Global well-posedness and blow-up for the damped perturbation of Camassa-Holm type equations

Abstract

In this paper, we prove global well-posedness of strong solutions to a class of perturbed Camassa-Holm type equations in Besov spaces. It is shown that the existence of global solutions depends only on the L 1 -integrability of the time-dependent parameters, but not on the shape or smoothness conditions on initial data. As a by-product, we obtain a new global-in-time result for the weakly dissipative Camassa-Holm type equations in Besov spaces, which considerably improves the results in [11,21,22]. Moreover, we derive two kinds of precise blow-up criteria for a certain perturbed Camassa-Holm type equations in Sobolev space, which in some sense tell us how the time-dependent parameters affect the singularity formation of strong solutions.

On Blow-Up of Solutions to a Weakly Dissipative Two-Component Camassa–Holm System

“…The system (1) appears initially in [1], then Constantin and Ivanov [2] give a demonstration about its derivation in view of the shallow water theory from the hydrodynamic point of view. Local well-posedness, blow-up, global existence, stability and other mathematical properties can be seen in [3][4][5][6][7][8][9][10][11][12][26][27][28][29][30][31][32] and references therein.…”