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| In this talk I shall investigate reaction-diffusion systems that satisfy the comparison principle and possess a mass conservation property.
Motivated from mathematical analysis of transport models by molecular motors and chemical reversible reaction models, recently we have obtained some fundamental results on the structure of stationary and time-periodic solutions in a rather general framework of order-preserving dynamical systems. More precisely, our general results state that:
(1) if there exists at least one fixed point, then there exist infinitely many of them, and the set of fixed points (which correspond to stationary or time-periodic solutions of the model equations) is totally ordered, connected and unbounded;
(2) any bounded orbit converges to some element of this continua of fixed points as time tends to infinity.
In particular, our general results imply that if the model equation possesses a trivial stationary or time-periodic solution (such as zero), then there are automatically infinitely many nontrivial stationary or time-periodic solutions.
In the present talk I shall present some new applications of our general theory to other problems of reaction-diffusion systems.
This is joint work with Danielle Hilhorst and Hiroshi Matano. |
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