Non-stabilizing solutions of semilinear hyperbolic and elliptic equations with damping

We consider two types of equations on a cylindrical domain Omega x (0, infinity), where Omega is a bounded domain in R-N, N greater than or equal to 2. The first type is a semilinear damped wave equation, in which the unbounded direction of Omega x (0, infinity) is reserved for time t. The second type is an elliptic equation with a singled-out unbounded variable t. In both cases, we consider solutions that are defined and bounded on Omega x (0, infinity) and satisfy a Dirichlet boundary condition on partial derivativeOmega x (0, infinity). We show that, for some nonlinearities, the equations have bounded solutions that do not stabilize to any single function phi : Omega --> R, as t --> infinity; rather, they approach a continuum of such functions. This happens despite the presence of damping in the equation that forces the t derivative of bounded solutions to converge to 0 as t --> infinity. Our results contrast with known stabilization properties of solutions of such equations in the case N = 1.