Decay Properties of Solutions to a 4-parameter Family of Wave Equations

TL;DR

This paper studies the decay and unique continuation properties of solutions to a broad family of wave equations, including well-known integrable cases, highlighting how initial data influences long-term behavior.

Contribution

It introduces a unified analysis of decay and continuation properties for a 4-parameter wave equation family, encompassing several notable integrable models.

Findings

Solutions decay at infinity if initial data decay

Unique continuation holds for specific parameter values

Includes analysis of Camassa-Holm, Degasperis-Procesi, Novikov, and Fokas-Olver-Rosenau-Qiao equations

Abstract

In this paper, persistence properties of solutions are investigated for a 4-parameter family ($k-abc$ equation) of evolution equations having $(k+1)$-degree non-linearities and containing as its integrable members the Camassa-Holm, the Degasperis-Procesi, Novikov and Fokas-Olver-Rosenau-Qiao equations. These properties will imply that strong solutions of the $k-abc$ equation will decay at infinity in the spatial variable provided that the initial data does. Furthermore, it is shown that the equation exhibits unique continuation for appropriate values of the parameters $k$, $a$, $b$, and $c$.