Well-posedness and blow up phenomena for the periodic generalized Camassa-Holm equation
The Cauchy problem for the periodic generalized Camassa-Holm equation \[ \begin{cases} u_t - u_{xx} + ku_x + [g(u)/2]_x = 2u_x u_{xx} + uu_{xxx} &t > 0, \quad x \in \mathbb{R}, \\ u({0,x}) = u_0(x), & x \in \mathbb{R}, \\ u({t,x + 1}) = u({t,x}), & t \geq 0, \quad x \in \mathbb{R}\end{cases} \] is considered in the paper. Here \(g:\mathbb{R} \to \mathbb{R}\) is a given \(C^m\)-function, \(m \geq 2\) , and \(k \in \mathbb{R}\) is a fixed constant. For \(g(u) = 3u^2\) and \(k \geq 0\) the periodic generalized equation becomes the periodic Camassa-Holm equation, modelling the unidirectional propagation of shallow water waves over a flat bottom, \(u\left( {t,x} \right)\) standing for the fluid velocity at time \(t > 0\) in the spatial \(x\) direction and \(k\) being a nonnegative parameter related to the critical shallow water speed. The Camassa-Holm equation has a bi-Hamiltonian structure and is completely integrable. For \(k = 0,\) the periodic Camassa-Holm equation is a re-expression of geodesic flow on the diffeomorphism group of the circle, fact that can be used to show that it satisfies the least action principle. That equation has global solutions and also solutions which blow up in finite time. The aim of the paper is to establish the local well-posedness of strong solutions to generalized Camassa-Holm equation for a large class of initial data, to give the precise blow up scenario, and to show that the equation has solutions which blow up, provided their initial data satisfy certain conditions. The paper is organized as follows. In Section 2, the author proves the local well-posedness of the initial value problem associated with the periodic generalized Camassa-Holm equation. In Section 3, the author gives an exact blow up scenario and gets an explosion result of strong solution to the equation.
scientific article; zbMATH DE number 2139415