefficient is related to the diffusively of the sound with . 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:

This paper is related to the following works (see [16, 39]). Now, when we talk about the 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: where are physical parameters and is a positive self-adjoint operator on a Hilbert space The convolution term 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 , where is a bounded domain with sufficiently smooth boundary . 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

Let and be the set spaces defined, respectively, by

Consider the equation where depend on the inner product in , is supposed to be a solution of (1), and . Upon using (6) and (1), we find

Now, we give two useful inequalities: (i)Gronwall inequality: if for any , we havewhere and are two nonnegative integrable functions on the interval with nondecreasing and is constant, then (ii)Trace inequality: when , we havewhere is a bounded domain in with smooth boundary , and is a positive constant.

Definition 1. If a function satisfies Equation (3), each 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. If and , , then there is at least one generalized solution in to problem (1).

Proof. Let be a fundamental system in , such that . Now, we will find an approximate solution of the problem (1) in the form where the constants are defined by the conditions and can be determined from the relations Invoking to (11) in (6) gives for . From (7), it follows that Let Then (8) can be written as A differentiation with respect to yields Thus, for every , there exists a function satisfying (6). Now, we will demonstrate that the sequence is bounded. To do this, we multiply each equation of (6) by the appropriate summing over from to then integrating the resultant equality with respect to from 0 to , with , which yields After simplification of the of (19), we observe that Taking into account the equalities (20) and (21) in (12), we obtain Now, multiplying each equation of (6) by the appropriate , we add them up from to and then integrate with respect to from to , with , and we obtain With the same reasoning in (12), we find Upon using (31) and (32) into (23), we have Now, multiplying each equation of (6) by the appropriate , we add them up from to and then integrate with respect to from to , with , and we obtain With the same reasoning in (12), we find A substitution of equalities (42) and (43) in (34) gives Multiplying (22) by , (33) by , and (44) by , we get We can estimate all the terms in the right-hand side of (45) as follows: Combining inequalities (46)–(79) and equality (45) and making use of the following inequality: where we have where Choosing , and sufficiently large the relation (80) reduces to where Applying the Gronwall inequality to (60) and then integrating from to , it appears that We deduce from (84) that Therefore, the sequence is bounded in , and we can extract from it a subsequence for which we use the same notation which converges weakly in to a limit function ; we have to show that is a generalized solution of (4). Since in and in , then
Now to prove that (3) holds, we multiply each relation in (15) by a function then add up the obtained equalities ranging from to , and integrate over on . If we let then we have for all of the form .
Since therefore, we have Thus, the limit function satisfies (3) for every . We denote by the totality of all functions of the form , with ,
But is dense in , and then relation (3) holds for all . Thus, we have shown that the limit function is a generalized solution of problem (4) in .