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

Article type: Research Article

Authors: Klevtsovskiy, Arsen V.^a | Mel’nyk, Taras A.^{b; *}

Affiliations: [a] National Pedagogical Dragomanov University, Pyrogova st., 9, Kyiv 01601, Ukraine. E-mail: avklevtsovskiy@gmail.com | [b] Department of Mathematical Physics, Taras Shevchenko National University of Kyiv, Volodymyrska st., 64/13, Kyiv 01601, Ukraine. E-mail: melnyk@imath.kiev.ua

Correspondence: [*] Corresponding author. Tel.: +380-44-259 0540; Fax: +380-44-259 0785; E-mail: melnyk@imath.kiev.ua.

Abstract: A thin graph-like junction Ωε⊂R3 consists of several thin curvilinear cylinders that are joined through a domain (node) of diameter O(ε). Here ε is a small parameter characterizing the thickness of the thin cylinders and the node. In Ωε 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 uε as ε→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 εα0 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 α0. 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.

Keywords: Approximation, semilinear parabolic problem, asymptotic estimate, thin graph-like junction

DOI: 10.3233/ASY-181511

Journal: Asymptotic Analysis, vol. 113, no. 1-2, pp. 87-121, 2019