Inertial manifolds associated to partly dissipative reaction-diffusion systems. (English) Zbl 0689.58039

Marion, Martine

J. Math. Anal. Appl. 143, No. 2, 295-326 (1989).

The purpose of the paper is to show that certain partly dissipative reactions-diffusin systems, namely systems with a polynomial growth nonlinearity and systems admitting an invariant region, possess inertial manifolds in space dimension \(n\leq 2\). These inertial manifolds contain the universal attractor, attract exponentially all the solutions as time goes to infinity, and, due to the weak dissipation, are infinite dimensional. The results are applied to certain classical systems, such like the Hodgkin-Huxley equations describing nerve impulse transmission, the Fitzhugh-Nagumo equations describing the signal transmission across axons and a system of solid combustion.

Reviewer: N.Jacob

MSC:

58J99	Partial differential equations on manifolds; differential operators
35Q99	Partial differential equations of mathematical physics and other areas of application
37-XX	Dynamical systems and ergodic theory

Keywords:

inertial manifolds; dissipative reactions-diffusin systems; Hodgkin- Huxley equations; Fitzhugh-Nagumo equations

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