A Liouville comparison principle for sub- and super-solutions of the equation $w_t-\Delta_p (w) = |w|^{q-1}w$

Abstract

We establish a Liouville comparison principle for entire weak sub- and super-solutions of the equation $(\ast)$ $w_t-\Delta_p (w) = |w|^{q-1}w$ in the half-space ${\mathbb S}= {\mathbb R}^1_+\times {\mathbb R}^n$, where $n\geq 1$, $q>0$ and $ \Delta_p (w):={div}_x(|\nabla_x w|^{p-2}\nabla_x w)$, $1<p\leq 2$. In our study we impose neither restrictions on the behaviour of entire weak sub- and super-solutions on the hyper-plane $t=0$, nor any growth conditions on their behaviour and on that of any of their partial derivatives at infinity. We prove that if $1<q\leq p-1+\frac pn$, and $u$ and $v$ are, respectively, an entire weak super-solution and an entire weak sub-solution of ($\ast$) in $\Bbb S$ which belong, only locally in $\Bbb S$, to the corresponding Sobolev space and are such that $u\geq v$, then $u\equiv v$. The result is sharp. As direct corollaries we obtain known Fujita-type and Liouville-type theorems.