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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-like (Korteweg-de Vries) equations 37K40 Soliton theory, asymptotic behavior of solutions 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction 76B25 Solitary waves (inviscid fluids)