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We study the fourth order Schrodinger 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.
We are concerned with the existence and asymptotic properties of solutions to the following fourth-order Schr{o}dinger equation begin{equation}label{1} {Delta}^{2}u+mu Delta u-{lambda}u={|u|}^{p-2}u, ~~~~x in R^{N} end{equation} under the normalized
We prove existence results for entire graphical translators of the mean curvature flow (the so-called bowl solitons) on Cartan-Hadamard manifolds. We show that the asymptotic behaviour of entire solitons depends heavily on the curvature of the manifo
We prove an existence result for the Poisson equation on non-compact Riemannian manifolds satisfying weighted Poincare inequalities outside compact sets. Our result applies to a large class of manifolds including, for instance, all non-parabolic mani
We prove global weighted Strichartz estimates for radial solutions of linear Schrodinger equation on a class of rotationally symmetric noncompact manifolds, generalizing the known results on hyperbolic and Damek-Ricci spaces. This yields classical St
We derive a priori second order estimates for fully nonlinear elliptic equations which depend on the gradients of solutions in critical ways on Hermitian manifolds. The global estimates we obtained apply to an equation arising from a conjecture by Ga