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 , namely
is the Fourier multiplier by . For the nonlinear fractional Schrödinger equation, we extend the previous results in [#!HongSire!#] for . These results cover the well-known results for Schrödinger equation given in [#!CazenaveWeissler!#]. In the case
, we show the local well-posedness in the sub-critical case for in contrast to when , and when of [#!HongSire!#]. These results also generalize the ones of [#!ChoHwangKwonLee!#] when and of [#!GuoHuo13!#] when , where the authors considered the cubic fractional Schrödinger equation with
. To our knowledge, the nonlinear fractional wave equation does not seem to have been much considered, up to [#!Wang!#] on the scattering operator with an even integer and [#!ChenFanZhang14!#], [#!ChenFanZhang15!#] in the context of the damped fractional wave equation.
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