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 \[ u_t = D\Delta u + f(u)\quad {\text{on }}\Omega \times \mathbb{R}_ + \] with bounded initial data and various boundary conditions are considered, where D is an $m \times m$ positive diagonal matrix, $\Omega $ is a smooth bounded domain in $\mathbb{R}^n $, and $f:\mathbb{R}^m \to \mathbb{R}^m $ is locally Lipschitz. These results are based upon f satisfying a Lyapunov-type condition. The theory is applied to some specific reaction-diffusion problems.