On the Cauchy problem for a weakly dissipative generalized $$\mu $$ μ -Hunter-Saxton equation

Abstract

In this paper, we study the Cauchy problem of a weakly dissipative generalized \(\mu \)-Hunter-Saxton equation in the periodic setting. We first establish the local well-posedness for the generalized equation in Sobolev spaces \(H^{s}\), \(s>\frac{3}{2}\). Then we obtain a wave-breaking criterion for strong solutions and some results of wave-breaking solutions with certain initial profiles. We also determine the exact blow-up rate of strong solutions. Moreover, we show that the solution map for the generalized equation is Holder continuous in \(H^{s}\), \(s\ge 2\), equipped with the \(H^{r}\)-topology for \(0\le r<s\). Finally, we give the global existence results for strong solutions and weak solutions.