Well-posedness of nonlinear fractional Schrödinger and wave equations in Sobolev spaces

van Duong Dinh (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

We prove the well-posed results in sub-critical and critical cases for the pure power-type nonlinear fractional Schrödinger equations on R d. These results extend the previous ones in [22] for σ ≥ 2. This covers the well-known result for the Schrödinger equation σ = 2 given in [4]. In the case σ ∈ (0, 2)\{1}, we give the local well-posedness in sub-critical case for all exponent ν > 1 in contrast of ones in [22]. This also generalizes the ones of [11] when d = 1 and of [17] when d ≥ 2 where the authors considered the cubic fractional Schrödinger equation with σ ∈ (1, 2). We also give the global existence in energy space under some assumptions. We finally prove the local well-posedness in sub-critical and critical cases for the pure power-type nonlinear fractional wave equations.

Domaines: Equations aux dérivées partielles [math.AP]

van Duong Dinh. Well-posedness of nonlinear fractional Schrödinger and wave equations in Sobolev spaces. 2017. ⟨hal-01426761v1⟩