Persistence Property and Asymptotic Description for DGH Equation with Strong Dissipation

where the constants α2 and γ/c 0 are squares of length scales and the constant c 0 > 0 is the critical shallow water wave speed for undisturbed water at rest at spatial infinity. Since this equation is derived by Dullin, Gottwald, and Holm, in what follows, we call this new integrable shallow water equation (1) DGH equation. If α = 0, (1) becomes the well-known KdV equation, whose solutions are global as long as the initial data is square integrable. This is proved by Bourgain [2]. If γ = 0 and α = 1, (1) reduces to the Camassa-Holm equation which was derived physically by Camassa and Holm in [3] by approximating directly the Hamiltonian for Euler’s equations in the shallow water regime, where u(x, t) represents the free surface above a flat bottom. The properties about the well-posedness, blow-up, global existence, and propagation speed have already been studied in recent works [4–13], and the generalized version of a family of dispersive equations related to Camassa-Holm equation was discussed in [14]. It is very interesting that (1) preserves the bi-Hamiltonian structure and has the following two conserved quantities: