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Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation. (English) Zbl 0958.35119
The main objective of the work is to rigorously prove the well-posedness of the Camassa-Holm equation, which is an integrable equation appearing in the theory of waves on a surface of an ideal fluid. Unlike the classical Korteweg-de Vries equation, the Camassa-Holm equation is a fully nonlinear one (in fact, it contains no linear terms except for the time derivative, \(\partial u\partial t\)), therefore it has both regular soliton solutions and the so-called cuspons, which are solitons with a finite amplitude but singular first derivative. The approach adopted in the work is based on regularizing the equation by adding to it a linear dispersive term, followed by consideration of the limit in which the coefficient in front of the regularizing term is vanishing. As a result, the authors prove that the Camassa-Holm equation is locally wellposed in the Sobolev space of initial conditions \(H^s\) with any \(s>3/2\). Despite this result, blow-up solutions, i.e., those which develop a singularity in a finite time, are found too, and the formation of the singularity by these solutions is also studied in the paper.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76B25 Solitary waves for incompressible inviscid fluids
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