# Meeting Details

Title: Dynamic spike solutions to a singular parabolic equation CCMA Luncheon Seminar Chongchun Zeng, Georgia Tech 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 $\p \Omega$.