Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Well-posedness of nolinear fractional Schr\"odinger and wave equations
  in Sobolev spaces   We prove the well-posed results in sub-critical and critical cases for the
pure power-type nonlinear fractional Schr\"odinger equations on $\mathbb{R}^d$.
These results extend the previous ones in \cite{HongSire} for $\sigma\geq 2$.
This covers the well-known result for the Schr\"odinger 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 $d \geq
2$ where the authors considered the cubic fractional Schr\"odinger 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.

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The categories of this paper are: alg-geom
3
The categories of this paper are: physics
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The categories of this paper are: q-bio
5
The categories of this paper are: gr-qc
4
The categories of this paper are: cond-mat
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The categories of this paper are: comp-gas
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: nucl-th
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: plasm-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: solv-int
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The categories of this paper are: chem-ph