Blow-up of solutions for the dissipative Dullin–Gottwald–Holm equation with arbitrary coefficients

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

“…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. This issue is extensively studied, e.g.…”

Global Weak Solutions for the Weakly Dissipative Dullin-Gottwald-Holm Equation

“…Lemma 3.1. [1] Given u0 ∈ H s (R), s > 3 2 , there exist a maximal T = T (u0, α, c0, γ, λ) > 0 and a unique solution u to Eq. (3.2), such that…”

“…Recently, Novruzov [1] studied the finite-time blowup criteria on the Cauchy problem for the weakly dissipative Dullin-Gottwald-Holm equation:…”

Classical symmetries and conservation laws for the dissipative Dullin‐Gottwald‐Holm equation with arbitrary coefficients

“…When and λ >0 the GDGH equation is the disipate Dullin‐Gottwald‐Holm equation studied by Guo and Ni, and later by Novruzov …”

“…This is a nonlinear partial differential equation in dimensionless time-space variables describing the unidirectional propagation of shallow water waves. The function u(x, t) is the fluid velocity at time t > 0 in the direction x; is the coefficient of the linear dispersion term; f(u) is a polinomial, which derivative can be used to model the linear wave speed for undisturbed water at rest at spacial infinite; 2 and k are squares of length scales, where k is the coefficient of u in f(u), this is the linear wave speed for undisturbed water resting at spatial infinity; and is a coefficient of the weakly dissipative term (u − 2 u xx ).…”