Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Blow up for some semilinear wave equations in multi-space dimensions   In this paper, we discuss a new nonlinear phenomenon. We find that in $n\geq
2$ space dimensions, there exists two indexes $p$ and $q$ such that the cauchy
problems for the nonlinear wave equations {equation} \label{0.1} \Box u(t,x) =
|u(t,x)|^{q}, \ \ x\in R^{n}, {equation} and {equation} \label{0.2} \Box u(t,x)
= |u_{t}(t,x)|^{p}, \ \ x\in R^{n} {equation} both have global existence for
small initial data, while for the combined nonlinearity, the solutions to the
Cauchy problem for the nonlinear wave equation {equation} \label{0.3} \Box
u(t,x) = | u_{t}(t,x)|^{p} + |u(t,x)|^{q}, \ \ x\in R^{n}, {equation} with
small initial data will blow up in finite time. In the two dimensional case, we
also find that if $ q=4$, the Cauchy problem for the equation \eqref{0.1} has
global existence, and the Cauchy problem for the equation {equation}
\label{0.4} \Box u(t,x) = u (t,x)u_{t}(t,x)^{2}, \ \ x\in R^{2} {equation} has
almost global existence, that is, the life span is at least $ \exp
(c\varepsilon^{-2}) $ for initial data of size $ \varepsilon$. However, in the
combined nonlinearity case, the Cauchy problem for the equation {equation}
\label{0.5} \Box u(t,x) = u(t,x) u_{t}(t,x)^{2} + u(t,x)^{4}, \ \ x\in R^{2}
{equation} has a life span which is of the order of $ \varepsilon^{-18} $ for
the initial data of size $ \varepsilon$, this is considerably shorter in
magnitude than that of the first two equations. This solves an open optimality
problem for general theory of fully nonlinear wave equations (see
\cite{Katayama}).

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The categories of this paper are: dg-ga
6
The categories of this paper are: astro-ph
33
The categories of this paper are: plasm-ph
28
The categories of this paper are: alg-geom
8
The categories of this paper are: hep-ex
3
The categories of this paper are: physics
32
The categories of this paper are: atom-ph
4
The categories of this paper are: cond-mat
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The categories of this paper are: acc-phys
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The categories of this paper are: nucl-ex
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The categories of this paper are: q-alg
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The categories of this paper are: chem-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: stat
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The categories of this paper are: solv-int
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The categories of this paper are: econ