Variable Exponent Function Spaces and their ApplicationsView this Special Issue
Research Article | Open Access
Kh. Zennir, H. Dridi, S. Alodhaibi, S. Alkhalaf, "Nonexistence of Global Solutions for Coupled System of Pseudoparabolic Equations with Variable Exponents and Weak Memories", Journal of Function Spaces, vol. 2021, Article ID 5573959, 11 pages, 2021. https://doi.org/10.1155/2021/5573959
Nonexistence of Global Solutions for Coupled System of Pseudoparabolic Equations with Variable Exponents and Weak Memories
The most important behavior for evolution system is the blow-up phenomena because of its wide applications in modern science. The article discusses the finite time blowup that arise under an appropriate conditions. The nonsolvability of boundary value problem for damped pseudoparabolic differential equations with variable exponents is investigated. Such problem has been previously studied in the case if and are constants. New here is the case of variables of nonlinearity and which make the problem has a scientific interest.
1. Introduction and Overview
Boundary value problems for evolutionary equations of parabolic in degenerate sense are well studied (see, for example, [1–4]). In this article, we study boundary value problems for coupled system of pseudoparabolic equations with -Laplacian in the presence of weak viscoelasticities. Such problems have not been studied in depth. To begin with, let is an open bounded domain in for with smooth boundary ; we then consider in for 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 are given by the nonlinearities respectively. The weak-viscoelastic term is .
There exists a function such that where and . There exist two positive constants and such that
With -Laplacian, which is nonlinear differential operator, in , a problem of elliptic equation is considered as
The variable exponents and are two continuous functions on such that with
We assume that satisfies the Zhikov-Fan condition, i.e., for all , with and
We state assumptions on and as follows:
Define positive constants , and by for some constants which will be specified later.
Fan et al. discussed the existence and multiplicity of solutions of (9) for , where is a function defined on .
Regarding nonlinear parabolic equation, we mention the work by . The author proposed the problem. where is a bounded domain of with smooth boundary and . Time existence of solutions of system (16) was proved.
Whereas, in , nonlinear pseudoparabolic was considered
Global in time nonexistence of (17) was shown under an appropriate conditions on and .
The paper is organized as follows. In Section 2, we state the properties of the -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 in Theorem 9. The paper is concluded by explanatory commentaries.
We try to list here some useful mathematical tools.
First, let be an open bounded domain in for with smooth boundary and be a measurable function. Denoting by
We define the modular of a measurable function as where
The special Orlicz Musielak space is a Lebesgue space with variable-exponent, and it consists of all the measurable function defined on for which
Let be the Luxembourg norm on this space (see ).
The Sobolev space consists of functions whose distributional gradient exists and satisfies . This space is a Banach with respect to the norm
Lemma 1 (Corollary 8.2.5 in ). (1)If (12) holds with , thenwhere is a bounded domain and is a positive constant. The norm of the space is given by (2)Ifis a measurable function and with Then with a continuous and compact embedding and where is an embedding constant.
Proposition 2 (Section 1 and Lemma 3.2.20 in ). Let . The spaces and are separable, uniformly convex, and reflexive Banach spaces. The conjugate space of is , where For and , we have
Lemma 3 (Lemma 3.2.4 in ). If is a measurable function on and , then and are equivalent. For , we have (1) implies (2) implies
Lemma 5 (Lemma 3.2.20 in ). If a.e. in , there is a continuous inclusion and for all , where The following notation will be used throughout this paper for and . We have the following technical lemma.
Lemma 6. Let . For any with , , we have
Proof. Since holds for any , we have Using the notation (36), we obtain the desired result.
3. Main Results
Definition 7. The pair is said to be weak solution to (1)–(4) on if it satisfies,
for all test functions .
Here, we present without proof, the first known result concerning the local existence (in time) for the problem (1)–(4).
To prove the previous theorem, we can adopt the Faedo-Galerkin method which is the same procedure used in .
Theorem 9. Assume that (10)–(14) hold. Given satisfying
Then, any solution of (1) with (3) and (4) blows up in finite time where
and and are given in (66).
In order to prove Theorem 9, we need to exploit some lemmas. First, we define the modified energy functional associated to the problem (1)–(4) by
Proof. Multiplying (1) by and (1) by , integrating by parts over , summing and using (14) and Lemma 6. Then, it is clear that and are bounded, so the potential exists and is given by (85) as follows Differentiating (49) with respect to time, it follows that Hence, the proof is finished .
Proof. Using Young’s inequality, we obtain which proves the lemma.
In condition (10), holds, which yields
We derive a shaper estimate than that in Lemma 13.
Proof. First of all, note that, for , Then, owing to Lemmas 1 and 4, we have Similarly, for , we have Hence, by Lemmas 3 and 4, (58) is estimated as The substitution of (60) in the estimate (53) ends the proof.
Now, we can see that (56) takes the following form where
Let be a function defined by
Then, is increasing in decreasing for , such that as , and we have for
Proof. Since holds by (61). Then, there exists and such that . Thus, we have which implies that Now, to establish (72), we suppose by contradiction that for some and by continuity of we can choose such that Again, the use of (61) leads to