TheH1(R)Space Global Weak Solutions to the Weakly Dissipative Camassa-Holm Equation

Abstract

The existence of global weak solutions to the Cauchy problem for a generalized Camassa-Holm equation with a dissipative term is investigated in the spaceC([0,∞)×R)∩L∞([0,∞);H1(R))provided that its initial valueu0(x)belongs to the spaceH1(R). A one-sided super bound estimate and a space-time higher-norm estimate on the first-order derivatives of the solution with respect to the space variable are derived.

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.…”