# Construction of multi-bubble solutions for the energy-critical wave equation in dimension 5

## TL;DR

This paper constructs global solutions to the 5D energy-critical focusing wave equation that blow up at multiple points over infinite time, with concentration rates depending on the distances between these points.

## Contribution

It introduces a novel method to construct multi-bubble solutions with prescribed blow-up points and rates in the 5D energy-critical wave equation.

## Key findings

- Existence of solutions blowing up at multiple points in infinite time.
- Concentration rates asymptotic to c_k t^{-2} depending on inter-point distances.
- Extension of previous multi-soliton and blow-up solutions to higher dimensions.

## Abstract

We prove the existence of a global solution of the energy-critical focusing wave equation in dimension $5$ blowing up in infinite time at any $K$ given points $z_k$ of $\mathbb{R}^5$, where $K\geq 2$. The concentration rate of each bubble is asymptotic to $c_k t^{-2}$ as $t\to \infty$, where the $c_k$ are positive constants depending on the distances between the blow-up points $z_k$. This result complements previous constructions of blow-up solutions and multi-solitons of the energy-critical wave equation in various dimensions $N\geq 3$.