Some new decay properties of solutions for a modified Camassa–Holm equation with cubic nonlinearity

Abstract

In this article, we derive some new asymptotic properties of solution for a modified Camassa–Holm equation with cubic nonlinearity in weighted Sobolev spaces. Specifically, we prove that the strong solutions of Eq. (1.2) will maintain the corresponding decay properties provided the initial data ω0(x), ω0,x(x) decay logarithmically, algebraically and exponentially, respectively, which improves and extends the persistence properties of strong solutions to Eq. (1.2) obtained by Wu and Guo in paper [J. Math. Phys. 55, 081504 (2014)]. This method can be used to study the asymptotic decay properties for a class of Camassa–Holm equations.