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

$$\begin{aligned} u(t,x)=t^\frac{\mu }{2}W(t^\mu x)+\eta (t,x) \end{aligned}$$

in the forward lightcone $\{(t,x)\in {\mathbb{R }}\times {\mathbb{R }}^3: |x|\le t, t\gg 1\}$ where $W(x)=(1+\frac{1}{3} |x|^2)^{-\frac{1}{2}}$ is the ground state soliton, $\mu$ is an arbitrary prescribed real number (positive or negative) with $|\mu |\ll 1$, and the error $\eta$ satisfies

$$\begin{aligned} \Vert \partial _t \eta (t,\cdot )\Vert _{L^2(B_t)} +\Vert \nabla \eta (t,\cdot )\Vert _{L^2(B_t)}\ll 1,\quad B_t:=\{x\in {\mathbb{R }}^3: |x|<t\} \end{aligned}$$

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 $t\rightarrow \infty$ (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.