Blow-up of solutions for a quasilinear system with degenerate damping terms

where k, l, θ , ≥ 0; j, s ≥ 1 for N = 1, 2, and 0 ≤ j, s ≤ N+2 N–2 for N ≥ 3; and η ≥ 0 for N = 1, 2 and 0 < η ≤ 2 N–2 for N ≥ 3, hi(·) : R+ → R+ (i = 1, 2) are positive relaxation functions which will be specified later. (|(·)|a + |(·)|b)|(·)t|τ–1(·)t and – (·)tt are the degenerate damping term and the dispersion term, respectively, and M(σ ) is a nonnegative locally Lipschitz function for γ ,σ ≥ 0 like M(σ ) = α1 + α2σ . Especially, we select α1 = α2 = 1, and


Introduction
In this paper, we consider the following problem: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ |u t | η u tt -M( ∇u 2 2 ) u + t 0 h 1 (ts) u(s) dsu tt + (|u| k + |v| l )|u t | j-1 u t = f 1 (u, v), (x, t) ∈ × (0, T), |v t | η v tt -M( ∇v 2 2 ) v + Physically, the relationship between the stress and strain history in the beam inspired by Boltzmann theory is called viscoelastic damping term, where the kernel of the term of memory is the function h (for further details, see the references [3, 5-9, 12, 17, 20]). If η ≥ 0, this type of problem has been studied by many authors. For more depth, here are some papers that focused on the study of this damping (for example, see ( [10,11,13,16,19,25,27,28]). The effect of the degenerate damping terms often appears in many applications and practical problems and turns a lot of systems into different problems worth studying. Recently, the stability, the asymptotic behavior, and blowing up of evolution systems with time degenerate damping have been studied by many authors, see [1,2,18]. The most important is the source term with nonlinear functions f 1 and f 2 satisfying appropriate conditions. In physics they appear in several issues and theories. Many researchers also touched on this type of problem in several different issues, where the global existence of solutions, stability, and blow-up of solutions were studied. For more information, the reader is referred to [1,11,15,18,[22][23][24]26]. Most recently, if γ = 0, α 1 = 1 our problem (1.1) was studied in [14]. Under some restrictions on the initial datum, standard conditions on relaxation functions, the authors established the global existence and proved the general decay of solutions. Based on all of the above results, we believe that the combination of these terms of damping (memory term, degenerate damping, dispersion, and the source terms) constitutes a new problem worthy of study and research, different from the above that we will try to shed light on. Our paper is divided into several sections: in the next section we lay down the hypotheses, concepts, and lemmas we need. In the last section we prove our main result.

Preliminaries
We prove the blow-up result under the following suitable assumptions.
(A1) h i : R + → R + are differentiable and decreasing functions such that we take a 1 = b 1 = 1 for convenience.

Lemma 2.2 [22]
There exist two positive constants c 0 and c 1 such that Then, for any initial data problem (1.1) has a unique solution for some T > 0 Proof By multiplying (1.1) 1 , (1.1) 2 by u t , v t and integrating over , we get d dt we obtain (2.5) and (2.6).

Blow-up
In this section, we prove the blow-up result of solution of problem (1.1). First, we define the functional 2) and (2.4) hold, and suppose that E(0) < 0 and Then the solution of problem (1.1) blows up in finite time.

Conclusion
In this paper, we are interested in the blow-up for a quasilinear system of viscoelastic equations with degenerate damping and general source terms according to some suitable hypothesis. This work is a general case of the recent results of Boulaaras' works in [11,21] using the energy method. Next we will prove the result of local existence of this studied problem based on the recent result in [4].