Nonexistence of Global Solutions for Coupled System of Pseudoparabolic Equations with Variable Exponents and Weak Memories

Department of Mathematics, College of Sciences and Arts, Qassim University, Ar Rass, Saudi Arabia Laboratoire de Mathématiques Appliquées et de Modélisation, Université 8 Mai 1945 Guelma, B.P. 401, Guelma 24000, Algeria Laboratory of Applied Mathematics Badji Mokhtar University-Annaba, P.O. Box 12, 23000 Annaba, Algeria Computer Department, College of Sciences and Arts, Qassim University, Ar Rass, Saudi Arabia


Introduction and Overview
Boundary value problems for evolutionary equations of parabolic in degenerate sense are well studied (see, for example, [1][2][3][4]). In this article, we study boundary value problems for coupled system of pseudoparabolic equations with pðxÞ-Laplacian in the presence of weak viscoelasticities. Such problems have not been studied in depth. To begin with, let Ω is an open bounded domain in ℝ n for n ≥ 2 with smooth boundary ∂Ω; we then consider in Ω × ð0, TÞ for 0 < T < ∞ with initial condition and boundary condition The lack of stability of solutions of partial differential equations is a huge restriction for qualitative studies. The terms responsible for the blow-up phenomenon in our system (1) is that of more complicated nonlinearities when they dominate the damped terms, especially when it comes with the existence of a large class of Laplacian operator The functions f j : ℝ 2 ⟶ ℝ, j = 1, 2 are given by the nonlinearities respectively. The weak-viscoelastic term is σðtÞ Ð t 0 ϖ j ðt − sÞ uðsÞ ds.
With pðxÞ-Laplacian, which is nonlinear differential operator, in [9], a problem of elliptic equation is considered as The variable exponents qð·Þ and pð·Þ are two continuous functions on Ω such that with We assume that qðxÞ satisfies the Zhikov-Fan condition, i.e., for all x, y ∈ Ω, with K > 0, 0 < κ < 1 and We state assumptions on ϖ j and σ as follows: Define positive constants α 0 , α 1 , E 1 , and E 2 by for some constants C 1 , C 2 , ρ 1 , ρ 2 > 0 which will be specified later. Fan et al. discussed the existence and multiplicity of solutions of (9) for u ∈ W 1,pðxÞ ðℝ n Þ, where n ≥ 2, pðxÞ is a function defined on ℝ n .
Regarding nonlinear parabolic equation, we mention the work by [1]. The author proposed the problem.
where Ω is a bounded domain of ℝ n with smooth boundary ∂Ω and p, q ≥ 2. Time existence of solutions of system (16) was proved. Whereas, in [10], nonlinear pseudoparabolic was considered Global in time nonexistence of (17) was shown under an appropriate conditions on ϖ, pðxÞ, and qðxÞ.
The paper is organized as follows. In Section 2, we state the properties of the pðxÞ-growth conditions and present assumptions of the kernel functions. In Section 3, we state our main results and prove some auxiliary lemmas. In Section 4, we prove the global nonexistence of solutions given 2 Journal of Function Spaces in Theorem 9. The paper is concluded by explanatory commentaries.

Preliminary
We try to list here some useful mathematical tools. First, let Ω be an open bounded domain in ℝ n for n ≥ 2 with smooth boundary ∂Ω and p : Ω ⟶ ½1,∞ be a measurable function. Denoting by We define the pð·Þ modular of a measurable function w : Ω ⟶ ℝ ∪ f±∞g as where The special Orlicz Musielak space L pð·Þ ðΩÞ is a Lebesgue space with variable-exponent, and it consists of all the measurable function w defined on Ω for which be the Luxembourg norm on this space (see [16]). The Sobolev space W 1,qð·Þ ðΩÞ consists of functions w ∈ L qð·Þ ðΩÞ whose distributional gradient ∇ x w exists and satisfies |∇ x w | ∈L qð·Þ ðΩÞ. This space is a Banach with respect to the norm Lemma 1 (Corollary 8.2.5 in [17]).
(1) If (12) holds with qðxÞ, then where Ω is a bounded domain and C is a positive constant.
The norm of the space W 1,qð·Þ 0 ðΩÞ is given by (2) If is a measurable function and Then with a continuous and compact embedding and where C S > 0 is an embedding constant.
Proof. Since holds for any Δ p Using the notation (36), we obtain the desired result.

Main Results
Before stating our main theorems, we define the weak solution for the problem (1)-(4).
To prove the previous theorem, we can adopt the Faedo-Galerkin method which is the same procedure used in [4].
We introduce the main result concerned with the finite time blowup of solutions of the problem (1)-(4).
In condition (10), Ω ∞ = ∅ holds, which yields We derive a shaper estimate than that in Lemma 13.

Conclusion
In the present paper, we examined the global nonexistence of solutions for a class of coupled pseudoparabolic equations with weak memories. In fact, the influence of the memory terms are unable to guarantee the stability of our problem. More precisely, we derived some conditions on the functions pðxÞ, qðxÞ, ϖ 1 ðtÞ, and ϖ 2 ðtÞ that could occurs blowing up solutions under conditions (10) and (45) with negative initial data. The novelty of our work is to outlined the effects of weak-memory terms, i.e., it lies primarily in the use of a new relation between the relaxation functions and the exponent of the nonlinear sources to get some necessary conditions, which make the problem very interesting from the application point of view, in particular related to the energy systems (including heat systems). The damping terms in this paper are composed with weak-viscoelastic terms and strong damping, which let our problem dissipative named pseudoparabolic. The functions f j : ℝ 2 ⟶ ℝ, j = 1, 2 are given in more complicated nonlinearities. We used a large class of Laplacian operator, with variable exponents. After restriction on initial data, we found that when the sources dominate, we get a blow-up finite time, even though the existence of the damping terms which well known to ensure the stability of solutions.

Data Availability
No data were used in this study.

Conflicts of Interest
The authors declare that they have no competing interests.