Well-posedness for a New Completely Integrable Shallow Water Wave Equation

Recently a kind of nonlinear dispersive shallow water equation has been researched. In this paper, we investigate well-posedness in classes of discontinues functions for the nonlin- ear and third order dispersive Degasperis-Procesi with dispersive term ut i utxx + 4uux + ∞(u i uxx)x = 3uxuxx + uuxxx: By using viscous approximations and prior estimates, we prove existence of at least one weak solution, satisfying a restricted set of entropy inequality in the class L 2 (R) \ L 4 (R) . And in L 1 (R) \ BV (R) there exists entropy weak solution.