Abstract
We study the isothermal Euler equations with friction and consider non-stationary solutions locally around a stationary subcritical state on a finite time interval. The considered control system is a quasilinear hyperbolic system with a source term. For the corresponding initial-boundary value problem we prove the existence of a continuously differentiable solution and present a method of boundary feedback stabilization. We introduce a Lyapunov function which is a weighted and squared $H^1$-norm of the difference between the non-stationary and the stationary state. We develop boundary feedback conditions which guarantee that the Lyapunov function and the $H^1$-norm of the difference between the non-stationary and the stationary state decay exponentially with time. This allows us also to prove exponential estimates for the $C^0$- and $C^1$-norm.
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