We consider the wave equation with a locally damping and a nonlinear source term in a bounded domain. $ y_{tt} - \Delta y + a(x) g(y_t) =|y|^{p-2} y, $ where $p>2$. The damping is nonlinear and is effective only in a neighborhood of a suitable subset of the boundary. We show, for certain initial data and suitable conditions on $g,\,\, a$ and $p$ that this solution is global we use the Faedo-Galerkin method. Also we established the exponential decay of the energy when the nonlinear damping grows linearly by introducing a suitable Lyapunov functional.