Life-Spans and Blow-Up Rates for a p-Laplacian Parabolic Equation with General Source

Abstract

This article investigates the blow-up results for the initial boundary value problem to the quasi-linear parabolic equation with p-Laplacian

$u_{t}-\nabla \cdot\left(|\nabla u|^{p-2} \nabla u\right)=f(u),$

where p≥2 and the function f(u) satisfies

$\alpha \int_{0}^{u} f(s) \mathrm{d} s \leq u f(u)+\beta u^{p}+\gamma, \quad u>0$

for some positive constants α,β,γ with $0<\beta \leq \frac{(\alpha-p) \lambda_{1, p}}{p}$, which has been studied under the initial condition $J_{p}\left(u_{0}\right)<0.$ This paper generalizes the above results on the following aspects: a new blow-up condition is given, which holds for all p>2; a new blow-up condition is given, which holds for p=2; some new lifespans and upper blow-up rates are given under certain conditions.

Keywords

p-Laplacian / parabolic equation / blow-ups / life-spans / blow-up rates