Norm inflation and ill-posedness for the Fornberg-Whitham equation

Abstract

In this paper, we prove that the Cauchy problem for the Fornberg-Whitham equation is not locally well-posed in $B_{p,r}^s(\mathbb{R})$ with $(s,p,r)\in (1,1+\frac{1}{p})\times [2,\infty )\times [1,\infty ]$ or $(s,p,r)\in \{1\}\times [2,\infty )\times [1,2]$ by showing norm inflation phenomena of the solution for some special initial data.