Global smooth solutions for hyperbolic systems with time-dependent damping

The Cauchy problem for hyperbolic systems of balance laws admits global smooth solutions near the constant states under stability condition. This was widely studied in previous works. In this paper, we concern hyperbolic systems with time-dependent damping $\mu {(1+t)}^{-\lambda }G\left(U\right)$ with $\mu >0,\lambda >0$. In the following two cases, $\left(i\right)\lambda =1,\mu >{\mu }_0,$ where ${\mu }_0>0$ is a constant depending only on the coefficients of the system; $\left(ii\right)0<\lambda <1,\mu >0,$ we prove that the smooth solutions exist globally when the initial data is small. To obtain these stability results, we establish uniform energy estimates and various dissipative estimates for all time and employ an induction argument on the order of derivatives of smooth solutions. Finally, we apply these results to some physical models.