Taiwanese Journal of Mathematics

Blow-up of a Degenerate Non-linear Heat Equation

Chi-Cheung Poon

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Abstract

We study the blowup behavior of non-negative solutions of the following problem: \begin{equation*} \begin{array}{rll} u_t & \hspace{-0.2cm} \displaystyle = u^{p}(\Delta u + u^q) & \quad {\rm in}\quad \Omega \times (0,T),\\ u(x,t) & \hspace{-0.2cm} \displaystyle = 0 & \quad {\rm whenever} \quad x \in \partial \Omega, \end{array} \end{equation*} with $p \gt 0$ and $q \gt 1$. We will show that it is possible to have solutions blowing up at only one point, and $$\limsup_{t \to T^-} \left( (T-t)^{1/(p+q-1)} \max_\Omega u(x,t) \right) = \infty.$$

Article information

Source
Taiwanese J. Math., Volume 15, Number 3 (2011), 1201-1225.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500406295

Digital Object Identifier
doi:10.11650/twjm/1500406295

Mathematical Reviews number (MathSciNet)
MR2829907

Zentralblatt MATH identifier
1235.35050

Subjects
Primary: 35K55: Nonlinear parabolic equations

Keywords
quasilinear parabolic equation blowup

Citation

Poon, Chi-Cheung. Blow-up of a Degenerate Non-linear Heat Equation. Taiwanese J. Math. 15 (2011), no. 3, 1201--1225. doi:10.11650/twjm/1500406295. https://projecteuclid.org/euclid.twjm/1500406295


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References

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