Stability analysis of asymptotic profiles for fast diffusion

Date

Dec 14, 2017

Time

3:15 PM - 4:15 PM

Speaker

Prof. Dr. Goro Akagi

Affiliation

Tohoku University, Mathematical Institute

Series

TUD Mathematik Oberseminar Analysis

Language

en

Main Topic

Mathematik

Other Topics

Mathematik

Host

Prof. Dr. S. Neukamm

Description

This talk is concerned with asymptotic profiles for solutions to the Cauchy-Dirichlet problem for the Fast Diffusion equation (FD) in smooth bounded domains under the so-called Sobolev subcritical condition. It is well-known that every solution of (FD) vanishes in finite time with a power rate; more precisely, it asymptotically approaches to a separable solution (Berryman and Holland '80). Then the asymptotic profile for each vanishing solution can be characterized as a non-trivial solution of the Emden-Fowler equation (EF). The (global) stability of asymptotic profiles was also discussed for the case that (EF) has a unique positive solution; on the other hand, the case that (EF) may have multiple (positive) solutions had not been studied for many years. In this talk, we shall first overview how to formulate notions of stability and instability of asymptotic profiles, and then, we shall exhibit criteria to distinguish the (in)stability of each asymptotic profile. Next, we shall move on to discussing how to treat non-isolated asymptotic profiles; indeed, (EF) may admit a one-parameter family of positive solutions, e.g., for sufficiently thin annular domains. In particular, for such thin annular domains, each non-radial asymptotic profile belonging to some one-parameter family turns out to be stable and the (unique) radial positive profile turns out to be unstable. The method of analysis relies on variational method, uniform extinction estimates for solutions to (FD), the Łojasiewicz-Simon inequality and energy techniques. Furthermore, we may also discuss other related issues, e.g., exponential stability of non-degenerate asymptotic profiles of least energy.

Links

• Oberseminar Analysis

Last modified: Nov 28, 2017, 3:57:44 PM