Asymptotics of the Fast Diffusion Equation Via Entropy Methods I: Linearization and Spectral Gap Analysis
by
Matteo Bonforte
Departamento de Matemáticas, Universidad Autónoma de Madrid, Spain
Coauthors: A. Blanchet, J. Dolbeault, G. Grillo and J. L. Vazquez

We consider non-negative solutions of the fast diffusion equation u_t=Du^m with m ∈ (0, 1), in the Euclidean space \RR^d, d ≥ 3, and study the asymptotic behavior of a natural class of solutions, in the limit corresponding to t→∞ for m ≥ m_c=(d-2)/d, or as t approaches the extinction time when m < m_c. For a class of initial data we prove that the solution converges with a polynomial rate to a self-similar solution, for t large enough if m ≥ m_c, or close enough to the extinction time if m < m_c. Such results are new in the range m ≤ m_c where previous approaches fail. In the range m_c < m < 1 we improve on known results.

In the first part, M. Bonforte will explain the functional analitic setup used for the linearized analysis. A new critical exponent appears in the spectral analysis when considering dimension d ≥ 5, namely m^*=(d-4)/(d-2). We prove Hardy-Poincarè Inequalities and we get sharp estimates for the constant, achieving the best constant in the lower range (0, m^*).

In the second part A. Blanchet will explain how to pass from the linearized functional inequalities to the nonlinear counterparts, i.e. how to pass from linear Hardy-Poincarè inequalities to entropy-entropy production estimates which finally give the asymptotic result.

Date received: July 3, 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 # cavj-85.