The Cauchy Problem for the Camassa-Holm Equation with Quartic Nonlinearity in Besov Spaces
Shan Zheng *
Department of Basic Courses, Guangzhou Maritime Institute, Guangzhou, Guangdong 510725, China.
*Author to whom correspondence should be addressed.
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 .
Keywords: Camassa-Holm equation, qurtic nonlinearity, well-posedness, Besov spaces, non-uniform dependence.