THE CAUCHY PROBLEM AND BLOW-UP PHENOMENA FOR A NEW INTEGRABLE TWO-COMPONENT PEAKON SYSTEM WITH CUBIC NONLINEARITIES

This paper studies the local well-posedness and blow-up phenomena for a new integrable two-component peakon system in the Besov space. Firstly, by utilizing the Littlewood-Paley theory, the logarithmic interpolation inequality and the Osgood's Lemma, we investigate the existence and uniqueness of the solution to the system in the critical Besov space B-2,1(1/2)(R) xB(2,1)(1/2) (R), and show that the data-to-solution mapping is Holder continuous. Secondly, we derive a blow-up criteria for the Cauchy problem in the critical Besov space. Finally, with suitable conditions on the initial data, a new blow-up criteria for the system is obtained by virtue of the global conservative property of the potential densities m and n along the characteristics and the blow-up criteria at hand.