Exponential growth for the wave equation with compact time-periodic positive potential

Abstract

We prove the existence of smooth positive potentials V ( t, x ), periodic in time and with compact support in x , for which the Cauchy problem for the wave equation u tt − Δ x u + V ( t, x ) u = 0 has solutions with exponentially growing global and local energy. Moreover, we show that there are resonances, z ∈ [Copf], [verbar] z [verbar] > 1, associated to V ( t, x ). © 2008 Wiley Periodicals, Inc.