On the behavior of the solution of the dissipative Camassa–Holm equation with the arbitrary dispersion coefficient

Abstract

In this paper, we consider the dissipative Camassa-Holm equation with arbitrary dispersion coefficient and compactly supported initial data. We demonstrate the simple conditions on the initial data that lead to finite time blow-up of the solution in finite time or guarantee that the solution exists globally. Also, propagation speed for the equation under consideration is investigated.

Global existence and propagation speed for a generalized Camassa–Holm model with both dissipation and dispersion

“…In particular, the propagation speed is seriously affected by both the dissipative parameter λ and the dispersion coefficient k. On the other hand, it is worthy to note that the main difficulty in establishing the above results lies in controlling certain norms of (N +1)-order nonlinearities. In addition, some of our results cover the earlier corresponding results studied in [29,37].…”

A new blow-up criterion for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion

“…Recently, Novruzov [1] studied the Cauchy problem for the weakly dissipative Dullin-Gottwald-Holm (DGH) equation (i.e (1.1) with N = , c = , b = , a = ) and establish certain conditions on the initial datum to guarantee that the corresponding positive strong solutions blow up in nite time. The same equation for arbitrary solution has been considered in [44]. Authors showed the simple conditions on the initial data that lead to the blow-up of the solutions in nite time or guarantee that the solutions exist globally.…”

Global existence and propagation speed for a Degasperis-Procesi equation with both dissipation and dispersion

“…It is found [13,15,20]that such kinds of considerable differnces were investigated for the different model (nonlinear wave equations). Recently, Novruzov and Hagverdiyev [17] analyzed the behavior of solutions to the dissipative Camassa-Holm equation with arbitrary dispersion coefficient. In the present paper, we discuss the global existence and the propagation speed of strong solutions to the equation (1).…”