Ill-posedness for the Hamilton-Jacobi equation in Besov spaces $B_{\infty ,q}^0$

Jinlu Li, Weipeng Zhu, Zhaoyang Yin

In this paper, we study the Cauchy problem for the following Hamilton-Jacobi equation \bbal\bca \pa_tu-\De u=|\na u|^2,\quad t>0, \ x\in \R^d,\\ u(0,x)=u_0, \quad \quad x\in \R^d.…