Stochastic quasilinear viscoelastic wave equation with nonlinear damping and source terms

Sangil Kim,Yong Han Kang,Jong-Yeoul Park

doi:10.1186/s13661-018-0932-z

Abstract

The goal of this study is to investigate an initial boundary value problem for the stochastic quasilinear viscoelastic wave equation involving the nonlinear damping vert u_{t} vert ^{q-2} u_{t} and a source term of the type vert u vert ^{p-2}u driven by additive noise. By an appropriate energy inequality, we prove that finite time blow-up is possible for equation (1.1) below if p > {q, rho +2 } and the initial data are large enough (that is, if the initial energy is sufficiently negative). Also, we show that if q geq p, the local solution can be extended for all time and is thus global.

Highlights

In this paper, we are concerned with the following stochastic viscoelastic wave equation: t|ut|ρ utt – u – utt + h(t – τ ) u(τ ) dτ + |ut|q–2ut= |u|p–2u + σ (x, t)∂tW (x, t) in D × (0, T), u = 0 on ∂D × (0, T), u(x, 0) = u0(x), ut(x, 0) = u1(x) in D, (1.1)where D is a bounded domain in Rn with smooth boundary ∂D, with given positive constants ρ > 0, q ≥ 2, and p ≥ 2
Where D is a bounded domain in Rn with smooth boundary ∂D, with given positive constants ρ > 0, q ≥ 2, and p ≥ 2
System (1.1) without the stochastic term is a model for quasilinear viscoelastic wave equation with nonlinear damping and source terms

Summary

Introduction

We are concerned with the following stochastic viscoelastic wave equation: t. Cheng et al [23] proved the existence of a global solution and blow-up solutions with positive probability for the nonlinear stochastic viscoelastic wave equation with linear damping (see [18, 22, 26]). = |u|p–2u + σ (x, t)∂tW (x, t) in D × (0, T), u = 0 on ∂D × (0, T), u(x, 0) = u0(x), ut(x, 0) = u1(x) in D , where D is a bounded domain in Rn with smooth boundary ∂D, q ≥ 2, p ≥ 2, is a given positive constant which measures the strength of noise; W (x, t) is an infinite dimensional Wiener process; σ (x, t, w) is L2(D)-valued progressively measurable; and h is a positive relaxation function. We prove that the stochastic quasilinear viscoelastic wave equation (1.1) can blow up with positive probability or it is explosive in energy sense for p > {q, ρ + 2} and obtain the existence of global solution by the Borel-Cantelli lemma.

Thus there exists a constant

Note that