Construction of blow-up solutions for the focusing energy-critical nonlinear wave equation in $\mathbb{R}^4$ and $\mathbb{R}^5$

TL;DR

This paper constructs finite-time blow-up solutions for the focusing energy-critical nonlinear wave equation in four and five dimensions, with specific polynomial blow-up rates, extending previous results and addressing dimension-specific analytical challenges.

Contribution

It develops a method to construct blow-up solutions in 4D and 5D, adapting techniques to handle dimension-dependent nonlinearities and error control.

Findings

Constructed solutions with prescribed blow-up rates in 4D and 5D

Extended previous blow-up solution frameworks to higher dimensions

Addressed nonlinear regularity issues in 5D case

Abstract

We construct solutions $u(x,t)$ to the focusing, energy-critical, nonlinear wave equation \begin{equation} \partial_{tt}u - \Delta u - |u|^{p-1}u = 0, \quad t \geq 0, \ x \in \mathbb{R}^d, \ d \geq 3, \ p = (d+2)/(d-2) \end{equation} 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, Schr\"odinger 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.