We consider the asymptotic properties of solutions to the mixed problems for the quasilinear nonautonomous first-order hyperbolic systems with two variables in the case of smoothing boundary conditions. We prove that all smooth solutions to the problem for a decoupled hyperbolic system stabilize to zero in finite time independently of the initial data. If the hyperbolic system is coupled then we show that the zero solution to the quasilinear problem is exponentially stable.

中文翻译：

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拟线性双曲系统的有限时间稳定到零和指数稳定性

我们考虑在平滑边界条件的情况下具有两个变量的拟线性非自治一阶双曲系统的混合问题解的渐近性质。我们证明了解耦双曲系统问题的所有平滑解在有限时间内稳定为零，与初始数据无关。如果双曲系统是耦合的，那么我们表明拟线性问题的零解是指数稳定的。