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

where ðu, vÞ = ðuðt, xÞ, vðt, xÞÞ, m and N are positive natural numbers, p, q, r, s > 1, σ and δ are nonnegative numbers that will be specified later, 0 < α, β < 1, and D0,t , 0 < κ < 1, is the Caputo fractional derivative (with respect to t) of order κ. 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:


Introduction
In this paper, we investigate the system of nonlinear wave equations with nonlinear time fractional damping: where ðu, vÞ = ðuðt, xÞ, vðt, xÞÞ, m and N are positive natural numbers, p, q, r, s > 1, σ and δ are nonnegative numbers that will be specified later, 0 < α, β < 1, and C D κ 0,t , 0 < κ < 1, is the Caputo fractional derivative (with respect to t) of order κ. 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 Ω is a bounded domain of ℝ N with smooth boundary ∂Ω, a > 0, p > 1, and 0 < α < 1. It was shown that, if u is a solution to (2), then there exist T * ≤ ∞ and sufficiently large initial data so that u blows up at T * . Problem (2) was also considered in [14]. Namely, it was shown that the solution of (2) is unbounded and grows up exponentially in the L p+1 -norm for sufficiently large initial data. In [15], the following problem was studied: where p > 1, 1 ≤ β ≤ 2, ð−ΔÞ β/2 is the fractional Laplacian of order β/2, 0 < α < 1, and D α 0,t is the Riemann-Liouville fractional derivative of order α. It was shown that for all p > 1, if lim inf jxj→+∞ u 1 ðxÞ = +∞,, then (3) does not admit a local weak solution for any T > 0.
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.

Lemma 1.
Let ς > 0, f ∈ L 1 ½0, T, and g ∈ L ∞ ½0, T. Then, The following lemma will be used later in the proof of our main result.

Main Result
We begin with the definition of the intended solutions of system (1). Given 0 < T < ∞, let Q T = ½0, T × ℝ N and Φ T be the set of functions φ = φðt, xÞ ∈ C 2,2m t,x ðQ T Þ satisfying the following conditions: is said to be a global weak solution of system (1), if for all 0 Journal of Function Spaces Next, we introduce the parameters Here, m and N are positive natural numbers, p, q, r, s > 1, σ and δ are positive numbers that will be specified later, 0 < α, β < 1, and C D κ 0,t , 0 < κ < 1. If then there exists no global weak solution of system (1). Consider now the case of a single equation. Namely, where 0 < α < 1, σ ≥ 0, and p, q > 1. A global weak solution of (18) can be defined in a similar way as in Definition 3. Taking in Theorem 4u = v, α = β, σ = δ, q = r, p = s, and ðu 0 , u 1 Þ = ðv 0 , v 1 Þ, one deduces the following corollary.

Proof of Theorem 4
Before proving Theorem 4, we need some preliminary results.
3 Journal of Function Spaces where A is defined by (7) with γ ≫ 1 and where μ ≫ 1, ℓ > 0 is a certain parameter that will be specified later, and Ξ ∈ C ∞ 0 ð½0,∞Þ is a decreasing function satisfying Proof. One can check easily that φ ∈ C 2,2m t,x ðQ T Þ and satisfies conditions (a) and (b). On the other hand, by (9), for all 0 < κ < 1, one obtains which yields This shows that φ satisfies condition (c).
The following estimate follows from elementary calculations.

Lemma 8. There exists a constant C > 0 such that
for any positive natural number m.