Logarithmic Sobolev inequalities and nonlinear evolution equations
by
Gabriele Grillo
Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

We discuss the role of techniques based on Sobolev and on logarithmic Sobolev inequalities in the study of asymptotics and of L^p-L^q regularization properties of certain nonlinear evolution equations. The class of evolution considered includes the equations [u\dot]=D_pu and [u\dot]=D(u^m) both in the degenerate and in the singular range. We discuss in particular, as a model issue, the equivalence between the validity of an euclidean-type Sobolev inequality on a Riemannian manifold and the bound ∥u(t)∥_∞ ≤ C∥u₀∥_q^g t^-b (q ≥ 1) required to be valid for all u₀ ∈  L^q, all t > 0 and suitable b, g. As a second example we mention a variant of a result of Gross. It says that, if a suitable single logarithmic Sobolev inequality involving a gradient-type nonlinear Dirichlet form holds in a probability space, then the p-Laplacian-driven evolution equation associated to the generator of the Dirichlet form regularizes instantaneously from L² to L^q for all q ∈ (2, ∞).

Date received: May 25, 2007

Copyright © 2007 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caum-54.