Global Regularity for Supercritical Nonlinear Dissipative Wave Equations in 3D

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 $H_{\mathrm{rad}}^k({\mathbf{R}}^3)\times H_{\mathrm{rad}}^{k-1}({\mathbf{R}}^3)$ for all $p\ge 3$ and $k\ge 3$. 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 $W^{k,q}(\mathbf{R})\times W^{k-1,q}(\mathbf{R})$ for all $k\in [1,2]$ and $q\in [1,\infty ]$.

13 pages