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Russian Journal of Mathematical Physics

, Volume 16, Issue 2, pp 208–227 | Cite as

Trajectory attractor of a reaction-diffusion system with a series of zero diffusion coefficients

  • V. V. Chepyzhov
  • M. I. Vishik
Article
  • 46 Downloads

Abstract

We study a reaction-diffusion system of N equations with k nonzero and Nk zero diffusion coefficients. More exactly, the first k equations of the system contain the terms a i Δu i f j (u, v), i = 1, …, k, with the diffusion coefficient a i > 0. The right-hand sides of the other Nk equations contain only nonlinear interaction functions −h j (u, v), j = k + 1, …, N, with zero diffusion. Here u = (u 1, …, u k ) and v = (υ k+1, …, υ N ) are unknown concentration vectors. Under appropriate assumptions on the interaction functions f(·) and h(·), we construct the trajectory attractor \( \mathfrak{A}^0 \) of this reaction-diffusion system. We also find the trajectory attractors \( \mathfrak{A}^\delta \), δ = (δ 1, …, δ k ), for the analogous reaction-diffusion systems having the terms δ j Δυ j h j (u, v), j = k + 1, …, N, with small diffusion coefficients δ j ⩾ 0 in the last Nk equations. We prove that the trajectory attractors \( \mathfrak{A}^\delta \) converge to \( \mathfrak{A}^0 \) (in an appropriate topology) as δ → 0+.

Keywords

Mathematical Physic Weak Solution Vector Function Global Attractor Diffusion System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Pleiades Publishing, Ltd. 2009

Authors and Affiliations

  1. 1.Institute for Information Transmission ProblemsRAS (Kharkevich Institute)MoscowRussia

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