Camassa-holm Equation, Initial Boundary Value Problems

In this paper, we obtain the existence and uniqueness of the local strong solutions to the initial boundary value problem for a one-dimensional shallow-water equation (Camassa-Holm equation) on the half space {x > 0} with initial data u₀ ∈ H²(R⁺)∩ H₀¹(R⁺). The solution is obtained as a limit of the solutions for a class of approximation problems. We also establish the global result of the corresponding solution, provided that the initial data u₀ satisfies certain positivity condition.