Fourth Order Schrödinger Equation with Mixed Dispersion on Certain Cartan-Hadamard Manifolds.

This paper is devoted to the study of the following fourth order Schrödinger equation with mixed dispersion on M N , an N-dimensional Cartan-Hadamard manifold. Namely we consider 4NLS i ∂ t ψ = - Δ M 2 ψ + β Δ M ψ + λ | ψ | 2 σ ψ in R × M , ψ (0 , ·) = ψ 0 ∈ X , where β ≥ 0 , λ = { - 1 , 1 } , 0 < σ < 4 / (N - 4) + , Δ M is the Laplace-Beltrami operator on M and X = L 2 (M) or X = H 2 (M) . At first, we focus on the case where M is the hyperbolic space H N . Using the fact that there exists a Fourier transform on this space, we prove the existence of a global solution to (4NLS) as well as scattering for small initial data provided that N ≥ 4 and 0 < σ < 4 / N if X = L 2 (H N) or 0 < σ < 4 / (N - 4) + if X = H 2 (H N) . Next, we obtained weighted Strichartz estimates for radial solutions to (4NLS) 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. [ABSTRACT FROM AUTHOR]