On the global attractor of the two-component ππ-Camassa–Holm equation with viscous terms

In the present paper, the global attractor of the two-component ππ-Camassa–Holm equation with viscous terms is concerned. The two-component ππ-Camassa–Holm equation describes a generalized formulation for the two-component Camassa–Holm shallow water wave equation, which was first established by J. Lenells as a geodesic equation on a Kählerian manifold. The viscosity terms are given by second order differential operators. The global existence of a solution to the viscous two-component ππ-Camassa–Holm equation with the periodic boundary condition is studied by using the Galerkin procedure. By advantage of some uniform prior estimates and many inequalities, one obtains the compact and bounded absorbing set and the existence of the global attractor in H2×H2/RH2×H2/R for the viscous two-component ππ-Camassa–Holm equation.