Seminár z kvalitatívnej teórie diferenciálnych rovníc

Seminar on Qualitative Theory of Differential Equations

Thursday 21.5.2015 at 14:00 and 15:00 Lecture room M-223

Kazuhiro Ishige (Tohoku University):

Concavity properties of solutions for parabolic equations

and

Tatsuki Kawakami (Osaka Prefecture University):

When does the heat equation have a solution with a sequence of similar level sets?

Abstracts:

K. Ishige:

We discuss parabolic concavity properties of parabolic boundary problems.

Furthermore, we introduce a new kind of convolution, which is a sort of parabolic version of the classical supremal convolution

of convex analysis, and we compare solutions of different parabolic problems in different domains.

This talk is based on a joint work with Paolo Salani (Univ. of Florence, Italy).

T. Kawakami:

We consider the unique bounded solution of the Cauchy problem for the heat equation in the whole space

with a non-trivial bounded non-negative initial data which has compact support.

It is well known that if initial data is radially symmetric, then the solution must be radially symmetric.

The overdetermined problems, which determine the shape of solutions by using some additional information of solutions,

are interesting ones in the study of qualitative properties of solutions of partial differential equations.

In this talk, as one of the overdetermined problems, we consider the Cauchy problem for the heat equation

which has a solution with a certain sequence of similar level sets.

This is a joint work with Shigeru Sakaguchi (Tohoku University).