Asymptotic approximations for semilinear parabolic convection-dominated transport problems in thin graph-like networks

We consider time-dependent convection-diffusion problems with high Péclet number of order O(ε−1) in thin three-dimensional graph-like networks consisting of cylinders that are interconnected by small domains (nodes) with diameters of order O(ε). On the lateral surfaces of the thin cylinders and the boundaries of the nodes we account for solution-dependent inhomogeneous Robin boundary conditions which render the associated initial-boundary problem to be nonlinear. The strength of the inhomogeneity is controlled by an intensity factor of order O(εα), α∈R. The asymptotic behavior of the solution is studied as ε→0, i.e., when the diffusion coefficients are eliminated and the thin three-dimensional network is shrunk into a graph. Three qualitatively different cases are discovered in the asymptotic behavior of the solution depending on the value of the intensity parameter α: α=1, α>1, and α