Global regularity for supercritical nonlinear dissipative wave equations in 3D

Abstract

The nonlinear wave equation utt−Δu+|ut|p−1ut=0 is shown to be globally well-posed in the Sobolev spaces of radially symmetric functions Hradk(R3)×Hradk−1(R3) for all p≥3 and k≥3. Moreover, global C∞ solutions are obtained when the initial data are C0∞ and exponent p is an odd integer.The radial symmetry allows a reduction to the one-dimensional case where an important observation of Haraux (2009) can be applied, i.e., dissipative nonlinear wave equations contract initial data in Wk,q(R)×Wk−1,q(R) for all k∈[1,2] and q∈[1,∞].