Universal bounds at the blow-up time for nonlinear parabolic equations

Abstract

We prove a priori supremum bounds for solutions to \begin{equation*} u_{t} - {\text{\rm div}} \big(u^{m-1} | {Du}| ^{\lambda -1} Du \big) = f(x) u^{p}\,, \end{equation*} as $t$ approaches the time when $u$ becomes unbounded. Such bounds are universal in the sense that they do not depend on $u$. Here $f$ may become unbounded, or vanish, as $x\to 0$. When $f\equiv1$, we also prove a bound below, as well as uniform localization of the support, for subsolutions to the corresponding Cauchy problem.