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

TL;DR

This paper proves a Liouville comparison principle for entire sub- and super-solutions of a p-Laplacian evolution equation in half-space, establishing conditions under which solutions coincide without growth restrictions.

Contribution

It introduces a sharp Liouville comparison principle for solutions of a nonlinear PDE without imposing growth or boundary conditions at infinity or on the hyper-plane t=0.

Findings

Proves that under certain conditions, sub- and super-solutions must be identical.

Derives new and known Fujita-type and Liouville-type results as corollaries.

Establishes the principle for 1<p≤2 and q within a specific range.

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 $n\geq 1$, $q>0$ 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.