The global Cauchy problem for the critical non-linear wave equation

AbstractWe study the global Cauchy problem for the non-linear wave equation □ϑ + ¦ϑ¦p−1 ϑ = 0 for the critical value p = (n + 2)(n − 2) in space dimension n ⩾ 3. We identify a weak space-time integrability property (STIP) of the solutions and prove that it is sufficient to ensure the uniqueness of weak solutions, the global existence of finite energy solutions with the naturally associated STIP, and the global existence of regular solutions (with some n-dependent restrictions on the regularity). For spherically symmetric solutions, we prove that the previous crucial STIP follows from the Morawetz inequality, actually in a much stronger form than necessary, thereby proving that all the previous results hold in the spherically symmetric case