Porous media equations with two weights: smoothing and decay properties of energy solutions via Poincar\'e inequalities

TL;DR

This paper investigates weighted porous media equations on various domains, analyzing existence, uniqueness, smoothing effects, and long-term behavior of solutions, with particular focus on Neumann boundary conditions and weighted measures.

Contribution

It provides new insights into the smoothing and decay properties of energy solutions for weighted porous media equations, especially for the less-studied Neumann problem and finite measure cases.

Findings

Existence and uniqueness of weak solutions established.

Short-time smoothing effects linked to weighted Poincaré inequalities.

Long-term convergence to weighted averages in finite measure cases.

Abstract

We study weighted porous media equations on domains $\Omega\subseteq{\mathbb R}^N$, either with Dirichlet or with Neumann homogeneous boundary conditions when $\Omega\not={\mathbb R}^N$. Existence of weak solutions and uniqueness in a suitable class is studied in detail. Moreover, $L^{q_0}$-$L^\varrho$ smoothing effects ($1\leq q_0<\varrho<\infty$) are discussed for short time, in connection with the validity of a Poincar\'e inequality in appropriate weighted Sobolev spaces, and the long-time asymptotic behaviour is also studied. Particular emphasis is given to the Neumann problem, which is much less studied in the literature, as well as to the case $\Omega={\mathbb R}^N$ when the corresponding weight makes its measure finite, so that solutions converge to their weighted average instead than to zero. Examples are given in terms of wide classes of weights.