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

Abstract

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

For more details, see [5–8] and references therein.

With -Laplacian, which is nonlinear differential operator, in [9], 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:

satisfies

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 [1]. 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 [10], nonlinear pseudoparabolic was considered

Global in time nonexistence of (17) was shown under an appropriate conditions on and .

System (1), where the exponents and , the existence/nonexistence results have been extensively studied (please see [11–15]).

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.

2. Preliminary

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 [16]).

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 [17]). (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 [17]). 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 [17]). If is a measurable function on and , then and are equivalent. For , we have (1) implies (2) implies

Lemma 4 (Section 2 in [18] and Lemma 3.2.5 in [17]). If is a measurable function on , then for all .

Lemma 5 (Lemma 3.2.20 in [17]). 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

Before stating our main theorems, we define the weak solution for the problem (1)–(4).

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).

Theorem 8. Assume that (10), (12), and (14) hold. Then, the problem (1)–(4) has a unique local solution satisfying for depending on .

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).

We define and find by (10) and (14).

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

Lemma 10. Let be the solution of (1)–(4) with (10)–(14). Then, the energy functional satisfies

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 .

Lemma 11. Let be a strong solution of (1)–(4) with (10)–(14). Then, we have

Proof. Using Young’s inequality, we obtain which proves the lemma.

Lemma 12. Let be a strong solution of (1)–(4) with (10)–(14). Then, we have

Proof. By using both equivalences (8) and (10), we deduce The proof is completed by direct use of (54).

In condition (10), holds, which yields

We derive a shaper estimate than that in Lemma 13.

Lemma 13. Let be a strong solution of (1)–(4) with (10)–(14). Then, we have where is an embedding constant defined in Lemma 1.

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

Since, by the fact (43), i.e., holds. Then, by (63), we can find for all

Lemma 14. Under the assumptions of Lemma 13. Then, the pair defined in (66) is negative.

Proof. Direct use of (63) and (66). Thus, we obtain By considering (68), (69). Then, we use Young’s inequality to get To provide this, we take sufficiently large and small enough.

Lemma 15. Let be a strong solution of (1)–(4) with (10)–(14) and initial condition satisfying (44) and (45). Then, there exists a constants and such that

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 This is impossible, since Hence, (72) established.

Lemma 16. Let be a strong solution of (1)–(4) with (10)–(14) and (44). Then, we have for all .

Proof. To prove (80), we exploit Lemmas 10 and 13 to obtain Then, by the definitions of , , and So, inequality (82) can be estimated as follows From the previous estimate, we deduce Thanks to (72) and (84) and the definition of different constants, then the desired results are proved.

4. Proof of Theorem 9

We are now ready to prove Theorem 9.

Proof. Let the auxiliary function defined as By differentiating the functional , we get By using (8), we get Thanks to Cauchy Schwarz’s inequality, we get holds for some positive constant to be determined later.
Substituting (89) in (86) to get Using Lemma 12 to estimate the terms and , we get Owing to (43), we can choose as where is sufficiently large such that by Lemma 14, we have where
We derive the differential inequality from (93). First, owing to Poincaré’s inequality, we have holds. For some constant , we have By using the following Minkowski’s inequality we can obtain Then, by Lemma 5, we have and by (82) and (93), we can assume that is sufficiently large for . Hence, thanks to (95), and hold for . Then, using Lemma 4 to get which gives where A direct integration of (100) over then yields, which implies that along with . Finally, we have where Then, the proof is now completed.

5. 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 , and 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 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.

Authors’ Contributions

The authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.