  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.
