Classification under mere energy-norm convergence

Establish the classification of positive-time multi-solitons for the five-dimensional focusing energy-critical wave equation under the weaker assumption that the energy-norm distance between the solution and the prescribed sum of collinear traveling solitons converges to zero as time tends to positive infinity, without assuming an explicit polynomial decay rate.

Background

The paper classifies solutions that approach a prescribed finite sum of traveling solitons at the explicit rate t{-1/2-δ} for some δ>0. Under that hypothesis, every such solution belongs to the K-parameter family of constructed multi-solitons.

The authors explicitly leave unresolved whether the same classification remains valid when the only asymptotic information is convergence to zero in the energy norm, with no specified decay rate. They contrast this issue with generalized Korteweg–de Vries and nonlinear Klein–Gordon equations, where additional monotonicity or dispersive estimates permit classification under weaker convergence assumptions.

References

For equation wave, it remains an open question to establish the classification result under the weaker assumption

\lim_{t\to+\infty} \left|\nabla_{t,x} \left(u -\sum_{k=1}K W_k\infty \right) (t)\right|_{L2} =0.

wave:

{∂t2u−Δu−∣u∣43u=0,(t,x)∈[0,∞)×R5,u∣t=0=u0∈H˙1(R5),∂tu∣t=0=u1∈L2(R5),\left\{ \begin{aligned} &\partial_t^2 u - \Delta u - |u|^{\frac 4{3}} u = 0, \quad (t,x)\in [0,\infty)\times R^5,\\ & u_{|t=0} = u_0\in \dot H^1(R^5),\\ & \partial_t u_{|t=0} = u_1\in L^2(R^5), \end{aligned}\right.

— Classification of multi-solitons in one sense of time for the 5D energy-critical wave equation  (2609.09967 - Martel et al., 9 Sep 2026) in Remark following Theorem 2, Section 1.1 (Main results)