Uniform decay rate estimates for Schrödinger and plate equations with nonlinear locally distributed damping

On a compact n-dimensional Riemannian manifold (M,g), we establish uniform decay rate estimates for the linear Schrödinger and plate equations subject to an internal nonlinear damping locally distributed on the manifold. Our approach can be also employed for other equations provided that inverse inequality for the linear model occurs. In the particular case of the wave equation, where the well-known geometric control condition (GCC) is equivalent to the observability inequality, our method generalizes the results due to Cavalcanti et al. (2010, 2009) [9,10] regarding the optimal choice of dissipative regions.