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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-criticalnonlinear wave equation $\Box u = u^5$ in $\R^{1+3}$, together with a spacetimebound $$ \| u \|_{L^4_t L^{12}_x(\R^{1+3})} \leq M(E(u))$$ for some finitequantity M(E(u)) depending only on the energy E(u) of u. We reprove thisresult, and show that this quantity obeys a bound of at most exponential typein the energy, and specifically $M(E) \leq C (1+E)^{C E^{105/2}}$ for someabsolute constant C > 0. The argument combines the quantitative local potentialenergy decay estimates of these previous papers with arguments used by Bourgainand the author for the analogous nonlinear Schr\"odinger equation.

Spacetime bounds for the energy-critical nonlinear wave equation in three spatial dimensions