Wave breaking and asymptotic analysis of solutions for a class of weakly dissipative nonlinear wave equations

Abstract

We study formation of singularities and persistence properties of solutions for a class of non-local and non-linear evolution equations with an arbitrary function and a non-negative (dissipation) parameter, which includes the famous Camassa-Holm equation and other recently reported hydrodynamic models as particular members. In case such a parameter is positive, solutions emanating from Cauchy problems have time decaying norms bounded from above by the norm of the corresponding initial datum. Whenever the function is even, for a given odd initial datum with slope satisfying a certain relation involving the dissipation parameter, then the corresponding solution of the problem breaks at finite time. We can also describe scenarios for the occurrence of wave breaking for more general initial data or functions, provided that those latter satisfy certain conditions on their first derivatives. We also study asymptotic behavior and unique continuation properties for solutions of the equation.