Title: Spike dynamics of a singular parabolic equation

Speaker: Prof. Chongchun Zeng (Georgia Institute of Technology)

Time: Aug 11, Thursday, 2011,10:00-11:00am

Venue: Room 1213 at BICMR, Resource Plaza, Peking University

Abstract: Consider a nonlinear parabolic equation $u_t = ep^2 Delta u - u + f(u)$ on a smooth bounded domain $Omega subset R^n$ with the zero Neumann boundary condition. In the past years, there had been extensive studies on steady spike solutions. Here a spike solution $u$ is one which is almost equal to zero everywhere except on a ball of radius $O(ep)$ where $u=O(1)$. In this talk, we show that there exist dynamic spike solutions which maintain the spike profile for all $t in R$ with the spike moving on $p Omega$. Moreover, these dynamic spike states form an invariant manifold in some appropriate function space, which is diffeomorphic to $partial Omega$. It is also proved that the leading order dynamics of the spike location follows the gradient flow of the mean curvature of $p Omega$.