L^q-Functional Inequalities and Weighted Porous Media Equations

Abstract

Using measure-capacity inequalities we study new functional inequalities, namely L ^q-Poincaré inequalities

${\bf{Var}}_{\mu } {\left( {f^{q} } \right)}^{{1 \mathord{\left/ {\vphantom {1 q}} \right. \kern-\nulldelimiterspace} q}} \leqslant {\text{C}}_{{\text{P}}} {\int {{\left| {\nabla f} \right|}^{2} dv} }$

and L ^q-logarithmic Sobolev inequalities

${\bf{Ent}}_{\mu } {\left( {f^{{2q}} } \right)}^{{1 \mathord{\left/ {\vphantom {1 q}} \right. \kern-\nulldelimiterspace} q}} \leqslant {\text{C}}_{{{\text{LS}}}} {\int {{\left| {\nabla f} \right|}^{2} dv} }$

for any q ∈ (0, 1]. As a consequence, we establish the asymptotic behavior of the solutions to the so-called weighted porous media equation

$\frac{{\partial u}}{{\partial t}} = \Delta u^{m} - \nabla \psi \cdot \nabla u^{m} $

for m ≥ 1, in terms of L ²-norms and entropies.