Blow-Up for a Stochastic Viscoelastic Lamé Equation with Logarithmic Nonlinearity

Amina Benramdane,Mohammed Sulaiman Alqawba,Salah Mahmoud Boulaaras,Bahri Belkacem Cherif,Nadia Mezouar,Chuanjun Chen

doi:10.1155/2021/9943969

Abstract

In this paper, we consider an initial boundary value problem of stochastic viscoelastic wave equation with nonlinear damping and logarithmic nonlinear source terms. We proved a blow-up result for the solution with decreasing kernel.

Highlights

Stochastic partial differential equations in a separable Hilbert space have been studied by many authors, and various results on the existence, uniqueness, stability, blow-up, and other quantitative and qualitative properties of solutions have been established
“A general stability result for second order stochastic quasilinear evolution equations with memory,” Boundary Value Problems, vol 2020, no. 1, Article ID 62, 2020
R. Kang, “Global existence and uniform decay for a nonlinear viscoelastic equation with damping,” Acta Applicandae Mathematicae, vol 110, no

Summary

Introduction

Stochastic partial differential equations in a separable Hilbert space have been studied by many authors, and various results on the existence, uniqueness, stability, blow-up, and other quantitative and qualitative properties of solutions have been established. ( utt = Δu + f ðuÞ + σðuÞWtðx, tÞ in x ∈ Rd, t > 0, ð2Þ uðx, 0Þ = gðxÞ, utðx, 0Þ = hðxÞ, where the initial data g and h are given functions and the nonlinear terms f ðuÞ and σðuÞ are assumed to be polynomials in u Four years later, he [4] established an energy inequality and the exponential bound for a linear stochastic equation and gave the existence theorem for a unique global solution for the randomly perturbed wave equation:. Yang et al [8] treated the following stochastic up in finite time They established the existence of global solution and asymptotic stability of the solution by using some properties of the convex function.

Preliminaries

Blow-Up