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

TL;DR

This paper constructs various types of solutions to the energy critical wave maps equation with topological degree one, including relaxation, blow-up, and oscillatory solutions, using matched asymptotic expansions.

Contribution

It introduces a method to explicitly construct solutions with complex asymptotic behaviors for the wave maps problem, including oscillations and different scaling regimes.

Findings

Constructed solutions with infinite time relaxation, blow-up, and intermediate behaviors.

Included solutions with oscillating soliton scales, both damped and undamped.

Extended the class of admissible length scales to include powers of time.

Abstract

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