Boundedness and Decay Results for Reaction-Diffusion Systems

Boundedness and decay results are obtained for semilinear parabolic systems of partial differential equations. m-component systems of the form \[ u_t = DΔ u + f(u) {on }Ω × R_ + \] with bounded initial data and various boundary conditions are considered, where D is an m × m positive diagonal matrix, Ω is a smooth bounded domain in Rⁿ, and f:Rᵐ → Rᵐ is locally Lipschitz. These results are based upon f satisfying a Lyapunov-type condition. The theory is applied to some specific reaction-diffusion problems.