## Exponential stability of the wave equation with boundary time-varying delay.(English)Zbl 1215.35030

The considered initial boundary value problem for the wave equation in $$\mathbb R^{n}$$ has a time-varying delay term in the boundary condition. The dependence on this term can be linear as well as nonlinear. Under some very special assumptions on the coefficients and delays the well posedness of both problems, linear and nonlinear, is proved. Techniques from semigroup theory are used. The second part of paper is dedicated to the proof of exponential stability results for both problems.

### MSC:

 35B35 Stability in context of PDEs 35L05 Wave equation 93D15 Stabilization of systems by feedback 35L20 Initial-boundary value problems for second-order hyperbolic equations

### Keywords:

delay feedbacks; stabilization; semigroup theory
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