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

For any $N\geq 2$, we construct a global solution of the energy-critical focusing wave equation in dimension four which blows up in infinite time at $N$ prescribed points $z_1,\ldots,z_N\in \mathbb R^4$, provided that the points form one orbit under a finite group of orthogonal symmetries. We denote by $c:=2\sum_{j\ne k}|z_j-z_k|^{-2}>0$ the corresponding interaction coefficient, which is independent of $k$. The common concentration scale satisfies \[ \log\frac{1}{\lambda(t)} = \left(\frac{9c}{4}\right)^{1/3}t^{2/3}+O(t^{1/3}) \qquad \text{as } t\to+\infty . \] This concentration rate comes from a genuinely four-dimensional effect: the borderline decay of the ground state makes the interaction between different bubbles enter the leading order parameter dynamics.