Global weak solutions for a generalized Camassa–Holm equation

Abstract

In this paper we mainly investigate the Cauchy problem of a generalized Camassa–Holm equation. We first derive two global existence results and two blow‐up results from the relationship between the Degasperis–Procesi equation and the generalized Camassa–Holm equation. We then prove the existence and uniqueness of global weak solutions under some certain sign conditions on the initial data.

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

“…Meanwhile, the differential operators are defined as u x � (zu/zx), u t � (zu/zt), u txx � (z 3 u/ zt zx 2 ), u xx � (z 2 u/zx 2 ), and u xxx � (z 3 u/zx 3 ). Particularly, when k � λ � c � 0 and a � 1, equation (1) is the wellknown completely integrable dispersive shallow water wave equation, noted as the Camassa-Holm (C-H) equation [1][2][3][4][5][6], that was firstly reckoned as a bi-Hamiltonian partial differential equation by Fokas and Fuchssteiner [7], which possesses infinitely many conservation laws. When λ � 0, equation (1) is the Dullin-Gottwald-Holm (D-G-H) equation [8,9] derived by asymptotic expansion in the shallow water regime, from the Hamiltonian for Euler's equation and also shown to be bi-Hamiltonian.…”

“…Based on above system, on the one hand, the author derives the infinite propagation speed for the corresponding solution with compactly supported initial value, which does not stand in its whole lifespan. Specifically, we assume u as a nontrivial solution to system (2), that has compact support on [a u 0 , b u 0 ] at the beginning, and T is the maximal existence time for u. Let ρ 0 be also initially compactly supported on the interval [a ρ 0 , b ρ 0 ].…”