We prove sharp asymptotic bounds for solutions to the porous media equation with homogeneous Dirichlet or Neumann boundary conditions on a bounded Euclidean domain, in dimension $N=1$ and $N=2$. This is achieved by making use of appropriate Gagliardo-Nirenberg inequalities only. The generality of the discussion allows to prove similar bounds for \emph{weighted} porous media equations, provided one deals with weights for which suitable Gagliardo-Nirenberg inequalities hold true. Moreover, we show equivalence between such functional inequalities and the mentioned asymptotic bounds for solutions.

Sharp asymptotics for the porous media equation in low dimensions via Gagliardo-Nirenberg inequalities

GRILLO, GABRIELE;MURATORI, MATTEO
2014

Abstract

We prove sharp asymptotic bounds for solutions to the porous media equation with homogeneous Dirichlet or Neumann boundary conditions on a bounded Euclidean domain, in dimension $N=1$ and $N=2$. This is achieved by making use of appropriate Gagliardo-Nirenberg inequalities only. The generality of the discussion allows to prove similar bounds for \emph{weighted} porous media equations, provided one deals with weights for which suitable Gagliardo-Nirenberg inequalities hold true. Moreover, we show equivalence between such functional inequalities and the mentioned asymptotic bounds for solutions.
Nonlinear diffusion, Gagliardo-Nirenberg inequalities, asymptotics
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11311/762750
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 5
  • ???jsp.display-item.citation.isi??? ND
social impact