A nonuniform Neumann boundary-value problem is considered for the Poisson equation in a thin domain Ω ε coinciding with two thin rectangles connected through a joint of diameter O(ε) . A rigorous procedure is developed to construct the complete asymptotic expansion for the solution as the small parameter ε → 0. Energetic and uniform pointwise estimates for the difference between the solution of the starting problem (ε > 0) and the solution of the corresponding limit problem (ε = 0) are proved, from which the influence of the geometric irregularity of the joint is observed.

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.

A semilinear parabolic problem is considered in a thin 3-D star-shaped junction that consists of several thin curvilinear cylinders that are joined through a domain (node) of diameter ( ).The purpose is to study the asymptotic behavior of the solution u as → 0, ie, when the star-shaped junction is transformed in a graph. In addition, the passage to the limit is accompanied by special intensity factors { i }and { i } in nonlinear perturbed Robin boundary conditions.We establish qualitatively different cases in the asymptotic behavior of the solution depending on the value of the parameters { i }and { i }. Using the multiscale analysis, the asymptotic approximation for the solution is constructed and justified as the parameter → 0. Namely, in each case, we derive the limit problem ( = 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.

A nonuniform Neumann boundary-value problem is considered for the Poisson equation in a thin 3D aneurysm-type domain that consists of thin curvilinear cylinders that are joined through an aneurysm of diameter O(ε). A rigorous procedure is developed to construct the complete asymptotic expansion for the solution as the parameter ε → 0.The asymptotic expansion consists of a regular part that is located inside of each cylinder, a boundary-layer part near the base of each cylinder, and an inner part discovered in a neighborhood of the aneurysm. Terms of the inner part of the asymptotics are special solutions of boundary-value problems in an unbounded domain with different outlets at infinity. It turns out that they have polynomial growth at infinity. By matching these parts, we derive the limit problem (ε = 0) in the corresponding graph and a recurrence procedure to determine all terms of the asymptotic expansion.Energetic and uniform pointwise estimates are proved. These estimates allow us to observe the impact of the aneurysm.

A semi-linear boundary-value problem with nonlinear Robin boundary conditions is considered in a thin 3D aneurysm-type domain that consists of thin curvilinear cylinders that are joined through an aneurysm of diameter 𝓞(ε). Using the multi-scale analysis, the asymptotic approximation for the solution is constructed and justified as the parameter ε → 0. Namely, we derive the limit problem (ε = 0) in the corresponding graph, define other terms of the asymptotic approximation and prove energetic and uniform pointwise estimates. These estimates allow us to observe the impact of the aneurysm on some properties of the solution.