We construct solutions u(x, t) to the focusing, energy-critical, nonlinear wave equation 0.1 \(\begin{aligned} \partial _{tt}u - \Delta u - |u|^{p-1}u = 0, \quad t \ge 0, \ x \in \mathbb {R}^d, \ d \ge 3, \ p = (d+2)/(d-2) \end{aligned}\) in dimension \(d \in \{4,5\}\) , exhibiting finite-time Type II blow-up precisely at \(x = t = 0\) with a prescribed polynomial blow-up rate of \(t^{-1-\nu }\) , where \(\nu > 1\) for \(d = 4\) and \(\nu > 3\) for \(d = 5\) . Such solutions have been constructed by Krieger–Schlag–Tataru for \(d = 3\) and by Jendrej for \(d = 5\) . The work of Jendrej includes the extremal case \(\nu = 3\) , which our method does not address, and the regime \(\nu > 8\) . The major difference between dimensions 4 and 5 consists in the renormalization procedure. In \(d = 4\) , we essentially follow the Krieger–Schlag–Tataru scheme developed for the 3-dimensional equation. This scheme has been applied with success for other equations such as the 3D-critical NLS, Schrödinger maps or wave maps. In all of these cases, the polynomial structure of the nonlinearity permits the use of simple algebraic manipulations to control error terms. By contrast, the case \(d = 5\) requires a modified setup due to the lower regularity of the nonlinearity, which complicates the treatment of nonlinear error terms.
