Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Blow-up and lifespan estimate to a nonlinear wave equation in
  Schwarzschild spacetime   In \cite{Luk}, Luk proved global existence for semilinear wave equations in
Kerr spacetime with small angular momentum($a\ll M$) \[
\Box_{g_K}\phi=F(\partial \phi), \] when the quadratic nonlinear term satisfies
the null condition. In this work, we will show that if the null condition does
not hold, at most we can have almost global existence for semilinear wave
equations with quadratic nonlinear term in Schwarzschild spacetime, which is
the special case of Kerr with $a=0$ \[ \Box_{g_S}\phi=(\partial_t\phi)^2, \]
where $\Box_{g_S}$ denotes the D'Alembert operator associated with
Schwarzschild metric. What is more, if the power of the nonlinear term is
replaced with $p$ satisfying $3/2\le p< 2$, we still can show blow-up, no
matter how small the initial data are. We do not have to assume that the
support of the data should be far away from the event horizon.

10
The categories of this paper are: q-bio
5
The categories of this paper are: gr-qc
28
The categories of this paper are: alg-geom
17
The categories of this paper are: q-fin
30
The categories of this paper are: q-alg
27
The categories of this paper are: comp-gas
24
The categories of this paper are: dg-ga
4
The categories of this paper are: cond-mat
19
The categories of this paper are: eess
18
The categories of this paper are: econ
11
The categories of this paper are: hep-lat
26
The categories of this paper are: bayes-an
3
The categories of this paper are: physics
31
The categories of this paper are: ao-sci
9
The categories of this paper are: nlin
15
The categories of this paper are: nucl-ex