# PDE Seminar

## Smoothness and Long-time Behavior of the Spreading Front Determined by a Nonlinear Free Boundary Problem

### Yihong Du 教授 (University of New England, Australia)

#### 2012年5月15日(周二)下午 3:30-4:30 闵行校区数学楼102报告厅

Abstract. We consider the spreading of species governed by a free boundary model. In one space dimension and in the radially symmetric case with a logistic

nonlinear term, it is known that this model exhibits a spreading-vanishing dichotomy. In this talk we discuss the non-radially symmetric case

with more general nonlinearities. By establishing suitable regularity of

the free boundary, we show that the spreading-vanishing dichotomy still

holds. Moreover, when spreading happens, the normalized free boundary

approaches the unit sphere as time goes to infinity. For logistic

nonlinearity we obtain a complete extension of the previous results

in the radially symmetric case to the general case. This is joint work

with Hiroshi Matano and Kelei Wang.

We consider the Neumann initial-boundary value problem for the following fully parabolic Keller-Segel system

\begin{equation}\left\{

\begin{array}{ll}

u_t=\Delta u-{\bf div} (u\nabla v), &x\in\Omega,t>0,\\[6pt]

v_t=\Delta v-v+u,&x\in\Omega,t>0.

\end{array}\right.

\end{equation}

where $\Omega$ is a ball in $\mathbb{R}^n$ with $n\ge 3$.

In this talk, I will prove that the set $S$ mentioned in the former talk is dense in $C^0(\bar\Omega)\times W^{1,\infty}(\Omega)$ with respect to the topology of $L^p(\Omega)\times W^{1,2}(\Omega)$ for any $p\in(1,\frac{2n}{n+2})$.