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Zbl 0709.35059
Martin, R.H.jun.; Smith, H.L.
Reaction-diffusion systems with time delays: Monotonicity, invariance, comparison and convergence. (English)
[J] J. Reine Angew. Math. 413, 1-35 (1991). ISSN 0075-4102; ISSN 1435-5345/e

We consider reaction diffusion systems containing time delays in the reaction terms. Sufficient conditions are obtained for the semiflow generated by mild solutions of such a system to leave invariant certain rectangular sets and to be monotone (eventually strongly monotone) with respect to the natural ordering on function space. Convergence to equilibrium is established for not necessarily monotone systems by application of the contracting rectangle method. Methods of monotone dynamical systems theory are applied in case of semiflow is eventually strongly monotone. In this case, most orbits converge to the set of equilibria. Also in the monotone case, with Neumann boundary conditions on a convex domain, spatially inhomogeneous equilibria are shown to be unstable in the linear approximation. Finally, applications of the results are made to n-species Lotka-Volterra competition systems with diffusion and time delays.
[R.H.Martin jun]

MSC 2000:: *35K57 Reaction-diffusion equations
35R10 Difference-partial differential equations
35B40 Asymptotic behavior of solutions of PDE

Keywords: comparison; time delays; mild solutions; invariant; monotone; Convergence to equilibrium; dynamical systems; Neumann boundary conditions; Lotka- Volterra competition systems

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Zentralblatt MATH Berlin [Germany]