Year: 2024
Author: QunFei Long
Journal of Partial Differential Equations, Vol. 37 (2024), Iss. 2 : pp. 187–197
Abstract
This article investigates the blow-up results for the initial boundary value problem to the quasi-linear parabolic equation with $p$-Laplacian $$u_t−∇·( |∇u|^{p−2}∇u)= f(u),$$ where $p≥2$ and the function $f(u)$ satisfies $$α\int^u_0f(s)ds≤u f(u)+βu^p+\gamma, u>0$$ for some positive constants $α,β,\gamma$ with $0<β≤ \frac{(α−p)λ_{1,p}}{p},$ which has been studied under the initial condition $J_p(u_0)<0.$ 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jpde.v37.n2.5
Journal of Partial Differential Equations, Vol. 37 (2024), Iss. 2 : pp. 187–197
Published online: 2024-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 11
Keywords: $p$-Laplacian parabolic equation blow-ups life-spans blow-up rates.