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

Abstract

In this paper, we investigate the Cauchy problem for the weakly dissipative Dullin-Gottwald-Holm equation. We first establish the local well-posedness result by Kato’s semigroup theory. Then, we obtain the precise blow-up scenario and present some blow-up results for strong solutions to the equation. Finally, we discuss the blow-up rate of the wave-breaking solutions. This result complements the early one in the literature, such as Novruzov [J. Math. Phys. 54, 092703 (2013)].

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

“…The local well-posedness of Cauchy problem for (1.1) with the initial data u (x) ∈ H s , s > , can be obtained by applying the Kato's theory, see [2,49]. It is easy to see that some results hold for (1.1).…”

“…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.…”

“…2). Observing that u = G * y with G(x) = e −|x| , x ∈ R, we haveu(t, x) = e −x x −∞ e ξ y(t, ξ )dξ + e x ∞ x e −ξ y(t, ξ )dξ , ux(t, x) = − e −x x −∞ e ξ y(t, ξ )dξ + e x ∞ x e −ξ y(t, ξ )dξ .…”

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

“…When = ε 0, → α 0 and = λ 0, (1.1) becomes the well-known KdV equation: 2) describes unidirectional propagation of waves at the free surface of shallow water under the influence of gravity. ( ) u x t , represents the wave height above a flat bottom, x is proportional to distance in the direction of propagation and t is proportional to the elapsed time, see, e.g., [1]. The Cauchy problem and long-time behavior of the KdV equation have been studied extensively, see, e.g., [2][3][4].…”

“…which was derived by Camassa and Holm in [7] by approximating directly the Hamiltonian for Euler equations in the shallow water regime. It turns out that it is also a model for the propagation of nonlinear waves in cylindrical hyperelastic rods, see [1]. Recently, the CH equation(s) has been investigated in [8][9][10][11][12][13][14][15][16][17][18].…”