WELL-POSEDNESS OF NONLINEAR FRACTIONAL SCHR\"ODINGER AND WAVE EQUATIONS IN SOBOLEV SPACES

Abstract

In this paper, we establish the local well-posedness results in sub-critical and critical cases for the pure power-type nonlinear fractional Schrodinger and wave equations on $\mathbb{R}^d$, namely $$ i\partial_t u + \Lambda^\sigma u + \mu |u|^{\nu−1}u = 0, \quad u_{\vert=0} = \phi, $$ and $$ \partial^2_t v + \Lambda^{2\sigma} v + \mu |v|^{\nu−1} v = 0, \quad v_{\vert t=0} = \phi,\quad \partial_t v_{\vert t=0} = \varphi, $$ where $\sigma \in (0,∞)\backslash \{1\}, \nu > 1,\mu \in \{\pm 1\}$ and $\Lambda =\sqrt{−\Delta}$ is the Fourier multiplier by $|\xi|$. For the nonlinear fractional Schrodinger equation, we extend the previous results in [22] for $\sigma \geq 2$. These results cover the well-known results for Schrodinger equation $\sigma = 2$ given in [4]. In the case $\sigma \in (0,2)\backslash \{1\}$, we show the local well-posedness in the sub-critical case for $\nu > 1$ in contrast to $\nu \geq 2$ when $d = 1$, and $\nu \geq 3$ when $d geq 2$ of [22]. These results also generalize the ones of [11] when $d = 1$ and of [18] when $d \geq 2$, where the authors considered the cubic fractional Schrodinger equation with $\sigma \in (1,2)$. To our knowledge, the nonlinear fractional wave equation does not seem to have been much considered, up to [37] on the scattering operator with $\sigma$ an even integer and [6], [7] in the context of the damped fractional wave equation.