We consider the energy-critical wave maps equation R 1+2 → S 2 in the equivariant case, with equivariance degree k ≥ 2. It is known that initial data of energy < 8πk and topological degree zero leads to global solutions that scatter in both time directions. We consider the threshold case of energy 8πk. We prove that the solution is defined for all time and either scatters in both time directions, or converges to a superposition of two harmonic maps in one time direction and scatters in the other time direction. In the latter case, we describe the asymptotic behavior of the scales of the two harmonic maps.The proof combines the classical concentration-compactness techniques of Kenig-Merle with a modulation analysis of interactions of two harmonic maps in the absence of excess radiation.Theorem 1.2 (Sequential Decomposition). [8,25] Let ψ(t) ∈ H ℓπ be a smooth solution to (1.5) on [0, T + ). Then there exists a sequence of times t n → T + , an integer J ∈ N, a regular map ϕ ∈ H 0 , sequences of scales λ n,j and signs ι j ∈ {−1, 1} for j ∈ {1, . . . , J}, so thatIn the case of finite time blow-up at least one scale λ n,1 → 0 as n → ∞ and ϕ(t) → ϕ(1) is a finite energy map with E( ϕ(1)) = E( ψ) − JE( Q). In the case of a global solution, ϕ(t) can be taken to be a solution to the linear wave equation (2.2) and signs ι j are required to match up so that lim r→∞ ψ(0, r) = ℓπ = lim r→∞ J j=1 ι j Q λn,j (r).Remark 1.3. A decomposition into bubbles for a sequence of times for the full nonequivariant model was obtained by Grinis [21] up to an error that vanishes in a weaker Besov-type norm. Duyckaerts, Jia, Kenig and Merle [12] proved that for energies slightly above E( Q) a one-bubble decomposition holds for continuous time.The same authors obtained in [13] a sequential decomposition into bubbles in the case of the focusing energy critical power-type nonlinear wave equation (NLW).Theorem 1.2 raises two natural questions:• Are there any solutions to (1.5) with J ≥ 2 in (1.7), i.e., are there any solutions that form more than one bubble? • And, if so, does the decomposition (1.7) hold continuously in time, i.e., does soliton resolution hold for (1.5)?

We construct pure two-bubbles for some energy-critical wave equations, that is solutions which in one time direction approach a superposition of two stationary states both centered at the origin, but asymptotically decoupled in scale. Our solution exists globally, with one bubble at a fixed scale and the other concentrating in infinite time, with an error tending to 0 in the energy space. We treat the cases of the power nonlinearity in space dimension 6, the radial Yang-Mills equation and the equivariant wave map equation with equivariance class k ≥ 3. The concentration speed of the second bubble is exponential for the first two models and a power function in the last case.Remark 1.1. More precisely, we will prove thatRemark 1.2. We construct here pure two-bubbles, that is the solution approaches a superposition of two stationary states, with no energy transformed into radiation. By the conservation of energy and the decoupling of the two bubbles, we necessarily have E(u(t)) = 2E(W ). Pure one-bubbles cannot concentrate and are completely classified, see [16].

We consider the semilinear wave equation with focusing energy-critical nonlinearity in space dimension N = 5 ∂ttu = ∆u + |u| 4/3 u, with radial data. It is known [7] that a solution (u, ∂tu) which blows up at t = 0 in a neighborhood (in the energy norm) of the family of solitons W λ , decomposes in the energy space asWe construct a blow-up solution of this type such that the asymptotic profile (u * 0 , u * 1 ) is any pair of sufficiently regular functions with u * 0 (0) > 0. For these solutions the concentration rate is λ(t) ∼ t 4 . We also provide examples of solutions with concentration rate λ(t) ∼ t ν+1 for ν > 8, related to the behaviour of the asymptotic profile near the origin. 3 8