The goal of this paper is to exhibit solutions u (x, t) of the focusing, critical energy, nonlinear wave equation equation ₓₓu - u - |u|^p-1u = 0, t 0, \ x Rᵈ, \ d 3, \ p = (d+2) / (d-2) equation in dimension d \4, 5\ where a finite-time Type II blow-up (meaning the solution blows-up while the energy remains bounded) occurs exactly at x = t = 0 with a prescribed polynomial blow-up rate t^-1- with > 1 when d = 4 and > 3 when d = 5. Such solutions have been constructed by Krieger-Schlag-Tataru in d = 3 and by Jendrej in d = 5. The result from Jendrej covers the extremal case = 3, which we are missing, and any > 8. Surprisingly, no one else has treated the d = 4 case yet. The major difference between dimensions 4 and 5 in our paper lies in this renormalization procedure. In d = 4, we essentially follow Krieger-Schlag-Tataru scheme used for the 3D equation. This scheme has been used with success in other equations such as the 3D-critical NLS, Schr\"odinger maps or wave maps. In all of these cases, the non-linearity is a polynomial, which allows to treat the error terms using simple algebra rules. In d = 5, the setup needs to be modified because the nonlinearity of the equation has a much lower regularity and it is not clear anymore how one should deal with the nonlinear error terms.
Dylan Samuelian (Thu,) studied this question.