Influence of the node on the asymptotic behaviour of the solution to a semilinear parabolic problem in a thin graph-like junction
Arsen V. Klevtsovskiy, Taras Mel’nyk · Asymptotic Analysis · 2019
A thin graph-like junction [Formula: see text] consists of several thin curvilinear cylinders that are joined through a domain (node) of diameter [Formula: see text]. Here ε is a small parameter characterizing the thickness of the thin cylinders and the node. In [Formula: see text] we consider a semilinear parabolic problem with nonlinear perturbed Robin boundary conditions both on the lateral surfaces of the cylinders and the node boundary. The purpose is to study the asymptotic behavior of the solution [Formula: see text] as [Formula: see text], i.e. when the thin graph-like junction is shrunk into a graph. The passage to the limit is accompanied by special intensity factor [Formula: see text] in the Robin condition on the node boundary. We establish qualitatively different cases in the asymptotic behaviour of the solution depending on the value of parameter [Formula: see text]. For each case we construct the asymptotic approximation for the solution up to the second terms of the asymptotics and prove the asymptotic estimates from which the influence of the local geometric heterogeneity of the node and physical processes inside are observed.