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Öğe Regularity and scattering for the wave equation with a critical nonlinear damping(2009) Todorova, Grozdena; Uğurlu, Davut; Yordanov, BorislavWe show that the nonlinear wave equation u + u 3 t = 0 is globally well-posed in radially symmetric Sobolev spaces H k rad(R 3) × H k-1 rad for all integers k > 2. This partially extends the well-posedness in H k(R 3) × H k-1(.R 3) 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,u t)| t=0 in radially symmetric Sobolev spaces. © 2009 The Mathematical Society of Japan.
