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

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 $K\ge 2$. The concentration rate of each bubble is asymptotic to $c_k{t}^{-2}$ as $t\to \mathrm{\infty }$, 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 $N\ge 3$.