Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Well-posedness, regularity and ill-posedness for the nonlinear
  fourth-order Schr\"odinger equation   We prove the local well-posedness for the nonlinear fourth-order
Schr\"odinger equation (NL4S) in Sobolev spaces. We also studied the regularity
of solutions in the sub-critical case. A direct consequence of this regularity
is the global well-posedness above mass and energy spaces under some
assumptions. Finally, we show the ill-posedness for (NL4S) in some cases of the
super-critical range.

30
The categories of this paper are: q-alg
5
The categories of this paper are: gr-qc
34
The categories of this paper are: chem-ph
24
The categories of this paper are: dg-ga
14
The categories of this paper are: math-ph
21
The categories of this paper are: adap-org
15
The categories of this paper are: nucl-ex
23
The categories of this paper are: solv-int
12
The categories of this paper are: nucl-th
37
The categories of this paper are: supr-con
4
The categories of this paper are: cond-mat
0
The categories of this paper are: hep-ph
33
The categories of this paper are: plasm-ph
29
The categories of this paper are: funct-an
25
The categories of this paper are: patt-sol
19
The categories of this paper are: eess