Abstract
Blow up and asymptotic behavior for a system of viscoelastic wave equations of Kirchhoff type with a delay term
Highlights
In this paper, we are concerned with the following problem:t utt − M ( ∇u 22)∆u + g(t − s)∆u(s)ds + μ1|ut(x, t)|m−1ut(x, t)+μ2|ut(x, t − τ )|m−1ut(x, t − τ ) = |u|p−1u, ut(x, t − τ ) = f0(x, t − τ ), x ∈ Ω, t ∈ (0, τ ), u(x, 0) = u0(x), ut(x, 0) = u1(x), x ∈ Ω, (1.1) (1.2) (1.3)Received February 04, 2018, Accepted: July 19, 2018, Online: August 18, 2018.F
The focus of the current paper is to investigate the initial boundary value problem for a system of viscoelastic wave equations of Kirchhoff type with a delay term in a bounded domain
Global existence and asymptotic behavior In order to prove the global existence result, we introduce the new variable z as in [12], z(x, k, t) = ut(x, t − τ k), x ∈ Ω, k ∈ (0, 1), which implies that τ zt(x, k, t) + zk(x, k, t) = 0, in Ω × (0, 1) × (0, ∞)
Summary
We are concerned with the following problem:. +μ2|ut(x, t − τ )|m−1ut(x, t − τ ) = |u|p−1u, ut(x, t − τ ) = f0(x, t − τ ), x ∈ Ω, t ∈ (0, τ ), u(x, 0) = u0(x), ut(x, 0) = u1(x), x ∈ Ω,. Let us recall some results regarding wave equations of Kirchhoff type This type of problem without delay (i.e., μ2 = 0) has been considered by many authors during the past decades and many results have been obtained (see [9], [18], [22], [24] ) and the references therein. Wu and Tsai [23] considered the global existence, asymptotic behavior and blow-up properties for the following equation t utt − M ( ∇u 22)∆u + g(t − s)∆u(s)ds − ∆ut = f (u),. Under suitable assumptions on the function g, the initial data and the parameters in the equations, we establish several results concerning asymptotic behavior and finite blow-up of solutions to (1.1)-(1.4) for both negative and positive initial energy.
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