Stability of singular solutions to the b-family of equations

Abstract

In this paper, we first construct some explicit solutions to the b-family of equations, which will become unbounded in a finite time. Then, we investigate the asymptotic stability of the aforementioned singular solutions of the b-family of equations in the Sobolev space \(H^s\) with \(s>\frac{7}{2}\). It is also interesting to point out that this stability highly depends on the values of parameter b, that is, \(b\in (-1,2]\). The proof is based on the detailed analysis on the estimates of the perturbed solutions and the properties of the corresponding linear operators.