Spatial homogeneity and invariant manifolds for damped hyperbolic equations

We discuss spatial homogeneity of semilinear damped hyperbolic equations using the theory of invariant manifolds. If the solutions of the corresponding reaction-diffusion equation tend to be spatially homogeneous, so do the solutions of the hyperbolic equation, provided the damping alpha is large. Moreover, the limit equations of the reaction-diffusion and the hyperbolic problems are identical for alpha approaching +infinity. An application is also presented for a hyperbolic problem in which the diffusion coefficient is large except in the neighborhood of a fixed point in the interval , where it becomes small. (Author)