We show that the nonlinear wave equation is globally well-posed in radially symmetric Sobolev spaces for all integers . This partially extends the well-posedness in for all , established by Lions and Strauss. As a consequence we obtain the global existence of solutions with radial data. The regularity problem requires smoothing and non-concentration estimates in addition to standard energy estimates, since the cubic damping is critical when . We also establish scattering results for initial data in radially symmetric Sobolev spaces.
Grozdena TODOROVA. Davut UĞURLU. Borislav YORDANOV. "Regularity and scattering for the wave equation with a critical nonlinear damping." J. Math. Soc. Japan 61 (2) 625 - 649, April, 2009. https://doi.org/10.2969/jmsj/06120625