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We consider the 1-equivariant energy critical wave maps problem withtwo-sphere target. Using a method based on matched asymptotic expansions, weconstruct infinite time relaxation, blow-up, and intermediate types ofsolutions that have topological degree one. More precisely, for a symbol classof admissible, time-dependent length scales, we construct solutions which canbe decomposed as a ground state harmonic map (soliton) re-scaled by anadmissible length scale, plus radiation, and small corrections which vanish (ina suitable sense) as time approaches infinity. Our class of admissible lengthscales includes positive and negative powers of t, with exponents sufficientlysmall in absolute value. In addition, we obtain solutions with soliton lengthscale undergoing damped or undamped oscillations in a bounded set, orundergoing unbounded oscillations, for all sufficiently large t.

Global, Non-scattering solutions to the energy critical wave maps equation