Blow-up phenomena and the local well-posedness and ill-posedness of the generalized Camassa–Holm equation in critical Besov spaces

In this paper, we first establish the local well-posednesss for the Cauchy problem of a generalized Camassa–Holm (gCH) equation in Besov spaces Bp,11+1p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B^{1+\frac{1}{p}}_{p,1}$$\end{document} with 1≤p<+∞.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le p<+\infty .$$\end{document} Then we gain two blow-up criterions, and present two new blow-up results. Finally, we prove the ill-posedness of the gCH equation in critical Besov spaces B2,r32,r∈(1,+∞].\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B^{\frac{3}{2}}_{2,r},~r\in (1,+\infty ].$$\end{document}