The study of global weak solutions for a generalized hyperelastic-rod wave equation

The global weak solution to the Cauchy problem for a generalized hyperelastic-rod wave equation (or the generalized Camassa–Holm equation) is investigated in the space C([0,∞)×R)⋂L∞([0,∞);H1(R))C([0,∞)×R)⋂L∞([0,∞);H1(R)) under the assumption that the initial value u0(x)u0(x) belongs to the space H1(R)H1(R). The limit of the viscous approximation for the equation is used to establish the existence of the global weak solution. The key elements in our analysis include 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.