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


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+.


Mathematical Physic Weak Solution Vector Function Global Attractor Diffusion System 
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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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