Asymptotic approximation of a nonlinear parabolic problem in a thin star-shaped junction

What is it about?

A semi-linear parabolic problem is considered in a thin 3-dimensional star-shaped junction that consists of several thin curvilinear cylinders that are joined through a domain (node) of the small diameter. We are interested in the study of evolution phenomena (the reaction-diffusion processes, heat-mass transfer, and flow motions) as this domain is shrunk into a graph.

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Why is it important?

A large amount of physical and mathematical articles and books dedicated to different models on graphs has been published lately. The main question arising in problems on graphs is point interactions at nodes of networks, i.e., the type of coupling conditions at vertices of the graph. Also, there is increasing interest in the investigation of the influence of a local geometric heterogeneity in vessels on the blood flow. This is both an aneurysm (a pathological extension of an artery like a bulge) and a stenosis (a pathological restriction of an artery). A natural approach to explain the meaning of point interactions at vertices of the graph is the use of the limiting procedure that we propose in this article.

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