Vesselin Petkov
Global Strichartz estimates for the wave equation with time-dependent potentials
(431K, postscript)
ABSTRACT. We obtain global Strichartz estimates for the solutions $u$ of the wave equation $(\partial_t^2 - \Delta_x + V(t,x))u = F(t,x)$ for time-periodic potentials $V(t,x)$ with compact support with respect to $x$. Our analysis is based on the analytic properties of the cut-off resolvent $R_{\chi}(z) = \chi (U(T) - zI)^{-1} \psi_1,$ where $U(T) = U(T, 0)$ is the monodromy operator and $T > 0$ the period of $V(t,x).$ We show that if $R_{\chi}(z)$ has no poles $z \in \C,\: |z| \geq 1$, then for $n \geq 3$, odd, we have a exponential decal of local energy. For $n \geq 2$, even, we obtain also an uniform decay of local energy assuming that $R_{\chi}(z)$ has no poles $z \in \C,\: |z| \geq 1,$ and $R_{\chi}(z)$ remains bounded for $z$ in a small neighborhood of $0$.