Regularizing Effects of Nonlinear Damping in Supercritical Defocusing Nonlinear Wave Equations
by
Grozdena Todorova
University of Tennessee, USA
Coauthors: Borislav Yordanov

There are many results on global well-posedness and regularity for the equation u_tt-Du+u⁵=0 in R³ (the so called critical case). In contrast, the global existence of smooth solutions in the supercritical case p > 1+[4/(n-2)] appears to be an open problem, even for the space dimension n=3. We show that semi-linear wave equations with a conveniently chosen nonlinear damping term g(u_t) and with defocusing smooth nonlinearities |u|^p-1u in the supercritical case p > 5 are globally well-posed in radially symmetric Sobolev spaces H^k_rad(R³)×H^k-1_rad(R³) for all integers k ≥ 3. The results apply to the case k=2 without the requirement of radial data. We emphasize that the damping is not stronger than the nonlinearity and does not depend on the supercritical growth of the nonlinearity. The results also extend to certain exponential nonlinearities. Finally, we obtain scattering results for radial initial data in Sobolev spaces with k ≥ 2.

Date received: April 29, 2007

Copyright © 2007 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caun-12.