Limit behavior of the global solutions to the Degasperis–Procesi-type equation

In this paper, we study the Degasperis–Procesi equation with a physically perturbation term—a linear dispersion. Based on the global existence result, we show that the solution of the Degasperis–Procesi equation with linear dispersion tends to the solution of the corresponding Degasperis–Procesi equation as the dispersive parameter goes to zero. Moreover, we prove that smooth solutions of the equation have finite propagation speed: they will have compact support if its initial data has this property.