Journal
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B
Volume 17, Issue 1, Pages 191-220Publisher
AMER INST MATHEMATICAL SCIENCES-AIMS
DOI: 10.3934/dcdsb.2012.17.191
Keywords
Reaction diffusion equations; stability; near-shadow system
Categories
Funding
- The Patrick F. Lett Graduate Students Bursary, Dalhousie
- NSERC, Canada
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We consider a class of one-dimensional reaction-diffusion systems, {u(t) = epsilon(2)u(xx) + f(u, w) tau w(t) = Dw(xx) + g(u, w) with homogeneous Neumann boundary conditions on a one dimensional interval. Under some generic conditions on the nonlinearities f,g and in the singular limit epsilon -> 0, such a system admits a steady state for which u consists of sharp back-to-back interfaces. For a sufficiently large D and for sufficiently small T, such a steady state is known to be stable in time. On the other hand, it is also known that in the so-called shadow limit D -> infinity, patterns having more than one interface are unstable. In this paper we analyse in detail the transition between the stable patterns when D = O(1) and the shadow system when D -> infinity. We show that this transition occurs when D is exponentially large in epsilon and we derive instability thresholds D-1 >> D-2 >> D-3 >> ... such that a periodic pattern with 2K interfaces is stable if D < D-K and is unstable when D > D-K. We also study the dynamics of the interfaces when D is exponentially large; this allows us to describe in detail the mechanism leading to the instability. Direct numerical computations of stability and dynamics are performed, and these results are in excellent agreement with corresponding results as predicted by the asymptotic theory.
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