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

Abstract

We prove the well-posed results in sub-critical and critical cases for the pure power-type nonlinear fractional Schrödinger equations on ${\mathbb{R}}^d$. These results extend the previous ones in \cite{HongSire} for $\sigma \ge 2$. This covers the well-known result for the Schrödinger equation $\sigma =2$ given in \cite{CazenaveWeissler}. In the case $\sigma \in (0,2)\backslash \{1\}$, we give the local well-posedness in sub-critical case for all exponent $\nu >1$ in contrast of ones in \cite{HongSire}. This also generalizes the ones of \cite{ChoHwangKwonLee} when $d=1$ and of \cite{GuoHuo} when $d\ge 2$ where the authors considered the cubic fractional Schrödinger equation with $\sigma \in (1,2)$. 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.

27 pages

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