Regularity and scattering for the wave equation with a critical nonlinear damping

Abstract

We show that the nonlinear wave equation $\square u+u_t^3=0$ is globally well-posed in radially symmetric Sobolev spaces $H_{\mathrm{rad}}^k({\mathit{R}}^3)\times H_{\mathrm{rad}}^{k-1}(R^3)$ for all integers $k>2$. This partially extends the well-posedness in $H^k({\mathit{R}}^3)\times H^{k-1}(R^3)$ for all $k\in [1,2]$, established by Lions and Strauss[12]. As a consequence we obtain the global existence of $C^{\infty }$ solutions with radial $C_0^{\infty }$ data. The regularity problem requires smoothing and non-concentration estimates in addition to standard energy estimates, since the cubic damping is critical when $k=2$. We also establish scattering results for initial data $(u,u_t){|}_{t=0}$ in radially symmetric Sobolev spaces.