  We consider the quintic, focusing semilinear wave equation on
$\mathbb{R}^{1+3}$, in the radially symmetric setting, and construct infinite
time blow-up, relaxation, and intermediate types of solutions. More precisely,
we first define an admissible class of time-dependent length scales, which
includes a symbol class of functions. Then, we construct solutions which can be
decomposed, for all sufficiently large time, into an Aubin-Talentini (soliton)
solution, re-scaled by an admissible length scale, plus radiation (which solves
the free 3 dimensional wave equation), plus corrections which decay as time
approaches infinity. The solutions include infinite time blow-up and relaxation
with rates including, but not limited to, positive and negative powers of time,
with exponents sufficiently small in absolute value. We also obtain solutions
whose soliton component has oscillatory length scales, including ones which
converge to zero along one sequence of times approaching infinity, but which
diverge to infinity along another such sequence of times. The method of proof
is similar to a recent wave maps work of the author, which is itself inspired
by matched asymptotic expansions.
