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

#### 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.
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Nonlinear diffusion, Gagliardo-Nirenberg inequalities, asymptotics
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11311/762750
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