Life span of small solutions to a system of wave equations. (July 2016)

• Record Type: Journal Article
• Authors: Hidano, Kunio; Yokoyama, Kazuyoshi
• Is Part Of: Nonlinear analysis. Volume 139(2016)
• Page Start: 106
• Page End: 130
• Publication Date: 2016-07
• Subjects: primary 35L05 35L15 -- secondary 35L71; Systems of wave equations -- Life span; 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.2016.02.020
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We study the Cauchy problem with small initial data for a system of semilinear wave equations □ u = | v | q, □ v = | ∂ t u | p in n -dimensional space. When n ≥ 2, we prove that blow-up can occur for arbitrarily small data if ( p, q ) lies below a curve in the p – q plane. On the other hand, we show a global existence result for n = 3 which asserts that a portion of the curve is in fact the borderline between global-in-time existence and finite time blow-up. We also estimate the maximal existence time and get its upper bound, which is sharp at least for ( n, p, q ) = ( 2, 2, 2 ) and ( 3, 2, 2 ) .