We derive a result on the limit of certain sequences of principal eigenvalues associated with some elliptic eigenvalue problems. This result is then used to give a complete description of the global structure of the curves of positive steady states of a parameter dependent diffusive version of the classical logistic equation. In particular, we characterize the bifurcation values from infinity to positive steady states. The stability of the positive steady states as well as the asymptotic behaviour of positive solutions is also discussed.

On the assumption of separated boundary conditions autonomous scalar reactiondiffusion equations do not admit periodic orbits. The relevance of the assumption of separatedness is shown by giving an example of non separated boundary conditions for which Hopf bifurcation occurs. The example is a model of a simple thermostat.1997 Academic Press

We show that a class of reaction diffusion systems on R N generates an asymptotically compact semiflow on the Banach space of bounded uniformly continuous functions. If such a semiflow is dissipative, then a unique, non-empty, compact minimal attractor is known to exist. We apply this abstract result to obtain the existence of the compact minimal attractor for reaction diffusion systems on R N that contain appropriate weight functions. We also state conditions, which guarantee that the attractor has finite Hausdorff-dimension.

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