On the Existence of Stationary and Periodic Solutions for a Class of Autonomous Reaction-diffusion Systems

Abstract. We examine the autonomous reaction-diffusion system ut = λ1uxx + f(u, v)u − v, vt = λ2vxx + f(u, v)v + u for t > 0 with Dirichlet boundary conditions on I = (0, 1), where λ1, λ2 are positive constants. If λ1 and λ2 coincide, then the zero solution is the only stationary solution. If the zero solution is unstable and f satisfies some monotonicity condition, then periodic motion can occur. In this case, we actually construct a periodic solution and show, in addition, that it has certain properties. If λ1 6= λ2, nontrivial stationary solutions might exist—we prove the existence for some values of λ1, λ2. On the other hand, we show that no such nontrivial stationary solution exists if the ratio λ1/λ2 is contained in an interval around 1 which we will state explicitly.