A general stability result for second order stochastic quasilinear evolution equations with memory

Hui Yang,Min Li,Shiyue Fang,Fei Liang

doi:10.1186/s13661-020-01359-8

Abstract

The goal of this paper is to discuss an initial boundary value problem for the stochastic quasilinear viscoelastic evolution equation with memory driven by additive noise. We prove the existence of global solution and asymptotic stability of the solution using some properties of the convex functions. Moreover, our result is established without imposing restrictive assumptions on the behavior of the relaxation function at infinity.

Highlights

The quasilinear viscoelastic wave equation of the following form: ⎧⎪⎪⎨|ut|ρ utt – u+t 0 g (t s) u(s) ds + h(ut) = f (u),(x, t) ∈ D × (0, T),⎪⎪⎩uu((xx, t) 0) = = 0, (x, u0(x), t)∈ ∂D × (0, ut(x, 0) =T ), u1(x), x ∈ D, (1.1)
⎩2 < p < 2(d – 1)/(d – 2), 2 < ρ ≤ d/(d – 2), if d ≥ 3. He proved the global nonexistence of the positive initial energy solutions of the quasilinear viscoelastic wave equation
In the case of σ = σ (x, t), they proved that the solution either blows up in finite time with positive probability or is explosive in L2 using the energy inequality

Summary

Introduction

The author proved that, for a certain class of relaxation functions and certain initial data in the stable set, the decay rate of the solution energy is similar to that of the relaxation function. He proved the global nonexistence of the positive initial energy solutions of the quasilinear viscoelastic wave equation. Let us firstly recall some results regarding ρ = 0 and g ≡ 0, (1.2) can be rewritten as the following stochastic wave equation: utt – u + h(ut) = f (u) + σ (x, t)∂tW (t, x), x ∈ D, t ∈ (0, T).

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Results

Conclusion