A Liouville comparison principle for entire sub- and super-solutions of the equation $u_t-{\Delta }_p(u)=|u{|}^{q-1}u$

Abstract

We establish a Liouville comparison principle for entire 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 $n\ge 1$, $q>0$ and ${\Delta }_p(w):={div}_x(|{\nabla }_{x}w{|}^{p-2}{\nabla }_{x}w)$, $1<p\le 2$. In our study we impose neither restrictions on the behaviour of entire sub- and super-solutions on the hyper-plane $t=0$, nor any growth conditions on the behavior of them or any of their partial derivatives at infinity. We prove that if $1<q\le p-1+\frac{p}{n}$, and $u$ and $v$ are, respectively, an entire weak super- and an entire weak sub-solution of ($\ast$) in $\mathbb{S}$ which belong, only locally in $\mathbb{S}$, to the corresponding Sobolev space and are such that $u\le v$, then $u\equiv v$. The result is sharp. As direct corollaries we obtain both new and known Fujita-type and Liouville-type results.