Reaction-diffusion systems on domains with thin channels (Q1904828)

The author deals with the reaction-diffusion equation \[ u_t= D \Delta u+ f(u),\quad (t, x)\in (0, \infty)\times \Omega \] with Neumann boundary conditions, where \(u\) is a vector-valued function and \(D\) is a positive diagonal matrix. He is interested in the dynamical system generated by this evolution equation and its characterization when the domain \(\Omega= \Omega_\varepsilon\) is a dumbbell shaped or a similar type domain. One of the important contributions in this field is Y. Morita's work [\textit{Y. Morita}, J. Dyn. Differ. Equations 2, No. 1, 69-115 (1990; Zbl 0702.35129)], who first obtained and characterized the inertial manifold of the reaction diffusion equation on Dumbbell domain and gave an interesting example. This paper tries to extend this work. The author carries out a theory in more general domains and situations. The technical part relies on the characterizations of the eigenvalue problem due to J. Arrieta's doctoral dissertation (Georgia institute of Technology 1991).