# Meeting Details

Title: Exponentially growing solutions of the wave equation with compact time-periodic potentials Computational and Applied Mathematics Colloquium Vesselin Petkov, University Bordeaux 1, Institute Mathematique de Bordeaux http://www.math.psu.edu/ltz/Petkov_CAM_Talk/abstractPennState.pdf We study the solutions of the Cauchy problem for the wave equation $\partial_t^2 u - \Delta_x u + V(t, x)u = 0$, where the potential $V(t,x)$ has compact support in $x$ and it is periodic with respect to $t$. If the potential $V(t,x)$ takes negative values, it is possible to construct solutions whose energy is exponentially growing. For positive potentials $V(t, x)$ this problem is more difficult since for time independent potentials $V(x)$ the energy is conserved. This problem was open for more than 30 years. In 1993 G. Majda and M. S. Wei by numerical computations suggested the existence of positive potentials for which it is possible to construct exponentially increasing solutions. In this talk we construct positive time-periodic potentials with compact support for which the wave equation has exponentially growing solutions. Moreover, we prove that the local energy of these solutions is also exponentially growing and we have so called {\it parametric resonance} phenomenon which is well known in the theory of the ordinary differential equations. This is a work in collaboration with F. Colombini and J. Rauch.