Existence and uniform decay of the wave equation with nonlinear boundary damping and boundary memory source term

Abstract

We consider the nonlinear model of the wave equation $$y_{tt}-\Delta y+f_0\left(\nabla y\right)=0$$ subject to the following nonlinear boundary conditions $$\frac{\partial y}{\partial\nu}+g(y_t)=\int_0^th(t-\tau )f_1(y( \tau ))\,d\tau .$$ We show existence of solutions by means of Faedo-Galerkin method and the uniform decay is obtained by using the multiplier technique.