Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Spacetime bounds for the energy-critical nonlinear wave equation in
  three spatial dimensions   Results of Struwe, Grillakis, Struwe-Shatah, Kapitanski, Bahouri-Shatah,
Bahouri-G\'erard and Nakanishi have established global wellposedness,
regularity, and scattering in the energy class for the energy-critical
nonlinear wave equation $\Box u = u^5$ in $\R^{1+3}$, together with a spacetime
bound $$ \| u \|_{L^4_t L^{12}_x(\R^{1+3})} \leq M(E(u))$$ for some finite
quantity M(E(u)) depending only on the energy E(u) of u. We reprove this
result, and show that this quantity obeys a bound of at most exponential type
in the energy, and specifically $M(E) \leq C (1+E)^{C E^{105/2}}$ for some
absolute constant C > 0. The argument combines the quantitative local potential
energy decay estimates of these previous papers with arguments used by Bourgain
and the author for the analogous nonlinear Schr\"odinger equation.

13
The categories of this paper are: quant-ph
7
The categories of this paper are: hep-th
6
The categories of this paper are: astro-ph
18
The categories of this paper are: econ
11
The categories of this paper are: hep-lat
28
The categories of this paper are: alg-geom
10
The categories of this paper are: q-bio
1
The categories of this paper are: cs
19
The categories of this paper are: eess
20
The categories of this paper are: acc-phys
4
The categories of this paper are: cond-mat
35
The categories of this paper are: mtrl-th
34
The categories of this paper are: chem-ph
32
The categories of this paper are: atom-ph
5
The categories of this paper are: gr-qc
29
The categories of this paper are: funct-an