Influence of the node on the asymptotic behaviour of the solution to a semilinear parabolic problem in a thin graph-like junction

Klevtsovskiy, Arsen V.; Mel'nyk, Taras A.

doi:10.3233/asy-181511

Cited by 6 publications

(5 citation statements)

References 21 publications

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“…However, the second terms {w Therefore, the problem (3.55) for the second term of the regular asymptotics should be considered in parallel with the limit spectral problem (3.33) to observe the node influence on the asymptotic behavior of the spectrum through the asymptotic estimate (3.82). This conclusion coincides with the main inference of [14].…”

Section: Lemma 31 ( [36]supporting

confidence: 91%

“…In [6] the authors studied the asymptotic behavior the high frequencies for the Laplace operator in a thin T-like shaped structure and gave a characterization of the asymptotic form of the eigenfunctions originating these vibrations. About the study of boundary-value problems in thin graph-like junctions in other contexts, we refer to last new papers [2,7,14,31,32] and references there.…”

Section: Introductionmentioning

confidence: 99%

“…To my knowledge, this is the first article in the literature concerning the asymptotics of eigenvibrations of a thin graph-like junction with a concentrated mass in the node. For this, the asymptotic approach developed in [12][13][14] is used. This approach makes it possible to take into account various factors (e.g.…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic approximations for eigenvalues and eigenfunctions of a spectral problem in a thin graph-like junction with a concentrated mass in the node

Mel'nyk

2020

Preprint

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A spectral problem is considered in a thin 3D graph-like junction that consists of three thin curvilinear cylinders that are joined through a domain (node) of the diameter O(ε), where ε is a small parameter. A concentrated mass with the density ε −α (α ≥ 0) is located in the node. The asymptotic behaviour of the eigenvalues and eigenfunctions is studied as ε → 0, i.e. when the thin junction is shrunk into a graph.There are five qualitatively different cases in the asymptotic behaviour (ε → 0) of the eigenelements depending on the value of the parameter α. In this paper three cases are considered, namely, α = 0, α ∈ (0, 1), and α = 1.Using multiscale analysis, asymptotic approximations for eigenvalues and eigenfunctions are constructed and justified with a predetermined accuracy with respect to the degree of ε. For irrational α ∈ (0, 1), a new kind of asymptotic expansions is introduced. These approximations show how to account the influence of local geometric inhomogeneity of the node and the concentrated mass in the corresponding limit spectral problems on the graph for different values of the parameter α. n0 4.4. Justification (α = m0 n0 ) 5. Asymptotic approximations in the case α = 1 5.1. Justification (α = 1

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Section: Lemma 31 ( [36]supporting

confidence: 91%

Section: Introductionmentioning

confidence: 99%

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Asymptotic approximations for eigenvalues and eigenfunctions of a spectral problem in a thin graph-like junction with a concentrated mass in the node

Mel'nyk

2020

Preprint

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“…2 and the assumption (3.86). When a problem has two parameters, it is necessary to change the asymptotic scale by adjusting it precisely for these parameters (see, e.g., [10,15]). For α ∈ ( 3 2 , 2) , we propose the following ansatzes: • the regular ansatz…”

Section: Asymptotic Approximation In the Case α >mentioning

confidence: 99%

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

Mel'nyk¹,

Rohde²

2023

Preprint

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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 can render the associated initial-boundary problem to be nonlinear. The strength of the inhomogeneity is controlled by an intensity factor of order O(ε α ) , α > 0 .The asymptotic behaviour of the solution is studied as ε → 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 α : α = 1, α > 1, and α ∈ (0, 1). We construct the asymptotic approximation of the solution, which provides us with the hyperbolic limit model for ε → 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.

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“…But what happens if α < 1 or β < 1 or δ < 0, is still unknown. For other boundary-value problems, such studies were carried out in [15,23].…”

mentioning

confidence: 99%

Asymptotic expansion for the solution of a convection-diffusion problem in a thin graph-like junction

Mel'nyk

Klevtsovskiy

2022

Asymptotic Analysis

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A steady-state convection-diffusion problem with a small diffusion of order O ( ε ) is considered in a thin three-dimensional graph-like junction consisting of thin cylinders connected through a domain (node) of diameter O ( ε ), where ε is a small parameter. Using multiscale analysis, the asymptotic expansion for the solution is constructed and justified. The asymptotic estimates in the norm of Sobolev space H 1 as well as in the uniform norm are proved for the difference between the solution and proposed approximations with a predetermined accuracy with respect to the degree of ε.

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Influence of the node on the asymptotic behaviour of the solution to a semilinear parabolic problem in a thin graph-like junction

Cited by 6 publications

References 21 publications

Asymptotic approximations for eigenvalues and eigenfunctions of a spectral problem in a thin graph-like junction with a concentrated mass in the node

Asymptotic approximations for eigenvalues and eigenfunctions of a spectral problem in a thin graph-like junction with a concentrated mass in the node

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

Asymptotic expansion for the solution of a convection-diffusion problem in a thin graph-like junction

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