Ill-posedness for the two component Degasperis-Procesi equation in critical Besov space

In this paper, we study the Cauchy problem for the two component Degasperis-Procesi equation in critical Besov space $B^1_{\infty,1}(\mathbb R)$. By presenting a new construction of initial data, we proved the norm inflation of the corresponding solutions in $B_{\mathrm{\infty },1}^1(\mathbb{R})$ and hence ill-posedness. This is quite different from the local well-posedness result for the Degasperis-Procesi equation in critical Besov space $B_{\mathrm{\infty },1}^1(\mathbb{R})$ due to the coupled structure of density function.