The Cauchy Problem for the Camassa-Holm Equation with Quartic Nonlinearity in Besov Spaces

In this paper, we study the Camassa-Holm equation with quartic nonlinearity. We prove thatthe Cauchy problem for this equation is locally well-posed in the critical Besov spaceB3=22;1or inBsp;rwith 1≤p, r≤+∞,s >max{1 + 1/p,3/2}. We also prove that if a weakerBqp;r-topology isused, then the solution map becomes H ̈older continuous. Furthermore, if the space variablexistaken to be periodic, we show that the solution map defined by the associated periodic boundaryproblem is not uniformly continuous inBs2;rwith 1≤r≤+∞, s >3/2 orr= 1, s=32.