Degenerate parabolic $p$-Laplacian equations: existence, uniqueness and asymptotic behavior of solutions

TL;DR

This paper investigates degenerate parabolic p-Laplacian equations with matrix and weight degeneracies, establishing existence, uniqueness, and conditions for finite time extinction and ultracontractive bounds.

Contribution

It provides new results on existence, uniqueness, and asymptotic behavior of solutions under mild assumptions, linking ultracontractive bounds to weighted Sobolev inequalities.

Findings

Existence and uniqueness of solutions under mild integrability conditions.

Finite time extinction and ultracontractive bounds under Sobolev inequality assumptions.

Equivalence between ultracontractive bounds and weighted Sobolev inequalities.

Abstract

In this paper we study the degenerate parabolic $p$-Laplacian,$ \partial_t u - v^{-1}{\rm div}(|\sqrt{Q} \nabla u|^{p-2} Q \nabla u)=0$, where the degeneracy is controlled by a matrix $Q$ and a weight $v$. With mild integrability assumptions on $Q$ and $v$, we prove the existence and uniqueness of solutions on any interval $[0,T]$. If we further assume the existence of a degenerate Sobolev inequality with gain, the degeneracy again controlled by $v$ and $Q$, then we can prove both finite time extinction and ultracontractive bounds. Moreover, we show that there is equivalence between the existence of ultracontractive bounds and the weighted Sobolev inequality.