The local well-posedness and existence of weak solutions for a generalized Camassa–Holm equation

Abstract

A generalization of the Camassa-Holm equation, a model for shallow water waves, is investigated. Using the pseudoparabolic regularization technique, its local well-posedness in Sobolev space H s (R) with s > 3 2 is established via a limiting procedure. In addition, a sufficient condition for the existence of weak solutions of the equation in lower order Sobolev space H s with 1 < s 3 2 is developed.

The Cauchy Problem for a Weakly Dissipative 2-Component Camassa-Holm System

“…which contradicts the assumption of the maximal existence time < +∞. Conversely, using Sobolev's embedding theorem → ∞ ( > 1/2), we derive that if condition (9) in Theorem 2 holds, the corresponding solution blows up in finite time, which completes the proof of Theorem 2.…”

“…Li and Olver [8] not only obtained the local posedness but also gave the conditions which could lead to some solutions blowing up in finite time in Sobolev space with > 3/2. For other methods to establish the local well-posedness and global existence of solutions to the Camassa-Holm equation or other shallow water models, the reader is referred to [9][10][11][12][13][14][15][16][17][18] and the references therein.…”

“…They obtained the global existence result and blow-up results for strong solutions in Sobolev space with > 3/2 by Kato's theory. In [9], Lai and Wu also investigated the weakly dissipative Camassa-Holm equation:…”

“…Motivated by the work in [9,21,23,27,33,[35][36][37][38], we study the weakly dissipative Camassa-Holm system (1). We note that the Cauchy problem of system (1) in Besov spaces has not been discussed yet.…”

On the Existence of Global Weak Solutions for a Weakly Dissipative Hyperelastic Rod Wave Equation

“…Coclite et al [6] used the analysis in [5] and established the existence of global weak solutions for a generalized hyperelastic rod wave equation (or a generalized Camassa-Holm equation); namely, = = 0 in (1). For the global or local solutions of the Camassa-Holm equation and other partial differential equations, the reader is referred to [7][8][9][10][11][12] and the references therein.…”