Asymptotic behavior of solutions to general hyperbolic-parabolic systems of balance laws in multi-space dimensions

We study time asymptotic behavior of solutions for a general system of hyperbolic-parabolic balance laws in $m$ space dimensions, $m \geq 2$. The system has physical viscosity matrices. Besides, there is a lower order term to account for relaxation, damping or chemical reaction. The viscosity matrices and the Jacobian matrix of the lower order term are rank deficient. We study Cauchy problem around a constant equilibrium state. Under a set of reasonable assumptions, existence of solution global in time has been established recently, and $L^p$ decay rates ($p \geq 2$) of the solution to the constant equilibrium state have been obtained. In this paper we further study the large time behavior of the solution. We show that it is time-asymptotically approximated by the solution of the corresponding linear system with the same initial data. For $p \geq 2$, optimal $L^p$ convergence rates to the asymptotic solution are obtained. These rates are faster by $(t+1)^{-1/2}$ (or $(t+1)^{-1/2} \: \mathrm{ln}(t+2)$ if $m=2$) when comparing to the convergence rates to the constant equilibrium state. Our result is general and applies to physical models such as gas flows with translational and vibrational non-equilibrium. Our result is new even for the special case of hyperbolic balance laws.

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

Received 27 June 2017

Published 2 April 2019