Fixed Point Theory and Applications for Function SpacesView this Special Issue
Blow-Up of Solutions for a Class Quasilinear Wave Equation with Nonlinearity Variable Exponents
This work deals with the blow-up of solutions for a new class of quasilinear wave equation with variable exponent nonlinearities. To clarify more, we prove in the presence of dispersion term a finite-time blow-up result for the solutions with negative initial energy and also for certain solutions with positive energy. Our results are extension of the recent work (Appl Anal. 2017; 96(9): 1509-1515).
We study in this paper the following nonlinear wave equation: Here, , be a bounded domain with a smooth boundary , and are constants, and the exponents , , and are given measurable functions on satisfying with
Also, we suppose that , , and satisfy the log-Holder continuity condition:
. In (4), if , the inequality is undefined because is undefined. The inequality is defined for not equal to , but the condition that is completely greater than zero always makes not equal to because . The term is called -Laplacian.
There are many studies that have studied the problem in the case of constant and variable-exponent nonlinearities.
In the case of constant exponent nonlinearity when , , and the explosion term forces the negative-energy solutions to explode in finite time ([1, 2]), whereas when , , and , the dissipation term guarantees the existence of global solutions for any initial data .
The problem was first treated by (Levine  and Vitillaro ) in the case when the both terms are present (the dissipation and source). He debated the case when , and determined the result of blow-up for solutions with negative initial energy. To extend Levine’s results in  considered a different method when and discussed the cases when and .
Chen et al. in  looked into the nonlinear Laplacian wave equation: when and are given functions. Under suitable conditions on the initial data and the functions , they realized global existence and uniqueness and also discussed the long-time behavior of the solution.
In , Benaissa and Mokeddem considered and they achieved an energy decay estimate for the solutions where , , is a positive function and expanded Yang  and Messaoudi  results. Recently, Mokeddem and Mansour  added some modification in the problem of Benaissa and Mokeddem  and established the same decay result.
Messaoudi and Houari  studied the nonlinear wave equation: where is a bounded domain in , and . They investigated with appropriate conditions imposed on , a global nonexistence result for solutions associated with negative initial energy. In , Kafini and Messaoudi treated a nonlinear wave equation with delay term and proved, under appropriate hypotheses on the initial data, that the energy of solutions explodes in a finite time. For more results, see the previous studies ([13–35]). In the case of variable-exponent nonlinearity, Antontsev  looked into the problem: when is a nonnegative constant, and are given functions. He discussed the case when and and demonstrated a blow-up result under particular hypothesis on .
Thereafter, Antontsev in  considered the same equation and established a local, global existence of weak solutions for specific conditions on and realized a blow-up results for solutions with nonpositive initial energy. In , Guo and Gao considered the same problem of Antontsev , they picked the constant and realized a blow-up result in finite time, and also, they alleged without any proof the same blow-up result for
Sun et al. in  studied the blow-up result for solutions with positive initial energy for the following equation:
They also gave lower and upper bounds for the blow-up time and provided a numerical illustrations for their result.
Lately, Messaoudi and Talahmeh  looked into where . They proved a blow-up result for certain solutions with arbitrary positive initial energy. Korpusov  generalized this result and established (10), with and are constants.
In this paper, we care to find sufficient conditions on and the initial data for which the blowup happens.
In addition to the introduction, our paper is divided into three sections. The second section deals with variable-exponent Lebesgue and Sobolev spaces and some of their characteristics. We also mention the result of existence, but without demonstration, and the second one deals with the result of blow-up for solutions with negative initial energy.
In the fourth one, we present and demonstrate the theorem of blow-up for certain solutions with positive initial energy.
2. Background and Preliminaries
This section contains some essential concepts and definitions about the Lebesgue and Sobolev spaces with variable exponents which will be useful to us later (see Fan and Zhao  and Lars et al. , Mezouar and Boulaaras ).
Let that is a domain of be a measurable function. We introduce the Lebesgue space with a variable exponent by where
Equipped with the Luxembourg-type norm,
is a Banach space (see Lars et al. ).
Now, we introduce the following variable-exponent Sobolev space :
This space is a Banach space with respect to the norm Otherwise, we put to be the closure of in
Remark 1. The space is generally defined in a different way for the variable exponent case.
However, the two definitions are equivalent under condition (4) (see Lars et al. ). We define as the dual of in the same way as the classical Sobolev spaces, where
Lemma 3. (Lars et al. ).
If is a measurable function and such that
Then, the embedding is continuous and compact.
Lemma 4. (Hölder’s inequality ).
Assume that are measurable functions defined on such that If and then with
Lemma 5. (Unit ball property ).
Assume that is a measurable function on . Then,
Remark 8. We can achieve the proof of the previous proposition by using the Galerkin method as in .
3. Blowing Up for Negative Initial Energy
To introduce and demonstrate our results, we first define our energy as follows:
Theorem 9. Assume that the assumptions of hold true and suppose that
Then, the solution of problem blows up in finite time.
To state and demonstrate our result, In order to prove our result, we give the following Lemma
Lemma 10. Suppose the conditions of Lemma 3 hold. Then, we have for any , where is a positive constant which depends on only.
Corollary 11. Let the assumptions of Lemma 10 hold. Then, for any , we get where and are a positive constant.
Corollary 12. Let the assumptions of Lemma 10 hold. Then, we have for any and
Corollary 13. Let the assumptions of Lemma 10 hold. Then, we have for any and
Lemma 15. Let be the solution of problem and assume that (2) holds. Then,
Remark 16. We can achieve the proof of the previous Lemmas and Corollaries as in the paper of Messaoudi and Talahmeh .
Lemma 17. Let be the solution of . Then, there exists a constant such that
Proof. Proof of Theorem 9.
As usual, multiplying by and integrating over in to get for almost every in since is absolutely continuous; hence, and for every in , by remembering the condition that . We then introduce for small to be chosen later and
Adding and subtracting the term , for , from the right side of (37), we get
So, for small enough, we obtain where
By using Young’s inequality, the last term in (40) yields
Thus, by picking such that for a large constant to be given later, and replacing in (41), we reach to
In this step, we choose so large that the coefficient
Once is fixed (thus ), we put sufficiently small so that
Next, we are in the position to obtain an inequality of the form Here, is a positive constant that depends on , (the constant of ).
To achieve (54), we estimate the term(32)Hence,
The proof is completed.
4. Blowing Up for Positive Initial Energy
Now, we are in the position to present and prove one of the main results of this section which is the blow-up for certain solutions with positive energy. For this goal, let be the best constant of the Sobolev embedding and set for that are to be specified later.
We state here the following theorem which will be our main result.
Theorem 18. Assume that the conditions of Proposition 7 hold true and suppose that
Then, the solution of blows up in a finite time.
To demonstrate our theorem, we refer the following two .
Lemma 19. Let the assumptions in Theorem 18 be fulfilled, and then there exists a constant such that
Proof. . Exploiting (21), we get where Let
By noting that , for , we can easily verify that the function is increasing for and decreasing for .
Because , there exists a positive constant such that . So, we get . This means that .
Lemma 20. Let the assumptions in Theorem 18 be fulfilled, and so we have
No data were used to support the study.
Conflicts of Interest
The author(s) declare(s) that they have no conflicts of interest.
The third-named author extends his appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through research groups program under grant (R.G.P-2/1/42). The idea and research project in this work have been presented by Prof. Salah Boulaaras to the authors of the paper. The authors also acknowledge to Prof. Salah Boulaaras for first and second revisions and kind comments on this work.
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