Monotone local semiflows with saddle-point dynamics and applications to semilinear diffusion equations

Consider a monotone local semiflow in the positive cone of a strongly ordered Banach space, for which 0 and ∞ are stable attractors, while all nontrivial equilibria are unstable. We prove that under suitable monotonicity, compactness, and smoothness assumptions, the two basins of attraction, B’ and B’’, are separated by a Lipschitz manifold M of co-dimension one that forms the common boundary of B’ and B’’. This abstract result is applied to a class of semilinear reaction-diffusion equations with superlinear, yet subcritical reaction terms.