WELL-POSEDNESS, BLOW-UP PHENOMENA AND GLOBAL EXISTENCE FOR THE GENERALIZED b-EQUATION WITH HIGHER-ORDER NONLINEARITIES AND WEAK DISSIPATION

This paper deals with the Cauchy problem for a weakly dissipative shallow water equation with high-order nonlinearities yt + u(m+1) y(x) + bu(m)u(x)y + lambda y = 0, where lambda; b are constants and m is an element of N, the notation y := (1 - partial derivative(2)(x))u, which includes the famous b-equation and Novikov equations as special cases. The local well-posedness of solutions for the Cauchy problem in Besov space B-p,r(s) with 1 <= p; r <= + infinity and s > max {1 + 1/p; 3/2} is obtained. Under some assumptions, the existence and uniqueness of the global solutions to the equation are shown, and conditions that lead to the development of singularities in finite time for the solutions are acquired, moreover, the propagation behaviors of compactly supported solutions are also established. Finally, the weak solution and analytic solution for the equation are considered.