English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We study a class of nonlinear diffusion equations whose model is theclassical porous media equation on domains $\Omega\subseteq{\mathbb R}^N$,$N\ge3$, with homogeneous Neumann boundary conditions. Firstly we improve someknown results in such model case, both as concerns sharp $L^{q_0}$-$L^\infty$regularizing properties of the evolution for short time and as concerns sharplong time asymptotics in the sense of $L^\infty$ convergence of solutions totheir mean value. The generality of the discussion allows to consider, almostat the same time, also weighted versions of the above equation provided anappropriate weighted Sobolev inequality is required to hold. \normalcolor Infact, we show that the validity of a slightly weaker functional inequality isequivalent to the validity of a suitable $L^{q_0}$-$L^\infty$ bound forsolutions to the associated weighted porous media equation. The long timeasymptotic analysis relies as well on the assumed weighted Sobolev inequalityonly, and allows to prove uniform convergence to the mean value, with the ratepredicted by linearization, in such generality. This fact was not known evenfor the explicit classes of weights previously considered in the literature.

Sharp short and long time $\mathbf L^{\boldsymbol \infty}$ bounds for solutions to porous media equations with Neumann boundary conditions