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

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, timedependent 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 oscillating in either a bounded or unbounded set, for all sufficiently large t.

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

“…On the other hand, the terms of the form rf 1 ptq in u ell and w 1,lm automatically match (in other words, they are the same). A similar kind of automatic matching of terms that are larger than those involving the degrees of freedom is observed in [20]. Therefore, we can choose v 2,0 so that the terms of the form f 0 ptq in the difference u ell,main ´w1,lm ´v2,lm vanish.…”

“…The method of proof of Theorem 1.1 is similar to that used in a wave maps work of the author, [20]. For completeness, we summarize the argument here.…”

“…the current work constructs a finite energy solution to (1.1) which can be decomposed as upt, rq " Q λptq prq `vrad pt, rq `ue pt, rq where v rad solves (1.3), Epu e , B t pQ λptq `ue qq Ñ 0 as t Ñ 8. (We remark that, aside from the precise smallness constraints on C u , C l , C 2 , the admissible class of λ of this work is the same as that of a wave maps work of the author, [20].) To the knowledge of the author, the existence of all of the solutions constructed in this work, aside from those with asymptotically constant λptq, or λptq equal to precisely a power of t (as in [2]) is new.…”

“…For each λ P Λ, there exists T 0 ą 0 and a finite energy solution, u, to (1.1), for t ą T 0 , satisfying upt, rq " Q λptq prq `vrad pt, rq `ue pt, rq where p´B 2 t `B2 r `2 r B r qv rad " 0, Epv rad , B t v rad q ă 8 and Epu e , B t pQ λptq `ue qq ď C ´sup xPrT λ ,ts a λpxq ¯2 t Remark 1. As in [20], the Cauchy data of v rad is explicit in terms of λ. In fact, v rad p0, rq " 0, B t v rad p0, rq " ψprq r ˜´15π a λprqλ 2 prq 8…”

“…The solutions in Theorem 1.1 include solutions with λptq as in (1.5). It is relatively straightforward to verify that the following functions are in Λ (and the complete details of this verification for the following functions is given in the remarks following Theorem 1.1 of [20]). For 0 ă λ 0 ď λ 1 P R, 0 ď α 0 , α 1 , α 0 `α1 ă 1 λptq " λ 0 log ´α0 ptq `λ1 log α 1 ptq 2 ``λ 1 log α 1 ptq ´λ0 log ´α0 ptq 2 sinplogplogptqqq For c ą 0 sufficiently small, and |a| sufficiently small, depending on c, λptq " t a p2 `c sinplogptqqq As per (1.10), there are many more possible forms of λ P Λ.…”