Fourth order Schrödinger equation with mixed dispersion on certain Cartan-Hadamard manifolds
arXiv:2105.13804 · doi:10.1007/s10884-022-10197-4
Abstract
We study the fourth order Schrödinger equation with mixed dispersion on an -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.
To appear in Journal of Dynamics and Differential Equations