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Volume 37, Issue 2
Life-Spans and Blow-Up Rates for a $p$-Laplacian Parabolic Equation with General Source

Qunfei Long

DOI: 10.4208/jpde.v37.n2.5

J. Part. Diff. Eq., 37 (2024), pp. 187-197.

Published online: 2024-06

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  • 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.

  • Keywords

$p$-Laplacian, parabolic equation, blow-ups, life-spans, blow-up rates.

  • AMS Subject Headings

35K92, 35B44, 35A23

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JPDE-37-187, author = {Long , Qunfei}, title = {Life-Spans and Blow-Up Rates for a $p$-Laplacian Parabolic Equation with General Source}, journal = {Journal of Partial Differential Equations}, year = {2024}, volume = {37}, number = {2}, pages = {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.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v37.n2.5}, url = {http://global-sci.org/intro/article_detail/jpde/23208.html} }
TY - JOUR T1 - Life-Spans and Blow-Up Rates for a $p$-Laplacian Parabolic Equation with General Source AU - Long , Qunfei JO - Journal of Partial Differential Equations VL - 2 SP - 187 EP - 197 PY - 2024 DA - 2024/06 SN - 37 DO - http://doi.org/10.4208/jpde.v37.n2.5 UR - https://global-sci.org/intro/article_detail/jpde/23208.html KW - $p$-Laplacian, parabolic equation, blow-ups, life-spans, blow-up rates. AB -

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.

Qunfei Long. (2024). Life-Spans and Blow-Up Rates for a $p$-Laplacian Parabolic Equation with General Source. Journal of Partial Differential Equations. 37 (2). 187-197. doi:10.4208/jpde.v37.n2.5
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