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Connecting orbit structure of monotone solutions in the shadow system. (English) Zbl 0902.35058

The authors study the dynamical system given by the following system of reaction-diffusion equations \[ {\partial u\over\partial t}= \varepsilon^2 {\partial^2u\over\partial x^2}+ f(u)- \xi,\quad {d\xi\over dt}= \int_I g(u,\xi)dx \] for \((t, x)\in [0,\infty)\times I\), where \(u\) satisfies the Neumann boundary condition on \([0,\infty)\times \partial I\). Here \(I= (0,1)\). The structure of the attractor (consisting of functions monotone in \(x\)) is investigated. The stability of equilibrium solutions and the connection orbits between them are classified.
Reviewer: S.Jimbo (Sapporo)

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
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References:

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