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

We consider time-dependent convection-diffusion problems with high P\'eclet number of order $\mathcal{O}(\varepsilon^{-1})$ in thin three-dimensional graph-like networks consisting of cylinders that are interconnected by small domains (nodes) with diameters of order $\mathcal{O}(\varepsilon).$ On the lateral surfaces of the thin cylinders and the boundaries of the nodes we account for solution-dependent inhomogeneous Robin boundary conditions which can render the associated initial-boundary problem to be nonlinear. The strength of the inhomogeneity is controlled by an intensity factor of order ${\mathcal{O}} (\varepsilon^\alpha)$, $\alpha >0$. The asymptotic behaviour of the solution is studied as $\varepsilon \to 0,$ i.e., when the diffusion coefficients are eliminated and the thin three-diemnsional network is shrunk into a graph. There are three qualitatively different cases in the asymptotic behaviour of the solution depending on the value of the intensity parameter $\alpha:$ $\alpha =1,$ $\alpha > 1,$ and $\alpha \in (0, 1).$ We construct the asymptotic approximation of the solution, which provides us with the hyperbolic limit model for $\varepsilon \to 0$ for the first two cases, and prove the corresponding uniform pointwise estimates and energy estimates. As the main result, we derive uniform pointwise estimates for the difference between the solutions of the convection-diffusion problem and the zero-order approximation that includes the solution of the corresponding hyperbolic limit problem.Comment: 33 pages, 4 fiure

Spatial-skin effect for a spectral problem with “slightly heavy” concentrated masses in a thick cascade junction

The asymptotic behavior (as ε→0) of eigenvalues and eigenfunctions of a boundary- value problem for the Laplace operator in a thick cascade junction with concentrated masses is studied. This cascade junction consists of the junction’s body and a great number 5N=(ε−1) of ε−alternating thin rods belonging to two classes. One class consists of rods of finite length and the second one consists of rods of small length of order O(ε). The density of the junction is of order O(ε−α) on the rods from the second class and O(1) outside of them. There exist five qualitatively different cases in the asymptotic behavior of eigenvibrations as ε→0, namely the cases of “light” concentrated masses (α∈(0,1)), “middle” concentrated masses (α=1), “slightly heavy” concentrated masses (α∈(1,2)), “intermediate heavy” concentrated masses (α=2), and “very heavy” concentrated masses (α>2). In the paper we study the influence of the concentrated masses on the asymptotic behavior of the eigen-magnitudes if α∈(1,2)

Enhanced spatial skin–effect for free vibrations of a thick cascade junction with “super heavy” concentrated masses

The asymptotic behavior (as ε→0) of eigenvalues and eigenfunctions of a boundary-value problem for the Laplace operator in a thick cascade junction with concentrated masses is studied. This cascade junction consists of the junction’s body and a great number 5N=O(ε−1) of ε-alternating thin rods belonging to two classes. One class consists of rods of finite length and the second one consists of rods of small length of order O(ε). The mass density is of order O(ε−α) on the rods from the second class and O(1) outside of them. There exist five qualitatively different cases in the asymptotic behavior of eigen-magnitudes as ε→0, namely the case of “light” concentrated masses (a∈(0,1)), “intermediate” concentrated masses (α=1) and “heavy” concentrated masses (α∈(1,+∞")) that we divide into “slightly heavy” concentrated masses (α∈(1,2)), “moderate heavy” concentrated masses (α=2), and “super heavy” concentrated masses (alpha>2). In the paper we study the influence of the concentrated masses on the asymptotic behavior of the eigen-magnitudes in the cases α=2 and α>2. The leading terms of asymptotic expansions both for the eigenvalues and eigenfunctions are constructed and the corresponding asymptotic estimates are proved. In addition, a new kind of high-frequency vibrations is found