Blow-up, global existence and propagation speed for a modified Camassa–Holm equation both dissipation and dispersion in \(H^{s,p}(\mathbb {R})\)

In this essay, we investigate the blow-up scenario, global solution and propagation speed for a modified Camassa–Holm (MCH) equation both dissipation and dispersion in Sobolev space \(H^{s,p} (\mathbb {R})\) $H^{s,p}(\mathbb{R})$ , \(s\ge 1\) $s\ge 1$ , \(p\in (1,\infty )\) $p\in (1,\infty )$ . First of all, by the mathematical induction of index s, we establish the precise blow-up criteria, which extends the result obtained by Gui et al. in article (Comm Math Phys 319: 731–759, 2013). Secondly, we derive the global existence of the strong solution of MCH equation both dissipation and dispersion. Eventually, the propagation speed of the equation is studied when the initial data are compactly supported.