Lyapunov function and spectrum comparison for a reaction–diffusion system with mass conservation

We are dealing with a two-component system of reaction–diffusion equations with mass conservation in a bounded domain with the Neumann boundary conditions. We prove the global boundedness of the solution in L∞L∞-norm for t⩾0t⩾0 under a condition, and then the existence of a Lyapunov function. Moreover, by studying the linearized eigenvalue problem of a nonconstant equilibrium solution, we provide a comparison theorem for the spectrum between the linearized operators of the system and an appropriate nonlocal scalar equation. As an application of the comparison result we obtain that any stable equilibrium solution must be monotone if the space dimension is one. It is also shown that a modified system with a new parameter, which covers the present model, possesses a Lyapunov function.