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Recent Trends in Nonlinear Diffusion

Asymptotics of Solutions of Weighted Porous Media Equations

speaker: Gabriele Grillo (Politecnico di Milano)

abstract: We consider weighted versions of the porous media equation, requiring appropriate versions of (homogeneous) Dirichlet of Neumann boundary conditions. Existence of solutions, and uniqueness in a suitable class, is investigated. The short and long time asymptotics of solutions is studied in connection with weighted functional inequalities naturally associated with the weights appearing in the equation. In particular, the equivalence between suitable weighted Poincaré inequalities (or weighted Sobolev inequalities) and short time regularizing properties of the evolution is proved. The long-time asymptotics can also be studied by using only appropriate functional inequalities. For example, in the case of Neumann boundary conditions, sharp bounds for the rate of uniform convergence to the (weighted) mean value can be proved as a consequence of weighted Sobolev inequalities. As a corollary, in all such cases the support of a solution corresponding to a smooth, positive, compactly supported datum becomes the whole space in finite time.

Tue 3 Jul, 10:00 - 11:00, Aula Dini


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