The basic existence theory of Kato and Majda enables us to obtain local-in-time classical solutions to generally quasilinear hyperbolic systems in the framework of Sobolev spaces (in x) with higher regularity. However, it remains a challenging open problem whether classical solutions still preserve well-posedness in the case of critical regularity. This paper is concerned with partially dissipative hyperbolic system of balance laws. Under the entropy dissipative assumption, we establish the local well-posedness and blow-up criterion of classical solutions in the framework of Besov spaces with critical regularity with the aid of the standard iteration argument and Friedrichs' regularization method. Then we explore the theory of function spaces and develop an elementary fact that indicates the relation between homogeneous and inhomogeneous Chemin-Lerner spaces (mixed space-time Besov spaces). This fact allows us to capture the dissipation rates generated from the partial dissipative source term and further obtain the global well-posedness and stability by assuming at all times the Shizuta-Kawashima algebraic condition. As a direct application, the corresponding well-posedness and stability of classical solutions to the compressible Euler equations with damping are also obtained.
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