Nonscattering solutions and blowup at infinity for the critical wave equation

TL;DR

This paper constructs special solutions to the critical wave equation that exhibit non-scattering and blowup behavior at infinity, challenging existing conjectures about the long-term dynamics of such equations.

Contribution

It introduces a new class of solutions with prescribed asymptotic behavior involving the ground state, demonstrating phenomena not predicted by the soliton resolution conjecture.

Findings

Existence of solutions with prescribed asymptotics involving the ground state

Solutions that concentrate or remain bounded without scattering

Contradiction of a strong soliton resolution conjecture

Abstract

We consider the critical focusing wave equation $(-\partial_t^2+\Delta)u+u^5=0$ in $\R^{1+3}$ and prove the existence of energy class solutions which are of the form [u(t,x)=t^\frac{\mu}{2}W(t^\mu x)+\eta(t,x)] in the forward lightcone ${(t,x)\in\R\times \R^3: |x|\leq t, t\gg 1}$ 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 \eta(t,\cdot)|_{L^2(B_t)} +|\nabla \eta(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.