General Decay of the Moore–Gibson–Thompson Equation with Viscoelastic Memory of Type II

Department of Mathematics, College of Sciences and Arts, ArRas, Qassim University, Saudi Arabia Laboratory of Operator "eory and PDEs: Foundations and Applications, Department of Mathematics, Faculty of Exact Sciences, University of El Oued, El Oued, Algeria Department of Matter Sciences, College of Sciences, Amar Teleji Laghouat University, Laghouat, Algeria Department of Mathematics and Computer Science, University of Catania, Catania, Italy


Introduction and Preliminaries
In this work, we are interested in the study of the general decay of solutions of the following problem: where a, b, β > 0 are physical parameters and c is the speed of sound. e convolution term t 0 h(t − s)Δu t (s)ds reflects the memory type-II effect of materials due to viscoelasticity, where h is the relaxation function.
Moore-Gibson-ompson (MGT) equation is based on the modeling of high amplitude sound waves. ere has been quite a bit of work in this area of research due to a wide range of applications such as medical and industrial use of high intensity ultrasound in lithotripsy, heat therapy, ultrasonic cleaning, etc. Classical models of nonlinear acoustics include the Kuznetsov equation, the Westervelt equation, and the Kokhlov-Zabulotskaya-Kuznetsov equation. An in-depth study of linear models is a good starting point for a better understanding of the approximate behaviors of nonlinear models. Indeed, even in the linear case, the works [1] have shown a rich dynamic. In [2], Marchand et al. presented a detailed analysis of this equation. Using a quasi-abstract group approach and refined spectral analysis, they establish the well posedness of the problem and define the accumulation point of eigenvalues. Kaltenbacher et al. [3] also studied the fully nonlinear version of the MGT equation and established the global well posedness and the exponential decay for the nonlinear equation under consideration.
In the case of a third-order equation, the modeling of memory effects is quite complex. e memory term may affect u, or u t , or even a combination of both. Accordingly, the corresponding models are called as memory of type I, type II, and type III. is classification raises a fundamental question on the impact of the memory terms on the stability properties of the corresponding evolutions. We mention that the memory has some stabilizing effects (see, for instance, [4,5]). On the contrary, a quantification of the stability results via decay rates shows that the memory may destroy the stability properties of a stable system.
In this paper, we are interested in studying the dynamics that results from the memory kernel of type II (see [6]) in our problem (1).
Natural questions can be asked based on the study of the viscoelastic memory and integral condition system (see [7][8][9][10]): could the addition of the memory kernel of type II harm the stability of this kind of problem in any way? If the answer to the question in the wave condition with friction damping are relatively simple, it is not easy to answer in the case of memory kernel of type II that we present below, especially in Fourier space. is work is part of the effort to understand this the MGT problem and memory kernel of type II. e coupling between the Fourier law of heat condition and different systems has been considered by many authors and there are many results. For example, Bresse system (Bresse-Fourier) has been studied in [11], the Bresse system coupled with the Cattaneo law of heat condition (Bresse-Cattaneo) has been studied in [12], and Timoshenko system with past history has been considered in [13]. For further details, we refer the reader to the following papers [14,15].
We mention also that several results related to viscoelasticity have been obtained by using the theory of Lie symmetries. Precisely, thanks to the symmetry reduction obtained by means of the classical Lie symmetries, it was possible to obtain exact solutions of considerable interest. For further details in this direction, we refer the reader to [16].
Based on all last mentioned works, especially [6,10], we would like to prove the general decay result in the Fourier space to problem (1). To the best of our knowledge, this is one of the first works that deals with the MGT problem with viscoelastic memory kernel of type II in the Fourier space. e paper has the following structure. In Section 1, we put our assumptions and preliminaries that will be employed in our main decay result. In Section 2, by using the energy method in the Fourier space, we construct the Lyapunov functional and find the estimate for the Fourier image. Section 3 is devoted to the conclusion.
To prove our main result, we need the following hypotheses and lemmas.
(2) (H2) ere exist positive numbers θ and λ ∈ (c 2 /b, β/a) such that λ/θ < κ and where roughout this paper, we use c, C, C i , i � 1, 2, to denote a generic positive constant. where Lemma 2. For any k ≥ 0 and c > 0, there exist a constant C > 0 such that, for all t ≥ 0, the following estimate holds:

The Energy Method in the Fourier Space
In this section, we get the decay estimate of the Fourier image of the solution for system (1). By using Plancherel's theorem together with some integral estimates such as (2), this method will allow us to give the decay rate of the solution in the energy space. To this goal, we use the energy method in the Fourier space and construct appropriate Lyapunov functionals. Finally, we prove our main result. Applying the Fourier transform to (1), we get the following problem: Journal of Function Spaces Let u(ξ, t) be the solution of (9). en, the energy functional E(t), given by satisfies where Proof. Firstly, multiplying (9) by (u tt ) and taking the real part, we obtain en, we have Journal of Function Spaces 3 and Substituting (13) and (14) into (12), we obtain Secondly, multiplying (9) by (u t ) and taking the real part, we obtain Re au ttt u t + βu tt u t + c 2 |ξ| 2 uu t + b|ξ| 2 u t u t − |ξ| 2 We have Substituting (17) and (18) into (16), we obtain At this point, by (4), we have β − c 2 a/b > 0, and by (3), we select λ such that c 2 /b < λ < β/a.
Let E(t) be the energy functional. en, we have By (15) and (19), we have

□
At this stage, we define the functional where N, N 1 , and N 2 are positive constants to be properly chosen later.