Random data theory for the cubic fourth-order nonlinear Schrödinger equation

Abstract

We consider the cubic nonlinear fourth-order Schrödinger equation \[ i\partial_t u - Δ^2 u + μΔu = \pm |u|^2 u, \quad μ\geq 0 \] on ${\mathbb{R}}^N,N\ge 5$ 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.

24 pages. arXiv admin note: substantial text overlap with arXiv:1405.7327 by other authors

References in corpus (3)

• Random data Cauchy theory for supercritical wave equations I: Local theory
• Global wellposedness and scattering for the focusing energy-critical nonlinear Schrodinger equations of fourth order in the radial case
• Random data Cauchy theory for the fourth order nonlinear Schrödinger equation with cubic nonlinearity