[For the entire collection see Zbl 0519.00009.]
The author considers a reaction diffusion system of the form \(u_ t-L_ 1u=f(x,u,v),\quad v_ t-L_ 2v=g(x,u,v)\) where \(L_ i\) are elliptic operators. He assumes that \(f_ v\leq 0\), \(g_ u\leq 0\) or \(f_ v\geq 0\), \(g_ u\geq 0\) for some appropriate range u,v (defined in terms of a sub and super solution). Then the solution u(x,t), v(x,t) converges monotonically (as \(t\uparrow \infty)\) to a steady-state solution provided that the initial data \((u_ 0,v_ 0)\) is a sub and super solution of the stationary problem.