Scattering and stability for ODE-type blow-up surfaces for focusing nonlinear wave equations

We study the focusing power nonlinear wave equation with any power, in Minkowski space of any spacetime dimension. We present a complete understanding of the local stability and scattering theory (both in high regularity spaces) for solutions exhibiting ODE type blow-up on spacelike hypersurfaces, with the blow-up at each point modelled by the explicit solution $ϕ_{\mathrm{model}} = c_p t^{-α_p}$. Given a sufficiently regular spacelike hypersurface $Σ_f$, together with auxiliary scattering data $ψ$, we construct the unique corresponding solution to the nonlinear wave equation that (locally) forms an ODE type singularity on $Σ_f$ attaining $ψ$ as scattering data. Conversely, we show that such ODE type singularities are (locally) stable to suitably regular perturbations away from the singularity, and that the blow-up surface and scattering data remain regular, in a continuously dependent manner, following such perturbations.

💡 Research Summary

The paper studies the focusing power nonlinear wave equation
  □ ϕ + |ϕ|^{p‑1} ϕ = 0, p > 1,
in Minkowski space of arbitrary spatial dimension n ≥ 1. The authors address two complementary problems in a high‑regularity Sobolev setting (s≫1).

First, they prove a “scattering construction” (Theorem 1.1). Given any sufficiently smooth spacelike hypersurface Σ_f = {t = f(x)} with |∇f| < 1/10 and any auxiliary scattering datum ψ belonging to H^{s‑⌊2κ_p⌋}, they construct a unique solution ϕ defined locally near Σ_f that blows up in an ODE‑type manner exactly on Σ_f. Near the blow‑up surface the solution admits the asymptotic expansion
  ϕ(t,x) = c_p (1‑|∇f|^2)^{1/(p‑1)} (t‑f(x))^{‑α_p} + o((t‑f)^{‑α_p}),
with α_p = 2/(p‑1) and c_p determined by (1.2). The construction proceeds by flattening the surface via the change of variables τ = t‑f(x), linearizing around the leading singular profile ϕ_0 = c_p τ^{‑α_p}, and solving the perturbed equation (∂_τ^2‑γ_p τ^{‑2}) \barϕ = N