Global Existence and General Decay of Solutions for a Quasilinear System with Degenerate Damping Terms

In this work, we consider a quasilinear system of viscoelastic equations with degenerate damping, dispersion, and source terms under Dirichlet boundary condition. Under some restrictions on the initial datum and standard conditions on relaxation functions, we study global existence and general decay of solutions. The results obtained here are generalization of the previous recent work.

By taking in which a > 0,b > 0, and 1 < κ < +∞if n = 1, 2 and 1 < κ ≤ 3 − n n − 2 if n ≥ 3: It is simple to show that where To motivate our problem (1), it can trace back to the initial boundary value problem for the single viscoelastic equation of the form This type problem appears a variety of mathematical models in applied science. For instance, in the theory of viscoelasticity, physics, and material science, problem (5) has been studied by various authors, and several results concerning blow-up and energy decay have been studied case (η ≥ 0). For example, Liu [1] studied a general decay of solutions case ðgðu, u t Þ = 0Þ. Messaoudi and Tatar [2] applied the potential well method to indicate the global existence and uniform decay of solutions (gðu, u t Þ = 0 instead of Δu t ). Furthermore, the authors obtained a blow-up result for positive initial energy. Wu [3] studied a general decay of solution case (gðu, u t Þ = ju t j m u t ). Later, Wu [4] studied the same problem case ðgðu, u t Þ = u t Þ and discussed the decay rate of solution energy. Recently, Yang et al. [5] proved the existence of global solution and asymptotic stability result without restrictive conditions on the relaxation function at infinity case (f ðuÞ = σðx, tÞW t ðt, xÞ).
In case gðu, u t Þ = 0 and without dispersion term, problem (5) has been investigated by Song [6], and the blow-up result for positive initial energy has been proved. For a coupled system, He [7] investigated the following problem where η > 0,j, s ≥ 2: The author proved general and optimal decay of solutions. Then, in [8], the author investigated the same problem without damping term and established a general decay of solutions. Furthermore, the author obtained a blow-up of solutions for negative initial energy. In addition, problem (1) with in case η = 0 and without dispersion term, Wu [9] proved a general decay of solutions. Later, Pișkin and Ekinci [10] studied a general decay and blow-up of solutions with nonpositive initial energy for problem (1) case (Kirchhoff-type instead of Δu and without dispersion term).
In recent years, some other authors investigate the hyperbolic type system with degenerate damping term (see [11][12][13][14]). The rest of the paper is arranged as follows: in Section 2, as preliminaries, we give necessary assumptions and lemmas that will be used later and local existence theorem without proof. In Section 3, we prove the global existence of solution. In the last section, we studied the general decay of solutions.

Preliminaries
We begin this section with some assumptions, notations, lemmas, and theorems. Denote the standart L 2 ðΩÞ norm by k:k = k:k L 2 ðΩÞ and L p ðΩÞ norm by k:k p = k:k L p ðΩÞ : To state and prove our result, we need some assumptions: (A1) Regarding h i : ½0, ∞Þ ⟶ ð0, ∞Þ,ði = 1, 2Þ is C 1 functions and satisfies and nonincreasing differentiable positive C 1 functions ς 1 and ς 2 such that (A2) For the nonlinearity, we assume that (A3) Assume that η satisfies In addition, we present some notations: Lemma 1 (Sobolev-Poincare inequality) [15]. Let q be a number with 2 ≤ q < ∞ðn = 1, 2Þ or 2 ≤ q ≤ 2n/ðn − 2Þðn ≥ 3Þ, and then there is a constant C * = C * ðΩ, qÞ such that Now, we state the local existence theorem that can be established by combining arguments of [7,10].
We define the energy function as follows: Also, we define By computation, we get

Global Existence
In this part, in order to state and prove the global existence of solution (1), we firstly give two lemmas.
Theorem 5. Suppose that the conditions of Lemma 4 hold, then the solution (1) is bounded and global in time.
Proof. We have Thus, where positive constant C depends only on κ, l 1 , l 2 : This implies that the solution of problem (1) is global in time.