Global regularity for supercritical nonlinear dissipative wave equations in 3D. (March 2017) - Wakasa, Kyouhei; Yordanov, Borislav
Abstract: The nonlinear wave equation u t t − Δ u + | u t | p − 1 u t = 0 is shown to be globally well-posed in the Sobolev spaces of radially symmetric functions H rad k ( R 3 ) × H rad k − 1 ( R 3 ) for all p ≥ 3 and k ≥ 3 . Moreover, global C ∞ solutions are obtained when the initial data are C 0 ∞ 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 W k, q ( R ) × W k − 1, q ( R ) for all k ∈ [ 1, 2 ] and q ∈ [ 1, ∞ ] .