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 $n\geq 1$, $q>0$ 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 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 $1<q\leq p-1+\frac pn$, and $u$ and $v$ are, respectively, an entire weak super- 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 $u\leq v$, then $u\equiv v$. The result is sharp. As direct corollaries we obtain both new and known Fujita-type and Liouville-type results.
Categories: math.AP