Blow-Up of Solutions for a Class of Nonlinear Pseudoparabolic Equations with a Memory Term

Huafei Di,Yadong Shang

doi:10.1155/2014/507494

Abstract

We consider the nonlinear pseudoparabolic equation with a memory termut-Δu-Δut+∫0tλt-τΔuτdτ=div∇up-2u+u1+α,x∈Ω,t>0, with an initial condition and Dirichlet boundary condition. Under negative initial energy and suitable conditions onp,α, and the relaxation functionλ(t), we prove a finite-time blow-up result by using the concavity method.

Highlights

He investigated the initial boundary value problem of (5) and established the global existence of a strong solution of the problem
In this paper, we consider the initial boundary value problem for a class of nonlinear pseudoparabolic equations with a memory term:t ut − βΔu − γΔut + ∫ λ (t − τ) Δu (x, τ) dτ= δ div (|∇u|p−2∇u) + f (u), x ∈ Ω, t > 0, (1)u (x, 0) = u0 (x), x ∈ Ω, u(x, t)|∂Ω = 0, x ∈ ∂Ω, t > 0, where Ω ⊂ Rn is a bounded domain with a smooth boundary ∂Ω, λ : R+ → R is a given continuous function, β, γ, and δ are all real constant parameters, p > 2, and div(|∇u|p−2∇u) is the so-called p-Laplace operator
By using the concavity method first introduced by Levine [5], under negative initial energy and suitable conditions on p, α, and the relaxation function λ(t), we prove that there exists finite-time blow-up solution

Summary

Introduction

He investigated the initial boundary value problem of (5) and established the global existence of a strong solution of the problem. He established the finite time blow-up result for the solution with negative or vanishing initial energy for nonlinear function f(u) = |u|p−2u. To the best of our knowledge, there are few works on the study of nonlinear pseudoparabolic equation with memory term of Volterra integral type.

Results

Conclusion