On the Existence and the Uniform Decay of a Hyperbolic Equation with Non-Linear Boundary Conditions

Abstract

In this paper, we study a hyperbolic model based on the equation $y_{tt}-\Delta_{y} + \sum_{j = 1}^{n}b_j(x,t)\frac{\partial y_{t}}{\partial x_j}= 0$ with nonlinear boundary conditions given by $\frac{\partial y}{\partial v}+ f(y) + g(y_t)= 0$.

We prove the existence and the uniqueness of global solutions. Also, we obtain the uniform decay of the energy without control of its derivative sign.