# Uniform Boundary Stabilization of the Wave Equation with a Nonlinear Delay Term in the Boundary Conditions

## Abstract

A wave equation in a bounded and smooth domain of \(\mathbb {R}^{n}\) with a delay term in the nonlinear boundary feedback is considered. Under suitable assumptions, global existence and uniform decay rates for the solutions are established by adopting an approach due to Lasiecka and Tataru (Differ Integral Equ 6:507–533, 1993). The proof of existence of solutions relies on a construction of a suitable approximating problem for which the existence of solution will be established using nonlinear semigroup theory and then passage to the limit gives the existence of solutions to the original problem. The uniform decay rates for the solutions are obtained by proving certain integral inequalities for the energy function and by establishing a comparison theorem which relates the asymptotic behaviour of the energy and of the solutions to an appropriate dissipative ordinary differential equation.

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