Porous media equations with two weights:  Smoothing and decay properties of energy  solutions via Poincaré inequalities

Gabriele Grillo; Matteo Muratori; Maria Michaela Porzio

doi:10.3934/dcds.2013.33.3599

Article Contents

2013, Volume 33, Issue 8: 3599-3640. Doi: 10.3934/dcds.2013.33.3599

This issue Previous Article On the moments of solutions to linear parabolic equations involving the biharmonic operator Next Article Partial hyperbolicity on 3-dimensional nilmanifolds

Porous media equations with two weights: Smoothing and decay properties of energy solutions via Poincaré inequalities

1.
Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano
2.
Dipartimento di Matematica, Università di Roma "La Sapienza", Piazzale A. Moro 2, 00185 Roma

Received: July 31, 2012

Revised: October 31, 2012

Published: July 2013

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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 Poincaré inequality in appropriate weighted Sobolev spaces, and the long-time asymptotic behaviour is also studied. In fact, we prove full equivalence between certain $L^{q_0}$-$L^\varrho$ smoothing effects and suitable weighted Poincaré-type inequalities. 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 mean value instead than to zero. Examples are given in terms of wide classes of weights.

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Mathematics Subject Classification: Primary: 35K55, 35B40; Secondary: 35K65, 39B62.

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