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

Abstract

We show that the nonlinear wave equation □u+u_t³=0 is globally well-posed in radially symmetric Sobolev spaces H^k_rad(R³)×H^k−1_rad(R³) for all integers k>2. This partially extends the well-posedness in H^k(R³)×H^k−1(R³) for all k∈[1,2], established by Lions and Strauss [12]. As a consequence we obtain the global existence of C^∞ solutions with radial C₀^∞ 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.