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

Abstract

AbstractThis paper is devoted to the study of the following fourth order Schrödinger equation with mixed dispersion on $$M^N$$ M N , an N-dimensional Cartan-Hadamard manifold. Namely we consider $$\begin{aligned} {\left\{ \begin{array}{ll} i\partial _t\psi = -\Delta _M^2\psi +\beta \Delta _M \psi +\lambda |\psi |^{2\sigma }\psi \quad \text {in }\mathbb {R}\times M,\\ \psi (0,\cdot )=\psi _0\in X, \end{array}\right. } \end{aligned}$$ 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 $$\beta \ge 0$$ β ≥ 0 , $$\lambda =\{-1,1\}$$ λ = { - 1 , 1 } , $$0<\sigma < 4/(N-4)_+$$ 0 < σ < 4 / ( N - 4 ) + , $$\Delta _M$$ Δ M is the Laplace-Beltrami operator on M and $$X=L^2 (M)$$ X = L 2 ( M ) or $$X=H^2 (M)$$ X = H 2 ( M ) . At first, we focus on the case where M is the hyperbolic space $$\mathbb {H}^N$$ 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 $$N\ge 4$$ N ≥ 4 and $$0<\sigma < 4/N$$ 0 < σ < 4 / N if $$X=L^2 (\mathbb {H}^N)$$ X = L 2 ( H N ) or $$0<\sigma < 4/(N-4)_+$$ 0 < σ < 4 / ( N - 4 ) + if $$X=H^2 (\mathbb {H}^N)$$ 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.