Ill-posedness and global solution for the b-equation

In this paper, we consider the Cauchy problem for the b-equation. Firstly, for s>32, if u₀(x)∈ Hˢ(R) and m₀(x)=u₀(x)-u₀ₓₓ(x)∈ L¹(R), the global solutions of the b-equation is established when b≥1 or b≤1. It's worth noting that our global result is a new result which doesn't need the condition that m₀(x) keeps its sign. For s<32, it is shown (see [13]) that the Cauchy problem of the b-equation is ill-posed in Sobolev space Hˢ(R) when $b>1$ or $b<1.$ In the present paper, for s=32, we prove that the Cauchy problem of the b-equation is also ill-posed in H³²(R) in the sense of norm inflation by constructing a class of special initial data when b≠1.