The Cauchy problem for a two-component generalized θθ-equations

In this paper we investigate a two-component θθ-system which arises in shallow water theory. The present work is mainly concerned with blow-up phenomenon, persistence properties and infinite propagation speed for this new kind of system. Firstly, we establish sufficient conditions on the initial data to guarantee wave breaking for this system. Moreover, the persistence properties of the strong solutions are also analyzed. Finally, we intend to investigate infinite propagation speed in the sense that the corresponding solution u(x,t)u(x,t) does not have compact spatial support for t>0t>0 though u0∈C0∞(R), while the other one ρ(x,t)ρ(x,t) is always compactly supported under the assumption of its initial compact support condition.