On multi-solitons for the energy-critical wave equation in dimension 5

Abstract

In this paper, we construct $K$-solitons of the focusing energy-critical nonlinear wave equation in five-dimensional space, i.e. solutions $u$ of the equation such that \begin{equation*} \|\nabla_{t,x}u(t)-\nabla_{t,x}\big(\sum_{k=1}^{K}W_{k}(t)\big)\|_{L^{2}}\to 0\quad \mathrm{as}\ t\to \infty, \end{equation*} where for any $k\in \{1,\dots ,K\}$, $W_k$ is Lorentz transform of the explicit standing soliton $W(x)=(1+|x{|}^2/15{)}^{-3/2}$, with any speed ${\mathbf{\ell }}_k\in {\mathbb{R}}^5$ ,$|{\mathbf{\ell }}_k|<1$ (${\mathbf{\ell }}_{k^{\prime }}\ne {\mathbf{\ell }}_k$ for $k^{\prime }\ne k$) satisfying an explicit smallness condition.

30 pages. minor revisions. arXiv admin note: substantial text overlap with arXiv:1708.09712, arXiv:1504.01595 by other authors