We study time asymptotic decay of solutions for a general system of hyperbolic-parabolic balance laws in multi space dimensions. The system has physical viscosity matrices and a lower order term for relaxation, damping or chemical reaction. The viscosity matrices and the Jacobian matrix of the lower order term are rank deficient. For Cauchy problem around a constant equilibrium state, existence of solution global in time has been established recently under a set of reasonable assumptions. In this paper we obtain optimal LP decay rates for p ≥ 2. Our result is general and applies to physical models such as gas ows with translational and vibrational non-equilibrium. Our result also recovers or improves the existing results in literature on the special cases of hyper.