A blow-up result for a system of coupled viscoelastic equations with arbitrary positive initial energy

Qian Li

doi:10.1186/s13661-021-01538-1

Abstract

This article is devoted to a study of the blow-up result for a system of coupled viscoelastic wave equations. By establishing a new auxiliary function and using the reduction to absurdity method, we obtain some sufficient conditions on initial data such that the solution blows up in finite time at arbitrarily high initial energy. This work generalizes and improves earlier results in the literature.

Highlights

1 Introduction In this article, we investigate the blow-up property of the coupled viscoelastic wave equations of the form u(τ ) dτ + |ut|m–2ut = f1(u, v), x∈
The author studied the interaction between the weak damping term ut|ut|m–2 and the nonlinear source term u|u|p–2, which was first considered by Levine [9, 10] when m = 1, and found, under suitable conditions on g and initial data, that the solutions exist globally for any initial data if m ≥ p and blow up in finite time with negative initial energy if p > m
Our aim in this paper is to extend the research method for the blow-up phenomena used in [8] to the couple viscoelastic wave system (1.1), while we should handle the additional difficulty caused by damping term, viscoelastic term and source term

Summary

Introduction

We investigate the blow-up property of the coupled viscoelastic wave equations of the form. The author studied the interaction between the weak damping term ut|ut|m–2 and the nonlinear source term u|u|p–2, which was first considered by Levine [9, 10] when m = 1, and found, under suitable conditions on g and initial data, that the solutions exist globally for any initial data if m ≥ p and blow up in finite time with negative initial energy if p > m. In the presence of the memory term, Han and Wang [5] considered the following system of viscoelastic equations: u(τ ) dτ + ut|ut|m–1 = f1(u, v), x∈ They obtained the local existence, global existence, uniqueness and a blow-up result for certain solutions with negative initial energy. G1(t – τ ) u(τ ) dτ u(t) dx + uut|ut|m–2 dx t g2(t – τ ) v(τ ) dτ v(t) dx + vvt|vt|m–2 dx (3.3) (3.4)

Taking the derivative of and

Then we choose ε small enough such that