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arXiv · 2105.13804

Fourth order Schr\"odinger equation with mixed dispersion on certain Cartan-Hadamard manifolds

Jean-Baptiste Casteras·Ilkka Holopainen
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Abstract

We study the fourth order Schr\"odinger 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.

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Jean-Baptiste Casteras, Ilkka Holopainen. 2021-05-28. Fourth order Schr\"odinger equation with mixed dispersion on certain Cartan-Hadamard manifolds. https://doi.org/10.1007/s10884-022-10197-4

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