Global regularity for supercritical nonlinear dissipative wave equations in 3D. (March 2017)

• Record Type: Journal Article
• Authors: Wakasa, Kyouhei; Yordanov, Borislav
• Is Part Of: Nonlinear analysis. Volume 152(2017)
• Page Start: 183
• Page End: 195
• Publication Date: 2017-03
• Subjects: primary 35L70 -- secondary 35A05; Wave equation -- Nonlinear damping -- Supercritical -- Regularity; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2016.12.013
• Languages: English
• ISSNs: 0362-546X

Abstract

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, ∞ ] .