A criterion for exponential stability of the zero solution to a weakly-coupled cooperation-diffusion parabolic system

The linearization principle is established for bounded (for all $t\in\mathbf{R}$) solutions to parabolic systems. A stability criterion for the linearized problem is obtained. The results are applied to study of exponentially stable solutions to the Neumann problem in a convex domain. Under some additional assumptions it is shown that such a solution is independent of the space variables.