The \(H^1(\mathbb R)\) space global weak solutions to the weakly dissipative Camassa-Holm equation (Q1925413)

Summary: 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 \(C([0, \infty) \times R) \cap L^\infty ([0, \infty); H^1(\mathbb R))\) provided that its initial value \(u_0(x)\) belongs to the space \(H^1(R)\). 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.