On blow-up for the supercritical defocusing nonlinear wave equation

TL;DR

This work constructs finite-time blow-up solutions for the defocusing supercritical NLW in dimensions $d\,\ge 4$ and odd exponents $p$, where the critical regularity exceeds the energy class. The authors recast the equation in modulus–phase form and introduce a front renormalization to reveal an underlying relativistic Euler dynamics, yielding a self-similar leading profile $(\rho_0,\phi_0)$ built from a radial velocity function $v(Z)$. They then iteratively build higher-order corrections $(\rho_n,\phi_n)$ by solving a family of linearized problems, using a surjectivity result for the linearized operator $\mathscr L$ in carefully chosen function spaces and a truncated approximate solution to control residuals. A backward-in-time energy method constructs a true solution near the approximation, producing a complex-valued blow-up with rate $|u(t)| \sim (T_*-t)^{-2\beta/(p-1)}$ and a corresponding growth in the critical Sobolev norm, under an Assumption verified in a companion paper for the stated $(d,p)$ ranges. The approach emphasizes a hydrodynamical perspective and preserves locality via finite-speed propagation, offering a new pathway to blow-up phenomena in defocusing supercritical wave equations.

Abstract

In this paper, we consider the defocusing nonlinear wave equation $-\partial_t^2u+Δu=|u|^{p-1}u$ in $\mathbb R\times \mathbb R^d$. Building on our companion work ({\it \small Self-similar imploding solutions of the relativistic Euler equations}), we prove that for $d=4, p\geq 29$ and $d\geq 5, p\geq 17$, there exists a smooth complex-valued solution that blows up in finite time.

Theorems & Definitions (80)

• Theorem 1.1
• Corollary 1.2
• Lemma 2.1: Shao-Wei-Zhang, Lemma 2.1
• Remark 2.2
• Lemma 2.3
• Proposition 2.4
• Lemma 2.5
• proof
• Proposition 2.6
• Proposition 2.7