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

By G. Grillo, Matteo Muratori, M. Porzio. 2012

56 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/28120897248c3c50b42b71a30fb69162fb33ab48

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

The Mathematical Theories of Diffusion: Nonlinear and Fractional Diffusion (2017, background): “Let us just mention some well-known references like [161, 207, 160], where it appears as diffusion with weights, see also the more recent [142] with two weights.”