# Jinhuan Wang : Sharp conditions for global existence to some PDEs and functional inequalities

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 flow 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. For example, the Hardy-Littlewood-Sobolev inequality vs parabolic-elliptic Keller-Segel model, Onofri's inequality vs parabolic-parabolic Keller-Segel model, and Sz. Nagy inequality vs 1-D thin film equation, and provide the results on the global existence and blow-up for above models under sharp conditions. Moreover, we obtain the uniqueness of the weak solution for the linear diffusion Keller-Segel model using the refined hyper-contractivity of the $L^p$ of the solution under the sharp initial condition, and prove the $L^{\infty}$ estimate of the solution utilizing the bootstrap method. We also provide some results on existence of the global smooth solution.

**Category**: Applied Math and Analysis**Duration**: 01:34:53**Date**: January 26, 2015 at 4:25 PM**Views**: 120-
**Tags:**seminar, Applied Math And Analysis Seminar

## 0 Comments