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

We study a reaction-diffusion system of N equations with k nonzero and N − k zero diffusion coefficients. More exactly, the first k equations of the system contain the terms aiΔui − fj(u, v), i = 1, …, k, with the diffusion coefficient ai > 0. The right-hand sides of the other N − k equations contain only nonlinear interaction functions −hj(u, v), j = k + 1, …, N, with zero diffusion. Here u = (u1, …, uk) and v = (υk+1, …, υN) are unknown concentration vectors. Under appropriate assumptions on the interaction functions f(·) and h(·), we construct the trajectory attractor \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathfrak{A}^0 $$\end{document} of this reaction-diffusion system. We also find the trajectory attractors \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathfrak{A}^\delta $$\end{document}, δ = (δ1, …, δk), for the analogous reaction-diffusion systems having the terms δjΔυj − hj (u, v), j = k + 1, …, N, with small diffusion coefficients δj ⩾ 0 in the last N − k equations. We prove that the trajectory attractors \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathfrak{A}^\delta $$\end{document} converge to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathfrak{A}^0 $$\end{document} (in an appropriate topology) as δ → 0+.