Title: The sharp initial condition for existence and blow-up to some PDEs vs the best constant of functional inequalities

Speaker:  Prof. Jinhuan Wang  ( Liaoning University )

Time: Tue, Nov.10 2020, PM: 14:00-15:00

Location:  Tencent Conference

Meeting Number: 438 103 995

Inviter: Prof. Haiyang Jin

Abstract:

     In many physical and biological systems, there are some competing effects such as focus and de-focus, attraction and repulsion, spread and concentration. These competing effects usually are represented by terms with different signs in a free energy. The dynamics of the physical system sometimes can be described by a gradient ow driven by the free energy. Some functional inequalities can be used to determine the domination among these competing effects in the free energy, and provided sharp conditions on initial data or coefficients in the system for the global existence. In this talk, we will introduce some important relations between functional inequalities and sharp conditions for the global existence to seme PDE. For example, the Hardy-Littlewood-Sobolev inequality vs parabolic-elliptic Keller-Segel (K-S) model, Onofri's inequality vs linear parabolic-parabolic K-S model, Sobolev inequality vs degenerate parabolic-parabolic K-S model, and Sz. Nagy inequality vs 1-D thin film equation. And we prove global existence and blow-up of solutions for above models under sharp conditions. Moreover, we obtain the unique ness of the weak solution for the linear diffusion Keller-Segel model using the refined hyper-contractivity of the Lp of the solution, and also obtain some results on the L1 estimate of the solution utilizing the bootstrap method.