Blow-up phenomena for the weakly dissipative Dullin–Gottwald–Holm equation

Abstract

In this paper, we study the Cauchy problem for the weakly dissipative Dullin–Gottwald–Holm equation. We establish certain conditions on the initial datum to guarantee that the corresponding positive strong solutions blow up in finite time.

Optimal control of a viscous generalized θ-type dispersive equation with weak dissipation

“…Later on, Zhang et al [1] obtained the precise blow-up scenario and established some blow-up results for strong solutions, as well as the blow-up rate of the wave-breaking solutions to (1.9). This result complements the early one in the literature, such as [35].…”

“…Wu and Yin [26] discussed the blow-up, blow-up rate and decay of solution to the weakly dissipative θ-type equation. Recently, Novruzov [35] considered blow-up phenomena the Cauchy problem for DGH Eq. (1.9) with weak dissipation and established certain conditions on the initial datum to guarantee that the corresponding positive strong solutions blow up in finite time.…”

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

“…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. Later on, Zhang et al [2] improved the results of [1]. In [45], Novruzov extended the obtained "blow-up" result to the DGH equation under some conditions on the initial data.…”

“…Zhou, Mu and Wang [43] considered the weakly dissipative gCH equation (i.e (1.1) with c = , a = b + , k = ). 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].…”

Local-in-space blow-up criteria for two-component nonlinear dispersive wave system

“…The infinite propagation speed for the Camassa-Holm equation (and the two-component extensions of it) for k = 0 was investigated in [22,31,32,33,34] (see also [47] for k = 0). Also, the wide range of problems for CH equation with non-zero dispersion coefficient was considered in [40,54,47,46]. In particular, certain conditions on the initial datum to guarantee that the corresponding solution exists globally or blows up in finite time were established.…”