Abstract: The Cauchy problem, associated with a class of higher-order Camassa-Holm equations which can be regarded as a generalization of Camassa-Holm equation, is shown. The long-time behavior of the solutions for this problem is discussed. It is proved that the algebraic decay exponent of the solutions u(x,t) at infinity is the same as that of the initial value data when |x| tends to infinity and the initial value data are algebraic decay.

Key words: higher-order Camassa-Holm equation, asymptotic behavior, algebraic decay