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

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.

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

“…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.…”

“…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).…”

Blow‐Up of Solutions for a Coupled Nonlinear Viscoelastic Equation with Degenerate Damping Terms: Without Kirchhoff Term

“…Lemma 3. Assume ( 5), (6), and ( 10) hold; let (u, v) be a solution of (1); then, E(t) is nonincreasing, that is,…”

“…Later, the same author in [5] considered the same problem but (g(u, u t ) � u t ) and discussed the decay rate of solution. Recently, in [6], the authors proved the existence of global solution and a general stability result.…”