Regularity and Scattering for the Wave Equation with a Critical Nonlinear Damping
نویسندگان
چکیده
We show that the nonlinear wave equation u + ut = 0 is globally well-posed in radially symmetric Sobolev spaces Hk rad(R 3) × Hk−1 rad (R 3) for all integers k > 2. This partially extends the well-posedness in Hk(R3) × Hk−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 C∞ 0 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, ut)|t=0 in radially symmetric Sobolev spaces.
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