Order-Preserving Skew-Product Flows and Nonautonomous Parabolic Systems

The paper deals with order-preserving (or monotone) skew-product flows in Banach spaces. Within a general framework, we study long-time behavior of their trajectories. We introduce the concepts of equilibria, sub- and super-equilibria and we give simple conditions that guarantee their existence. Attention is mainly paid to order-preserving skew-product flows which have an additional concavity property called sublinearity, frequently encountered in applications. For these flows we prove the uniqueness and stability of equilibria and also a limit set trichotomy, stating that either (i) all orbits are unbounded, or (ii) all orbits are bounded but their closure reaches out to the boundary of the part, or (iii) there exists a unique, globally attracting equilibrium. Our main examples are quasilinear systems of parabolic equations with almost-periodic coefficients.