Global existence and propagation speed for a generalized Camassa-Holm model with both dissipation and dispersion

In this paper, we study a generalized Camassa–Holm (gCH) model with both dissipation and dispersion, which has (N+1)-order nonlinearities and includes the following three integrable equations: the Camassa–Holm, the Degasperis–Procesi, and the Novikov equations, as its reductions. We first present the local well-posedness and a precise blow-up scenario of the Cauchy problem for the gCH equation. Then, we provide several sufficient conditions that guarantee the global existence of the strong solutions to the gCH equation. Finally, we investigate the propagation speed for the gCH equation when the initial data are compactly supported

Global Weak Solutions for the Weakly Dissipative μ-Hunter–Saxton Equation

The paper deals with the global existence of weak solutions for a weakly dissipative μ-Hunter–Saxton equation. The problem is analyzed by using smooth data approximating the initial data and Helly’s theorem.Розглянуто проблему глобального існування слабких розв'язків слабкодисипативного μ-рівняння Хантера- Сакстона за допомогою гладких даних, що є наближенням до початкових даних, та теорему Хеллі