Blow-up for p-Laplacian parabolic equations

Abstract

In this article we give a complete picture of the blow-up criteria for weak solutions of the Dirichlet problem $$ u_t= abla(| abla u|^{p-2} abla u)+lambda |u|^{q-2}u,quad hbox{in } Omega_T, $$ 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 $$ - abla(| abla psi|^{p-2} abla psi)=lambda |psi|^{p-2}psi,quadhbox{in } Omega;quad psi|_{partialOmega}=0. $$

Keywords

• p-Laplacian parabolic equations
• blow-up
• global existence
• first eigenvalue.