The well-posedness, ill-posedness and non-uniform dependence on initial data for the Fornberg–Whitham equation in Besov spaces

Abstract

In this paper, we first establish the local well-posedness (existence, uniqueness and continuous dependence) for the Fornberg–Whitham equation in both supercritical Besov spaces Bp,rs,s>1+1p,1≤p,r≤+∞ and critical Besov spaces Bp,11+1p,1≤p<+∞, which improves the previous work (Holmes and Thompson, 2017; Holmes, 2016; Yin, 2007). Then, we prove the solution is not uniformly continuous dependence on the initial data in the Besov spaces Bp,rs,s>1+1p,1≤p≤+∞,1≤r<+∞ or s=1+1p,1≤p<+∞,r=1. At last, we show that the Cauchy problem for the Fornberg–Whitham equation is ill-posed in Bp,∞σ with σ>3+1p,1≤p≤+∞.