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

Abstract

In this paper, we consider the dissipative Degasperis-Procesi equation with arbitrary dispersion coefficient and compactly supported initial data. We establish the simple condition on the initial data which lead to guarantee that the solution exists globally. We also investigate the propagation speed for the equation under the initial data is compactly supported.

Global Dissipative Solution for an Extended Dullin–Gottwald–Holm Equation

“…On the one hand, the authors [38] established the global well posedness of the strong solution based on certain condition of the initial datum. en, the propagation speed with compact supported initial datum was investigated, which improves early results [40][41][42]. On the other hand, the authors [39] constructed blowup threshold for a positive solution in finite time under certain condition of initial datum, that complements early works [36,37].…”

Global well-posedness for the Cauchy problem of generalized Boussinesq equations in the control problem regarding initial data

“…describes propagation of waves in weakly nonlinear and weakly dispersive media [3], and then a lot of variations of Boussinesq equation were considered in many works [5], [20], [7], [3], [18] and so on. In order to deal with more general cases in physics and mathematics, the general Boussinesq equation like (1) was introduced and studied in the aspects of global well-posedness as showed in [2], [1], [11] and [10].…”

Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation

“…If the viscosity is small, e.g. ν < l 2 /(4π 2 ), then λ 1 < 0 and thus the equation is not dissipative (which is different from dissipative equations in [11]).…”