Global well-posedness for systems of hyperbolic-parabolic composite type with center manifold

Concerning the global existence of classical solution to systems of hyperbolic-parabolic composite type, a well-known general theory was established by Kawashima in [4], where the dissipation condition (Kawashima-Shizuta condition) to the linearized system plays a fundamental role. Recently, systems with much weaker dissipations have attracted a lot of attentions, see [1,2,10,11] among others. The typical feature of this kind of system is that the corresponding linearized system has one eigenvalue with the real part equals to zero. This violates the Kawashima-Shizuta stability conditions. In this paper, we develop a general global well-posedness theory for this kind of system. Moreover, as the applications of the general theory, several examples are given.