# Postdoc Seminar

## Report on stability properties of shadow systems I

### 李芳 研究员 (华师大PDE中心)

#### 2013年9月17日(周二)下午 1:00-4:00 闵行校区行政楼12楼PDE中心1202报告厅

Abstract. It is known that $2\times 2$ reaction-diffusion system is formally reduced to the so called shadow system

\begin{equation}

\begin{cases}

\;u_t=d\Delta u+f(x, u,\xi) & \textrm{in }\Omega\times(0, T),\\

\;\tau\xi_t=\frac{1}{\vert\Omega\vert}\int_\Omega g(x, u,\xi)dx & \textrm{in } (0, T),\\

\;\frac{\partial u}{\partial \nu}=0 & \textrm{on } \partial \Omega\times(0, T).

\end{cases}

\end{equation}

when one of the diffusion rates goes to $+\infty$. Here $\Omega\subset \mathbb R^N$ is bounded and smooth, $N\geq 1$ and $\nu$ denotes the unit outer normal vector on $\partial \Omega$.

In this talk, we focus on the stability properties of steady states of the above shadow system using linearized analysis. Assume that $(u(x),\xi)$ is a nontrivial steady state.A complete classification of all linearized eigenvalues at $(u(x),\xi)$ is provided first. Then, based on this classification, we survey the qualitative properties of stable steady states of autonomous shadow systems. To be more specific, it is shown that

(i) When $N=1$, if $(u(x),\xi)$ is stable, then $u(x)$ is monotone in $x$.

(ii) When $N=2$ and $\Omega$ is a unit disk, if $(u(x),\xi)$ is stable, then the maximum(minimum) of $u$ is attained at exactly one point on $\partial \Omega$ provided that certain technical conditions are imposed on $f$ and $g$.

(iii) When $N\geq 2$ and $\Omega$ is a unit ball. If $u(x)$ is radially symmetric, then $(u(x),\xi)$ is unstable.

The results discussed in this talk are from the following references: F. Li, K. Nakashima and W.-M. Ni, DCDS-A, 2008;

Y. Miyamoto, J. Diff. Eqns., 2006; Y. Miyamoto, J. Diff. Eqns., 2007; W.-M. Ni, P. Polacik and E. Yanagida, Trans. Amer. Math. Soc., 2001.