Global Regularity for Supercritical Nonlinear Dissipative Wave Equations in 3D

TL;DR

This paper proves global well-posedness and smooth solutions for a supercritical nonlinear dissipative wave equation in 3D, leveraging radial symmetry to reduce the problem to a one-dimensional case.

Contribution

It establishes global regularity results for supercritical nonlinear wave equations in 3D using radial symmetry and contraction properties, extending previous understanding.

Findings

Global well-posedness in Sobolev spaces for all p ≥ 3

Existence of smooth solutions for odd integer p with smooth initial data

Radial symmetry reduces the problem to a one-dimensional analysis

Abstract

The nonlinear wave equation $u_{tt}-\Delta u +|u_t|^{p-1}u_t=0$ is shown to be globally well-posed in the Sobolev spaces of radially symmetric functions $H^k_{\rm rad}({\bf R}^3)\times H^{k-1}_{\rm rad}({\bf R}^3)$ for all $p\geq 3$ and $k\geq 3$. Moreover, global $C^\infty $ solutions are obtained when the initial data are $C_0^\infty$ and exponent $p$ is an odd integer. The radial symmetry allows a reduction to the one-dimensional case where an important observation of A. Haraux (2009) can be applied, i.e., dissipative nonlinear wave equations contract initial data in $W^{k,q}({\bf R})\times W^{k-1,q}({\bf R})$ for all $k\in[1,2]$ and $q\in [1,\infty]$.