We investigate the long-time behaviour of solutions of quasilinear hyperbolic systems with transparent boundary conditions when small source terms are incorporated in the system. Even if the finite-time stability of the system is not preserved, it is shown here that an exponential convergence towards the steady state still holds with a decay rate which is proportional to the logarithm of the amplitude of the source term. The result is stated for a system with dynamical boundary conditions in order to deal with initial data that are free of any compatibility condition. The proof of the existence and uniqueness of a solution defined for all positive times is also provided in this paper.
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Gugat, M., Perrollaz, V. & Rosier, L. Boundary stabilization of quasilinear hyperbolic systems of balance laws: exponential decay for small source terms. J. Evol. Equ. 18, 1471–1500 (2018). https://doi.org/10.1007/s00028-018-0449-z
- System of balance laws
- Shallow water equations
- Telegraph equation
- Finite-time stability
- Dynamical boundary conditions
- Exponential stability
- Decay rate
Mathematics Subject Classification