Nonexistence of Global Weak Solutions of a System of Nonlinear Wave Equations with Nonlinear Fractional Damping

Abstract

We consider the system of nonlinear wave equations with nonlinear time fractional damping where , and are positive natural numbers, , , , and , , is the Caputo fractional derivative of order . Namely, sufficient criteria are derived so that the system admits no global weak solution. To the best of our knowledge, the considered system was not previously studied in the literature.

1. Introduction

In this paper, we investigate the system of nonlinear wave equations with nonlinear time fractional damping: where , and are positive natural numbers, , and are nonnegative numbers that will be specified later, , and , , is the Caputo fractional derivative (with respect to ) of order . Namely, we are interested in obtaining sufficient conditions for which the considered system admits no global weak solution. Our approach is based on the nonlinear capacity method (see, e.g., [1]).

Before we state and prove our result, let us dwell on existence literature. Single wave equations or systems of wave equations have been studied in large; we may mention the books of Lions [2], Reed [3], Georgiev [4], and Strauss [5] and the papers of Aliev et al. [6], Said-Houari [7], Takeda [8], Goergiev and Todorova [9], Todorova and Yordanov [10], Zhang [11], and Kirane and Qafsaoui [12] for equations and systems with classical linear or nonlinear damping and Tatar [13], Kirane and Tatar [14], and Kirane and Laskri [15] for wave equations with fractional damping. In particular, in [13], the following problem was considered: where is a bounded domain of with smooth boundary , , , and . It was shown that, if is a solution to (2), then there exist and sufficiently large initial data so that blows up at . Problem (2) was also considered in [14]. Namely, it was shown that the solution of (2) is unbounded and grows up exponentially in the -norm for sufficiently large initial data. In [15], the following problem was studied: where , , is the fractional Laplacian of order , , and is the Riemann-Liouville fractional derivative of order . It was shown that for all , if , then (3) does not admit a local weak solution for any .

In the next section, we recall some notions on fractional calculus. In Section 3, we define global weak solutions of system (1) and state our main result. Moreover, as a consequence, we deduce a nonexistence result in the case of a single equation. Finally, the proof of the main result is given in Section 4.

2. Preliminaries

Some preliminaries on fractional calculus (see, e.g., [16–21]) are provided in this section. Given , the fractional integrals and of are defined by where denotes the gamma function.

Lemma 1. Let , , and . Then, Given , the Caputo fractional derivative of is defined by The following lemma will be used later in the proof of our main result.

Lemma 2. Let and be the function defined by where . Then, for all .

Proof. For , one has Taking , one obtains (8). Next, (9) follows immediately from (8).

3. Main Result

We begin with the definition of the intended solutions of system (1). Given , let and be the set of functions satisfying the following conditions: (a)(b), i.e., uniformly in , where is a compact(c), for all

Definition 3. Let , . A pair of functions is said to be a global weak solution of system (1), if for all and , Next, we introduce the parameters Here, and are positive natural numbers, , and are positive numbers that will be specified later, , and .

Theorem 4. Let , , , and . Suppose that , , and If then there exists no global weak solution of system (1).
Consider now the case of a single equation. Namely, where , , and . A global weak solution of (18) can be defined in a similar way as in Definition 3. Taking in Theorem 4, , , , , and , one deduces the following corollary.

Corollary 5. Let and , . Suppose that If then there exists no global weak solution of problem (18).

Example 6. Consider the equation Observe that (21) is a special case of (18) with , , , , , , , and . One has , , and On the other hand, one can check easily that Since it follows from Corollary 5 where (21) admits no global weak solution.

4. Proof of Theorem 4

Before proving Theorem 4, we need some preliminary results.

Given , let where is defined by (7) with and where , is a certain parameter that will be specified later, and is a decreasing function satisfying

Lemma 7. For all , the function defined by (25) belongs to .

Proof. One can check easily that and satisfies conditions (a) and (b). On the other hand, by (9), for all , one obtains which yields This shows that satisfies condition (c).

The following estimate follows from elementary calculations.

Lemma 8. There exists a constant such that for any positive natural number .

Proof of Theorem 4. Suppose that is a global weak solution of system (1). For , using (12) and Lemma 7, one obtains where is defined by (25). Similarly, using (13), one obtains Using Hölder’s inequality, the following holds: Similarly, one has It follows from (32), (34), (35), and (36) that Similarly, using (33) and Hölder’s inequality, one obtains Now, set For , let For , , and , let From (37) and (38), we may write On the other hand, from (25), for , one has Since , , by the dominated convergence theorem, one obtains Similarly, since , , the following holds: Again, from (25), one has Using (16), (45), and (47), one obtains Similarly, using (16), (46), and (48), one obtains Consequently, one has Next, it follows from (42) and (51) that Similarly, (43) and (52) yield Using the inequality it follows from (53) that where upon Similarly, it follows from (54) that Using (54), (57), and (58), the following holds: where is a constant that depends only on and . Here and below, any positive constant independent of is denoted by . Next, using the-Young inequality withwhich is small enough, one deduces from (59) that Similarly, one obtains Further, we shall estimate and for and for , , and . From (25), one has which yields Similarly, using (25) and Lemma 8, one obtains so Next, using (9) and (25), for , , and , one obtains whereupon Next, taking , it follows from (63), (65), and (67) that Using (60), (68), and (69), the following holds: where , , are defined by (14). Similarly, using (61), (68), and (69), one obtains where , , are defined by (15).

Now, from condition (17), we have two cases.

Case 1. If , in this case, taking the infimum limit as in (70) and using Fatou’s lemma, the following holds: Hence, by (33), the following holds: which contradicts (52).

Case 2. If , as in the previous case, taking the infimum limit as in (71), one obtains Hence, by (32), the following holds: which contradicts (51).

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare no conflict of interest.

Authors’ Contributions

All authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript.

Acknowledgments

M. Jleli is supported by Researchers Supporting Project RSP-2019/57, King Saud University, Riyadh, Saudi Arabia. M. Kirane has been supported by the “RUDN University program 5-100.”