Blow-Up Analysis for a Quasilinear Degenerate Parabolic Equation with Strongly Nonlinear Source

Abstract

We investigate the blow-up properties of the positive solution of the Cauchy problem for a quasilinear degenerate parabolic equation with strongly nonlinear source $u_t=div(|\nabla u^m{|}^{p-2}\nabla u^l)+u^q,(x$ $,t)\in R^N\times (0,T)$, where $N\ge 1$, $p>2$ , and $m$, $l$,$q>1$, and give a secondary critical exponent on the decay asymptotic behavior of an initial value at infinity for the existence and nonexistence of global solutions of the Cauchy problem. Moreover, under some suitable conditions we prove single-point blow-up for a large class of radial decreasing solutions.