Electron. J. Diff. Eqns., Vol. 2003(2003), No. 20, pp. 1-12.

Blow-up for p-Laplacian parabolic equations

Yuxiang Li & Chunhong Xie

Abstract:
In this article we give a complete picture of the blow-up criteria for weak solutions of the Dirichlet problem
$$ u_t=\nabla(|\nabla u|^{p-2}\nabla u)+\lambda |u|^{q-2}u,\quad \hbox{in } \Omega_T, $$
where $p greater than 1$. In particular, for $p greater than 2$, $q=p$ is the blow-up critical exponent and we show that the sharp blow-up condition involves the first eigenvalue of the problem
$$ -\nabla(|\nabla \psi|^{p-2}\nabla \psi)=\lambda |\psi|^{p-2}\psi,\quad\hbox{in } \Omega;\quad \psi|_{\partial\Omega}=0. $$

Submitted October 20, 2002. Published February 28, 2003.
Math Subject Classifications: 35K20, 35K55, 35K57, 35K65.
Key Words: p-Laplacian parabolic equations, blow-up, global existence, first eigenvalue.

Show me the PDF file (236K), TEX file, and other files for this article.

Yuxiang Li
Department of Mathematics
Nanjing University
Nanjing 210093, China
email: lieyuxiang@yahoo.com.cn
Chunhong Xie
Department of Mathematics
Nanjing University
Nanjing 210093, China

Return to the EJDE web page