The Cauchy problem for a 4-parameter family of equations with peakon traveling waves. (March 2016)

• Record Type: Journal Article
• Authors: Alexandrou Himonas, A.; Mantzavinos, Dionyssios
• Is Part Of: Nonlinear analysis. Volume 133(2016)
• Page Start: 161
• Page End: 199
• Publication Date: 2016-03
• Subjects: primary 35Q53 37K10 35C07; Fokas–Olver–Rosenau–Qiao equation -- Peakon traveling waves -- Novikov equation -- Integrable equations -- Camassa–Holm equation -- Degasperis–Procesi equation -- Cauchy problem -- Well-posedness in Sobolev spaces -- Non-uniform dependence on initial data -- Hölder continuity -- Approximate solutions -- Commutator estimate -- Conserved quantities; 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.2015.12.012
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: The initial value problem for a novel 4-parameter family of evolution equations, which are nonlinear and nonlocal and possess peakon traveling wave solutions, is studied on both the line and the circle. It is proved that this family of equations is well-posed in the sense of Hadamard when the initial data belong to the Sobolev spaces H s with s > 5 / 2 . Also, it is shown that the data-to-solution map is not uniformly continuous. However, if H s, s > 5 / 2, is equipped with a weaker H r norm, 0 ⩽ r < s, then the solution map becomes Hölder continuous.