Near Sharp Strichartz estimates with loss in the presence of degenerate hyperbolic trapping

Abstract

We consider an $n$-dimensional spherically symmetric, asymptotically Euclidean manifold with two ends and a codimension 1 trapped set which is degenerately hyperbolic. By separating variables and constructing a semiclassical parametrix for a time scale polynomially beyond Ehrenfest time, we show that solutions to the linear Schrödiner equation with initial conditions localized on a spherical harmonic satisfy Strichartz estimates with a loss depending only on the dimension $n$ and independent of the degeneracy. The Strichartz estimates are sharp up to an arbitrary $\beta >0$ loss. This is in contrast to \cite{ChWu-lsm}, where it is shown that solutions satisfy a sharp local smoothing estimate with loss depending only on the degeneracy of the trapped set, independent of the dimension.

References in corpus (2)

• Semilinear wave equations on the Schwarzschild manifold I: Local decay estimates
• Local smoothing for the Schrödinger equation with a prescribed loss

Cited by in corpus (2)

• Semiclassical limit of orthonormal Strichartz estimates on scattering manifolds
• Global endpoint Strichartz estimates for Schrödinger equations on the cylinder $\mathbb{R}\times \mathbb{T}$