Strichartz estimates for Klein–Gordon equations with moving potentials

Abstract

We study linear Klein–Gordon equations with several potentials whose centers move along nonlinear trajectories, and an external forcing. This model arises naturally in the perturbative analysis of the interaction of solitons and the stability of multi-solitons, the potentials capturing the leading order effect of the solitons, and lower order effects being treated as forcing. Our main result is a proof of local energy decay and Strichartz estimates, under the assumption that the solution is orthogonal to discrete spectral modes. We introduce Galilean transformations in our Lorentz-invariant setting, which lead to a nonselfadjoint matrix operator (with a single potential). We develop the spectral theory and obtain local energy decay for this matrix operator. To control the effect of multiple simultaneously moving potentials, we use a channel decomposition adapted to their trajectories. By combining the aforementioned single-potential estimates with refined bounds on the interaction between channels, we obtain the desired estimates for the full evolution. As an application of the Strichartz estimates, we consider the linear scattering problem in the absence of external forcing. Using a Lyapunov–Schmidt argument, we establish asymptotic completeness on the center-stable space.