The properties of solutions for a generalized b-family equation with peakons

摘要

This paper deals with the Cauchy problem for a shallow water equation with high-order nonlinearities, y _t +u ~(m+1) y _x +bu ~m u _x y=0, where b is a constant, m∈ N, and we have the notation y:= (1-?_x ~2) u, which includes the famous Camassa-Holm equation, the Degasperis-Procesi equation, and the Novikov equation as special cases. The local well-posedness of strong solutions for the equation in each of the Sobolev spaces H~s(R) with s>3/2 is obtained, and persistence properties of the strong solutions are studied. Furthermore, although the H~1(R)-norm of the solution to the nonlinear model does not remain constant, the existence of its weak solutions in each of the low order Sobolev spaces H~s(R) with 1