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 $t\to \infty$, 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 $N\ge 3$.

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5维能量临界波动方程多重气泡解的构造

我们证明了维数为5的能量临界聚焦波方程的整体解的存在，它在任意给定的K个点上无限长地爆炸${ž}_{ķ}$ 的 ${\mathbb{[R}}^5$，在哪里 $ķ\ge 2$。每个气泡的集中率渐近于$C_{ķ}{Ť}^{-2}$ 如 $Ť\to \infty$，其中 $C_{ķ}$ 是正常数，取决于爆炸点之间的距离 ${ž}_{ķ}$。该结果补充了各种尺寸的能量临界波动方程的爆破解决方案和多孤子的先前构造$ñ\ge 3$。