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Random data theory for the cubic fourth-order nonlinear Schrödinger equation (2009.14453v1)
Published 29 Sep 2020 in math.AP
Abstract: We consider the cubic nonlinear fourth-order Schr\"odinger equation [ i\partial_t u - \Delta2 u + \mu \Delta u = \pm |u|2 u, \quad \mu \geq 0 ] on $\mathbb{R}N, N \geq 5$ with random initial data. We prove almost sure local well-posedness below the scaling critical regularity. We also prove probabilistic small data global well-posedness and scattering. Finally, we prove the global well-posedness and scattering with a large probability for initial data randomized on dilated cubes.