Electronic Journal of Differential Equations,
Vol. 2003(2003), No. 20, pp. 1-12.
Title: Blow-up for p-Laplacian parabolic equations
Authors: Yuxiang Li (Nanjing University, China)
Chunhong Xie (Nanjing University, China)
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 $p>1$. In particular, for $p>2$, $q=p$ 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.