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

Global, Non-scattering solutions to the quintic, focusing semilinear wave equation on $\mathbb{R}^{1+3}$