Ill-posedness and global solution for the $b$-equation

Abstract

In this paper, we consider the Cauchy problem for the $b$-equation. Firstly, for $s>\frac{3}{2},$ if $u_0(x)\in H^s(\mathbb{R})$ and $m_0(x)=u_0(x)-u_{0xx}(x)\in L^1(\mathbb{R}),$ the global solutions of the $b$-equation is established when $b\ge 1$ or $b\le 1.$ It's worth noting that our global result is a new result which doesn't need the condition that $m_0(x)$ keeps its sign. For $s<\frac{3}{2},$ it is shown (see [13]) that the Cauchy problem of the $b$-equation is ill-posed in Sobolev space $H^s(\mathbb{R})$ when $b>1$ or $b<1.$ In the present paper, for $s=\frac{3}{2},$ we prove that the Cauchy problem of the $b$-equation is also ill-posed in $H^{\frac{3}{2}}(\mathbb{R})$ in the sense of norm inflation by constructing a class of special initial data when $b\ne 1.$

10 pages