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

Abstract

In this paper, we study the Camassa-Holm equation with quartic nonlinearity. We prove that the Cauchy problem for this equation is locally well-posed in the critical Besov space  or in  with 1 ≤ p, r ≤ +∞, s > max{1+1/p, 3/2}. We also prove that if a weaker -topology is used, then the solution map becomes H¨older continuous. Furthermore, if the space variable x is taken to be periodic, we show that the solution map defined by the associated periodic boundary problem is not uniformly continuous in   with 1 ≤ r ≤ +∞, s > 3/2 or r = 1, s = 3/2 .