# American Institute of Mathematical Sciences

January  2018, 38(1): 363-396. doi: 10.3934/dcds.2018018

## LP decay for general hyperbolic-parabolic systems of balance laws

 Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294-1170, USA

Received  May 2017 Revised  August 2017 Published  September 2017

Fund Project: This work was partially supported by a grant from the Simons Foundation (#244905 to Yanni Zeng).

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 $L^p$ decay rates for $p≥2$. Our result is general and applies to physical models such as gas flows with translational and vibrational non-equilibrium. Our result also recovers or improves the existing results in literature on the special cases of hyperbolic-parabolic conservation laws and hyperbolic balance laws, respectively.

Citation: Yanni Zeng. LP decay for general hyperbolic-parabolic systems of balance laws. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 363-396. doi: 10.3934/dcds.2018018
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