The H1(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 space C([0,∞) × R)∩L∞([0,∞); H1(R)) provided that its initial value u0(x) belongs to the space H1(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.