In this paper, we establish the local well-posedness results in sub-critical and critical cases for the pure power-type nonlinear fractional Schrödinger and wave equations on $\R^d$, namely

i\partial_t u + \Lambda^\sigma u + \mu \vert u\vert^{\nu-1} u=0, \quad u_{\vert t=0} =\varphi,


\partial^2_t v +\Lambda^{2\sigma} v + \mu \vert v\vert^{\nu...
...v_{\vert t=0}=\varphi, \quad \partial_t v_{\vert t=0} = \phi,

where $\sigma \in (0,\infty)\backslash\{1\}, \nu>1, \mu \in \{\pm 1\}$ and $\Lambda =\sqrt{-\Delta}$ is the Fourier multiplier by $\vert\xi\vert$. For the nonlinear fractional Schrödinger equation, we extend the previous results in [#!HongSire!#] for $\sigma \geq 2$. These results cover the well-known results for Schrödinger equation $\sigma =2$ given in [#!CazenaveWeissler!#]. 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 [#!HongSire!#]. These results also generalize the ones of [#!ChoHwangKwonLee!#] when $d=1$ and of [#!GuoHuo13!#] when $d\geq 2$, where the authors considered the cubic fractional Schrödinger equation with $\sigma \in (1,2)$. To our knowledge, the nonlinear fractional wave equation does not seem to have been much considered, up to [#!Wang!#] on the scattering operator with $\sigma$ an even integer and [#!ChenFanZhang14!#], [#!ChenFanZhang15!#] in the context of the damped fractional wave equation.

Citation details of the article

Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 31
Issue: 4
Year: 2018

DOI: 10.12732/ijam.v31i4.1

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