Fourth order Schr\"odinger equation with mixed dispersion on certain Cartan-Hadamard manifolds

TL;DR

This paper investigates the fourth order Schr"odinger equation with mixed dispersion on Cartan-Hadamard manifolds, establishing global solutions, scattering, weighted Strichartz estimates, and blow-up results.

Contribution

It introduces new analysis of the equation on non-compact manifolds, including existence, scattering, and blow-up phenomena, extending prior work to curved geometries.

Findings

Global solutions and scattering for small initial data on hyperbolic space

Weighted Strichartz estimates for radial solutions on symmetric manifolds

Blow-up results using virial arguments on certain manifolds

Abstract

We study the fourth order Schr\"odinger equation with mixed dispersion on an $N$-dimensional Cartan-Hadamard manifold. At first, we focus on the case of the hyperbolic space. Using the fact that there exists a Fourier transform on this space, we prove the existence of a global solution to our equation as well as scattering for small initial data. Next, we obtain weighted Strichartz estimates for radial solutions on a large class of rotationally symmetric manifolds by adapting the method of Banica and Duyckaerts (Dyn. Partial Differ. Equ., 07). Finally, we give a blow-up result for a rotationally symmetric manifold relying on a localized virial argument.