Nonscattering solutions and blowup at infinity for the critical wave equation

We consider the critical focusing wave equation $(-\partial_t^2+Δ)u+u^5=0$ in $\R^{1+3}$ and prove the existence of energy class solutions which are of the form [u(t,x)=t^\fracμ{2}W(t^μx)+η(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, $μ$ is an arbitrary prescribed real number (positive or negative) with $|μ|\ll 1$, and the error $η$ 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 $μ$, 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.

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