Nonlinear profile decomposition for the H˙12×H˙−12(Rd) energy subcritical wave equation

We consider the nonlinear wave equation utt−Δu±u|u|4d−1=0 in dimensions d≥2. This equation is critical in H˙12×H˙−12(Rd) and subcritical with respect to the energy. We prove the nonlinear profile decomposition. The proof must address the lack of compactness due to the Lorentz symmetry.