Global nonexistence of solutions for the viscoelastic wave equation of Kirchhoff type with high energy

In this paper we consider the viscoelastic wave equation of Kirchhoff type: $$ u_{tt}-M(\|\nabla u\|_{2}^{2})\Delta u+\int_{0}^{t}g(t-s)\Delta u(s){\rm d}s+u_{t}=|u|^{p-1}u $$ with Dirichlet boundary conditions. Under some suitable assumptions on $g$ and the initial data, we established a global nonexistence result for certain solutions with arbitrarily high energy.


Introduction
In this paper we consider the following problem where Ω is a bounded domain in R n (n ≥ 1) with a smooth boundary ∂Ω, p > 1, M (s) is a nonnegative C 1 function like M (s) = a + bs γ for s ≥ 0, a ≥ 0, b ≥ 0, a + b > 0, γ > 0 and g(t) represents the kernel of memory term.
Problem (1.1) without the viscoelastic term (i.e., g = 0) has been extensively studied and many results concerning global existence, decay and blow-up have been established. For example, the following equation has been considered by Matsuyama and Ikehata in [10], for g(u t ) = δ|u t | p−1 u t and f (u) = µ|u| q−1 u. The authors proved existence of the global solutions by using Faedo-Galerkin method and the decay of energy based on the method of Nakao [13]- [15]. Later, Ono [16] investigated equation (1.2) for M (s) = bs γ and f (u) = |u| p−2 u. When g(u t ) = −∆u t , u t or |u t | β u t , the author showed that the solutions blow up in finite time with E(0) ≤ 0. For M (s) = a+bs γ and g(u t ) = u t , this model was considered by the same author in [17]. By applying the potential well method he obtained the blow-up properties with positive initial energy E(0). Recently, Zeng et al. [27] studied equation (1.2) for the case g(u t ) = u t with initial condition and zero Dirichlet boundary condition. By using the concavity argument, they proved that the solutions to equation (1.2) blow up in finite time with arbitrarily high energy.
In the case of M ≡ 1 and in the presence of the viscoelastic term (i.e., g = 0), the equation was studied by Messaoudi in [11], where the author proved that any weak solution with negative initial energy blows up in finite time if p > m and while the solution continue to exist globally for any initial data in the appropriate space if m ≥ p.
This blow-up result was improved by the same author in [12] for positive initial energy under suitable conditions on g, m and p. More recently, Wang [22] investigated equation (1.3) and established a blow-up result with arbitrary positive initial energy. In the related work, Cavalcanti et al. [1] studied the following equation where a : Ω −→ R + is a function which may be null on a part of Ω. Under the condition that a(x) ≥ a 0 > 0 on ω ⊂ Ω, with ω satisfying some geometric restrictions and −ξ 1 g(t) ≤ g ′ (t) ≤ −ξ 2 g(t), t ≥ 0 to guarantee g L 1 ((0,∞)) is small enough, they proved an exponential decay rate.
When g = 0 and M is not a constant function, problems related to (1.1) have been treated by many authors. Wu and Tsai [24] considered the global existence, asymptotic behavior and blow-up properties for the following equation with the same initial and boundary conditions as that of problem (1.1). To obtain the decay result, they assumed that the nonnegative kernel g ′ (t) ≤ −rg(t), ∀ t ≥ 0 for some r > 0. In [23], Wu then extended the result of [24] under a weaker condition on g (i.e., g ′ (t) ≤ 0 for t ≥ 0).
Motivated by the above research, we consider problem (1.1) for m = 1 in this paper and establish a global nonexistence result for certain solutions with arbitrarily high energy. In this way, we can extend the result of [27] to nonzero term g and the result of [22] to nonconstant M (s). We also obtain the new result for blow-up properties of local solution with arbitrarily high energy. Throughout the rest of this paper, we always assume that m = 1.
The structure of this paper is as follows. In section 2, we present some assumptions, notations and main result. Section 3 is devoted to the proof of the main result.

Preliminaries and main result
In this section, we shall give some assumptions, notations and main result. We first give the following assumptions: ) is a non-negative and non-increasing function satisfying is of positive type in the following sense: In order to prove our result, we make the following assumption on M and g : (A3) There exist two positive constants, m 1 and α, such that if a > 0 and b ≥ 0, which is the same as the one in [22, Theorem 1.1] for the case a = 1 and b = 0, where C p is the constant from the Poincaré inequality u(t) 2 2 ≤ C p ∇u(t) 2 2 . Indeed, by straightforward calculation, we obtain If a = 0 and b > 0, then Taking s = ∇u(t) 2 2 , applying Lemma 3.3 below and Poincaré's inequality, we can get So, we can choose m 1 = p+1 Next, we introduce some notations. The energy functional E(t) and an auxiliary functional I(u) of the solution u(t) of problem (1.1) are defined as follows: and where (g • w)(t) = t 0 g(t − s) w(t, ·) − w(s, ·) 2 2 ds.
As in [22,27], we can get for t ≥ 0. Then we have Now we are in a position to state our main result.

Proof of main result
Before we start to prove Theorem 2.1, it is necessary to state the local existence theorem for problem (1.1), whose proof follows the arguments in [19,24].
The proof of Theorem 2.1 relies on the following lemmas.   Proof. Since u(t) is the solution of problem (1.1), by a simple computation, we have where the last inequality is derived by (3.3). Then we get Therefore, by using Lemma 3.2, we finish our proof.
A straightforward calculation gives consequently, We will use the following Young inequality to estimate the fifth term of the right hand side of (3.18), rs ≤ r 2 2ǫ + ǫs 2 2 where ǫ = 1 2 , r ≥ 0 and s ≥ 0. We obtain