Stability of singular solutions to the b-family of equations

In this paper, we first construct some explicit solutions to theb-family of equations,which will become unbounded in a finite time. Then, we investigate the asymptoticstability of the aforementioned singular solutions of the b-family of equations in theSobolev space H-s with s >(7)/(2). It is also interesting to point out that this stability highlydepends on the values of parameterb, that is, b is an element of (-1,2]. The proof is based on thedetailed analysis on the estimates of the perturbed solutions and the properties of thecorresponding linear operators.