Nonscattering solutions and blowup at infinity for the critical wave equation

Abstract

We consider the critical focusing wave equation $(-{\partial }_t^2+\Delta )u+u^5=0$ in ${\mathbb{R}}^{1+3}$ and prove the existence of energy class solutions which are of the form [u(t,x)=t^\fracμ{2}W(t^μx)+η(t,x)] in the forward lightcone $(t,x)\in \mathbb{R}\times {\mathbb{R}}^3:|x|\le t,t\gg 1$ where $W(x)=(1+(1/3)|x{|}^2{)}^{-(1/2)}$ is the ground state soliton, $\mu$ is an arbitrary prescribed real number (positive or negative) with $|\mu |\ll 1$, and the error $\eta$ satisfies [|\partial_t η(t,\cdot)|_{L^2(B_t)} +|\nabla η(t,\cdot)|_{L^2(B_t)}\ll 1,\quad B_t:={x\in\R^3: |x|<t}] for all $t\gg 1$. Furthermore, the kinetic energy of $u$ outside the cone is small. Consequently, depending on the sign of $\mu$, we obtain two new types of solutions which either concentrate as $t\to \infty$ (with a continuum of rates) or stay bounded but do not scatter. In particular, these solutions contradict a strong version of the soliton resolution conjecture.

53 pages, final version, some additional typos have been fixed