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.