On global solutions of the nonlinear Moore-Gibson-Thompson equation

: This work is devoted to study the global solutions of a class of nonlinear Moore-Gibson-Thompson equation. By applying the Galerkin and compact methods, we derive some sufficient conditions on the nonlinear terms, which lead to the existence and uniqueness of the global solution.

In recent years, increasing attention has been paid to the well-posedness and asymptotic behavior of the Moore-Gibson-Thompson (MGT) equation, see [1][2][3][4][5][6][7]. The MGT model is considered through third-order (in time), strictly hyperbolic partial differential equation as follows it is one of the nonlinear acoustic models describing the propagation of acoustics wave in gases and liquid, it has a wide range of applications in medical and industry. In the physical context of the acoustic waves, u is the velocity potential of the acoustic phenomena, α denotes the thermal relaxation time, c denotes the speed of sound, β denotes friction, and b denotes a parameter of diffusivity. It is often convenient to write MGT equation as an abstract form and it has been shown [8,9] that the linear part of Eq. (5) generates a strongly continuous semigroup as long as r > 0. In [10], the authors provided a brief overview of well-posedness results, both local and global, pertinent to various configurations of MGT equations. Especially, the authors in [11] considered the following model with nonlinear control feedback where the parameter β > 0, p(u) denotes an active force and the operator A is strictly positive. By semigroup method, it was proved in [11] we that (6) with initial data of arbitrary size in H is locally and globally well-posed under the following assumption: p ∈ C 1 (R) and its derivative satisfies −δ ≤ p ′ (s) ≤ m for some positive constants δ and m. Kaltenbacher et al., [12] established the well-posedness by Galerkin approximations and then employ fixed-point arguments for well-posedness of the Jordan-Moore-Gibson-Thompson (JMGT) equation More recently, Boulaaras et al., [13] proved the existence and uniqueness of the weak solution of the Moore-Gibson-Thompson equation with the integral condition by applying the Galerkin method.
In this paper, we extend the results in [11] to Problem (1)-(3) by applying the Galerkin method and compact method. The contents of this paper are organized as follows; In §2, we prepare some materials needed for our proof. Finally, in §3, we give the main result and the proof.

Preliminaries
Throughout this paper, the domain Ω is assumed to be sufficiently smooth to admit integration by parts and second-order elliptic regularity. We use C to denote a universal positive constant that may have different values in different places. W m,2 (Ω) = H m (Ω) and W m,2 0 (Ω) = H m 0 (Ω) denote the well-known Soblev space. We denote by ||.|| p the L p (Ω) norm and by ||∇.|| the norm in H 1 0 (Ω). In particular, we denote ||.|| = ||.|| 2 By a weak solution u(x, t) of Problem (1) in Ω, and In this paper, we assume α, β, c 2 , r > 0 and Lemma 1. [14] Let Ω ∈ R n be a bounded domain and w j be a base of L 2 (Ω). Then for any ϵ > 0 there exist a positive constant N ϵ , such that

Solvability of the problem
In this section, by using Galerkin's method and compactness method, we shall prove the existence of global solutions of Problem (1)-(3).
Let {w j (x)} j∈N be the eigenfunctions of the following boundary problem corresponding to the eigenvalue λ j (j = 1, 2, 3, ...). Then {w j (x)} j∈N can be normalized to from an orthogonal basis of H 2 (Ω) ∩ H 1 0 (Ω) and to be orthnormal with respect to the L 2 (Ω) scalar product. Now, we seek an approximate solution of Problem (1)-(3) in the form of where the constants T jN are defined by the conditions T jN (t) = (u N (x, t), w j (x)) and can be determined from the relation , and u 2 ∈ L 2 (Ω), then for any T > 0, Problem (11)-(12) possesses a solution u N on [0, T], and the following estimate holds in the class Proof. Problem (11)-(12) leads to a system of ODEs for unknown functions T jN (t). Based on standard existence theory for ODE, one can obtain functions T jN (t) : [0, t k ) → R, j = 1, 2, ..., k, which satisfy approximate Problem (11)-(12) in a maximal interval [0, t k ), t k ∈ (0, T]. This solution is then extended to the closed interval [0, T] by using the estimate below. Multiplying (11) by T jNtt (t), summing up the products for j = 1, 2, ..., N and integrating by parts, we get Integrating (14) with respect to t from 0 to t, we obtain We observe that Adding 2[(u N , u N t ) + (u N t , u N tt ) + (∇u N , ∇u N t )] to both sides of (15) and a substitution of the equalities (16) and (17) in (15) gives Then, by Hölder inequality and the fact | f (s)| = | t 0 f ′ (s)ds| ≤ C 1 |s| by (A1), we arrive at Taking into account that as N → ∞, then applying the Gronwall inequality to (19) and then integrating from 0 to t appears that Multiplying (11) by λ j T jN (t) summing up the products for j = 1, 2, ...N, integrating by parts and integrating with respect to t, we get Combining Cauchy inequality, the fact ||∆u N (0)|| 2 → ||∆u 0 || 2 , and | f (s)| ≤ C 1 |s|, and making use of the following inequality we have Choosing ϵ 1 sufficiently small and ϵ 2 sufficiently large such that ϵ 2 > 2c 2 , then it follows from (22) and (20) that Thus, applying Gronwall's inequality to (23), we deduce Combining (20) and (24), we get Furthermore, by (25), we have that (11)-(12) possesses a global solution.
This implies that w = 0 for all t ∈ [0, T]. Thus the uniqueness is proved.