Influence of the node on the asymptotic behaviour of the solution to a semilinear parabolic problem in a thin graph-like junction

Abstract

A thin graph-like junction ${\mathrm{\Omega }}_{\mathit{\epsilon }}\subset {\mathbb{R}}^3$ consists of several thin curvilinear cylinders that are joined through a domain (node) of diameter $\mathcal{O}\mathrm{(}\mathit{\epsilon }\mathrm{)}$ . Here ε is a small parameter characterizing the thickness of the thin cylinders and the node. In ${\mathrm{\Omega }}_{\mathit{\epsilon }}$ 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 ${\mathit{u}}_{\mathit{\epsilon }}$ as $\mathit{\epsilon }\to 0$ , i.e. when the thin graph-like junction is shrunk into a graph. The passage to the limit is accompanied by special intensity factor ${\mathit{\epsilon }}^{{\mathit{\alpha }}_0}$ 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 ${\mathit{\alpha }}_0$ . 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.