Bounds for blow-up time in a semilinear pseudo-parabolic equation with nonlocal source

Yang Lu,Liang Fei

doi:10.1186/s13660-016-1171-4

Abstract

This paper considers the following semilinear pseudo-parabolic equation with a nonlocal source: $$ u_{t}-\triangle u_{t}-\triangle u=u^{p}(x,t) \int_{\Omega}k(x,y)u^{p+1}(y,t)\,dy, $$ and it explores the characters of blow-up time for solutions, obtaining a lower bound as well as an upper bound for the blow-up time under different conditions, respectively. Also, we investigate a nonblow-up criterion and compute an exact exponential decay.

Highlights

1 Introduction In this paper, we deal with the blow-up problem for the following equation:
( . ), which is a new problem and has not been considered, by means of the potential well method, Yang [ ] obtained the global existence and asymptotic behavior of solutions with deducing exponential decay, and got the existence of solutions that blow up in finite time in H ( )-norm with energy J(u ) ≤ d
To obtain the main results, we introduce the functionals

Summary

Introduction

We deal with the blow-up problem for the following equation:. where ⊂ Rn (n ≥ ) is a bounded domain with smooth boundary, u (x) ∈ H ( ), T ∈ And bounds for the blow-up time have been explored [ , ]. ), which is a new problem and has not been considered, by means of the potential well method, Yang [ ] obtained the global existence and asymptotic behavior of solutions with deducing exponential decay, and got the existence of solutions that blow up in finite time in H ( )-norm with energy J(u ) ≤ d. The authors have trouble getting lower bounds for the blow-up time, and they received little attention. We use the means of a differential inequality technique and present some results on the bounds for the blow-up time to problem

In fact

The Poincaré inequality gives λ