Fourth order Schrödinger equation with mixed dispersion on certain Cartan-Hadamard manifolds
Jean-Baptiste Casteras, Ilkka Holopainen
TL;DR
The paper analyzes a nonlinear biharmonic Schrödinger equation with mixed dispersion on Cartan–Hadamard manifolds, proving global existence and scattering on hyperbolic space by leveraging a hyperbolic Fourier transform and TT* Strichartz theory, and extending the framework to rotationally symmetric manifolds through resolvent smoothing for radial data. It establishes Strichartz estimates in these curved settings, including weighted bounds for radial solutions, and demonstrates a blow-up criterion via a localized virial identity. Collectively, the results extend dispersive and scattering theory for fourth-order Schrödinger equations beyond Euclidean spaces, providing tools for both global behavior and finite-time blow-up on curved geometries. The methods connect harmonic analysis on hyperbolic spaces, resolvent-based smoothing, and virial techniques to address well-posedness, scattering, and blow-up in curved manifolds.
Abstract
We study the fourth order Schrödinger 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.