Abstract:
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.
Keywords:first-order quasilinear hyperbolic system, smoothing boundary conditions, stabilization to zero in finite time, exponential stability.