Best Possible Upper Bound for Blowup Solutions of the Quasilinear Heat Conduction Equation with Source

Abstract

This article studies the behavior of the weak solution of the Cauchy problem to quasilinear degenerate parabolic equation $u_t = (u^\sigma u_x )_x + u^{\sigma + 1} $, where $\sigma > 0$ is a fixed constant, with a nonnegative bounded compactly supported initial function. Let $T_0 $ be the finite blowup time for solution $u(x,t)$. The estimate $\sup _x u(x,t) \leqq M(t)$ for all $t \in (0,T_0 )$, where $u(x,t)$ is defined from some nonlinear ordinary differential equation, is proved by the comparison of intersection methods (based on the theory of lap number or zero set) with explicit noninvariant blowup solution.