Asymptotic approximation for the solution to a semi-linear parabolic problem in a thin star-shaped junction

Abstract

A semi-linear parabolic problem is considered in a thin $3D$ star-shaped junction that consists of several thin curvilinear cylinders that are joined through a domain (node) of diameter $\mathcal{O}(\varepsilon).$ The purpose is to study the asymptotic behavior of the solution $u_\varepsilon$ as $\varepsilon \to 0,$ i.e. when the star-shaped junction is transformed in a graph. In addition, the passage to the limit is accompanied by special intensity factors $\{\varepsilon^{\alpha_i}\}$ and $\{\varepsilon^{\beta_i}\}$ in nonlinear perturbed Robin boundary conditions. We establish qualitatively different cases in the asymptotic behaviour of the solution depending on the value of the parameters $\{{\alpha_i}\}$ and $\{\beta_i\}.$ Using the multi-scale analysis, the asymptotic approximation for the solution is constructed and justified as the parameter $\varepsilon \to 0.$ Namely, in each case we derive the limit problem $(\varepsilon =0)$ on the graph with the corresponding Kirchhoff transmission conditions (untypical in some cases) at the vertex, define other terms of the asymptotic approximation and prove appropriate asymptotic estimates that justify these coupling conditions at the vertex and show the impact of the local geometric heterogeneity of the node and physical processes in the node on some properties of the solution.