Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Weakly non-radiative radial solutions to 3D energy subcritical wave
  equations   In this work we consider the energy subcritical 3D wave equation
$\partial_t^2 u - \Delta u = \pm |u|^{p-1} u$ and discuss its (weakly)
non-radiative solutions, i.e. the solutions defined in an exterior region
$\{(x,t): |x|>|t|+R\}$ with $R\geq 0$ satisfying
  \[
  \lim_{t\rightarrow \pm\infty} \int_{|x|>|t|+R} \left(|\nabla u(x,t)|^2 +
|u_t(x,t)|^2\right) dx = 0.
  \] It has been known that any radial weakly non-radiative solution to the
linear wave equation is a multiple of $1/|x|$. In addition, any radial weakly
non-radiative solutions $u$ to the energy critical wave equation must possess a
similar asymptotic behaviour, i.e. $u(x,t)\simeq C/|x|$ when $|x|$ is large. In
this work we give examples to show that radial weakly non-radiative solutions
to energy subcritical equation ($3<p<5$) may possess a much different
asymptotic behaviour. However, a radial weakly non-radiative solution $u$ with
initial data in the critical Sobolev space $\dot{H}^{s_p}\times
\dot{H}^{s_p-1}(\mathbb{R}^3)$ must coincide with a $C^2$ solution $W$ to the
elliptic equation $-\Delta W = -|W|^{p-1} W$ so that $u(x,t) \equiv W(x) \simeq
C/|x|$ when $|x|$ is large.

23
The categories of this paper are: solv-int
8
The categories of this paper are: hep-ex
26
The categories of this paper are: bayes-an
12
The categories of this paper are: nucl-th
34
The categories of this paper are: chem-ph
30
The categories of this paper are: q-alg
27
The categories of this paper are: comp-gas
14
The categories of this paper are: math-ph
10
The categories of this paper are: q-bio
25
The categories of this paper are: patt-sol
35
The categories of this paper are: mtrl-th
17
The categories of this paper are: q-fin
21
The categories of this paper are: adap-org
7
The categories of this paper are: hep-th
16
The categories of this paper are: stat
28
The categories of this paper are: alg-geom