Riv. Mat. Univ. Parma, Vol. 5, No. 1, 2014

Gabriele Grillo [1] and Matteo Muratori [1]

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

Pages: 15-38
Received: 25 March 2013   
Accepted: 30 April 2013
Mathematics Subject Classification (2010): Primary: 35K55, 35B40; Secondary: 35K65, 39B62.

Keywords: Weighted porous media equation, smoothing effect, asymptotic behaviour, weighted Poincaré, Sobolev and Gagliardo-Nirenberg inequalities, nonlinear diffusion equations.
Authors addresses:
[1] : Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy

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 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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