On blow-up for the supercritical defocusing nonlinear wave equation

TL;DR

This paper demonstrates finite-time blow-up solutions for a supercritical defocusing nonlinear wave equation in specific dimensions and power ranges, extending understanding of solution behaviors in nonlinear wave dynamics.

Contribution

It constructs explicit smooth solutions that blow up in finite time for certain supercritical regimes, advancing the theory of nonlinear wave equations.

Findings

Existence of finite-time blow-up solutions in 4D for p ≥ 29

Existence of finite-time blow-up solutions in higher dimensions for p ≥ 17

Extension of blow-up phenomena to supercritical defocusing nonlinear wave equations

Abstract

In this paper, we consider the defocusing nonlinear wave equation $-\partial_t^2u+\Delta u=|u|^{p-1}u$ 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.