Ordinary differential equations (ODEs) on inertial manifolds for reaction-diffusion systems in a singularly perturbed domain with several thin channels

Abstract

Dynamics of solutions to a reaction-diffusion system in a domain of specific shape is investigated under the homogeneous Neumann boundary conditions. It is assumed that the domain hasN large regionsD _i ,i=1,...,N, and thin channelsQ _i,j (ɛ) connectingD _i andD _j , which approach a line segment asɛ → 0 in some sense. In such a domain the firstN eigenvalues of −Δ with the Neumann boundary conditions tend to zero as ɛ→ 0, while the (N + 1)-th eigenvalue is bounded away from zero. By virtue of this gap of the eigenvalues, an inertial manifold which is invariant and attracts every solution exponentially can be constructed under a certain condition. Moreover, the ODE describing the dynamics on the inertial manifold can be given in quite an explicit form through the analysis of the limit of the manifold asε → 0.