Asymptotic approximation for the solution to a semi-linear parabolic problem in a thick junction with the branched structure

We consider a linear parabolic problem in a thick junction domain which is the union of a fixed domain and a collection of periodic branched trees of height of order 1 and small width connected on a part of the boundary. We consider a threebranched structure, but the analysis can be extended to n-branched structures. We use unfolding operator to study the asymptotic behavior of the solution of the problem. In the limit problem, we get a multi-sheeted function in which each sheet is the limit of restriction of the solution to various branches of the domain. Homogenization of an optimal control problem posed on the above setting is also investigated. One of the novelty of the paper is the characterization of the optimal control via the appropriately defined unfolding operators. Finally, we obtain the limit of the optimal control problem. IntroductionIn this article, we consider a parabolic problem in a thick junction domain , > 0, a small parameter, and also the corresponding optimal control problem. Various materials with complex structures including multi-layer thick junctions are widely used in many fields of science. Such structures are usually known as complex structures because of its complexity both in construction and analysis. Other complex structures are perforated domains, composite materials, grid domains, and domains with oscillating boundaries to name a few.As mentioned earlier, constructions with thick junction (also multi-level) are used in many technologies, like microstrip radiator, nano technologies ([1, 2]), biological systems, fractal-type constructions, etc. Studying PDE problems in such complex structures has paramount importance. We refer to the work in [3][4][5][6] and the references therein for the study in multi-branched structures. Although the importance of optimal control may be at the junctions, we consider the controls on the entire oscillating part from which we can also understand the contribution from each branch at each level. One can apply need based controls at the appropriate junctions.The domain under consideration consists of multiple layer thick junctions known as branched structure (Figure 1). Such a domain has a fixed part and lot of thin periodically distributed parts (or handle trees) attached along certain part of the boundary of the domain at different levels. The trees have finite number of branching levels and in this paper, we take three branching levels, but one can consider any finite number of branches. The height of each branch is of O.1/, whereas the thickness is of O. /. We consider the domain in two-dimensional space. Such a domain has already been considered by Mel'nyk ([6]). Indeed the results can be extended to three dimensional problem and higher dimensions as well. Asymptotic analysis for a Robin problem in a thick junction has been investigated in [5]. In fact, our work is motivated from the work of Mel'nyk, where he considers a semi-linear parabolic problem with the source term vanishes on the oscillating interior part. He has studied the problem usin...

Section: Introductionmentioning

confidence: 92%

“…To give an appropriate meaning to the weak solution of homogenized problem, let us introduce the function space

scriptH

. We use the ideas introduced in . We say, a multi‐sheeted function of the form

ϕ : = ({, \begin{matrix} array \\ array {ϕ_{2, 1}, ϕ_{2, 2}, ϕ_{2, 3}, ϕ_{2, 4}} i f x \in {normalΩ}_{2}, \\ array {ϕ_{1, 1}, ϕ_{1, 2}} i f x \in {normalΩ}_{1}; ϕ_{0, 1} i f x \in {normalΩ}_{0}, ϕ^{-} i f x \in {normalΩ}^{-} \end{matrix})

belongs to

scriptH

, if ϕ − ∈ H 1 (Ω − ), for each i = 0,1,2, the functions ϕ i , m ∈ L 2 (0, L ; H 1 ( M i , M i + 1 ) for m = 1,⋯,2 i and on the boundaries (in the sense of trace), they satisfy

\begin{matrix} right ϕ^{-} |_{{normalΓ}_{0}} = ϕ_{0, 1} |_{{normalΓ}_{0}}, ϕ_{0, 1} |_{{normalΓ}_{1}} = ϕ_{1, 1} |_{{normalΓ}_{1}} = ϕ_{1, 2} |_{{normalΓ}_{1}}, \\ right ϕ_{1, 1} |_{{normalΓ}_{2}} \end{matrix}

…”

Section: Convergence Analysismentioning

confidence: 99%

“…The m u l t i ‐ s h e e t e d function u can be represented as

u (t, x) = ({, \begin{matrix} array \\ array u_{i, m} & array i f (t, x) \in Q_{i}, f o r i = 0, 1, 2; m = 1, \dots, 2^{i} \\ array u^{-} & array i f (t, x) \in Q^{-} . \end{matrix})

Such problems are considered in Mel'nyk in the context of problems in branched structures. The author has studied homogenization problems, based on asymptotic expansion.…”

Section: Convergence Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Optimal control problem in a domain with branched structure and homogenization

Aiyappan

Nandakumaran

2016

Asymptotic analysis of a boundary optimal control problem on a general branched structure

Aiyappan

Nandakumaran

Sufian

2019

We consider an optimal control problem posed on a domain with a highly oscillating smooth boundary where the controls are applied on the oscillating part of the boundary. There are many results on domains with oscillating boundaries where the oscillations are pillar-type (non-smooth) while the literature on smooth oscillating boundary is very few. In this article, we use appropriate scaling on the controls acting on the oscillating boundary leading to different limit control problems, namely, boundary optimal control and interior optimal control problem. In the last part of the article, we visualize the domains as a branched structure, and we introduce unfolding operators to get contributions from each level at every branch. KEYWORDS asymptotic analysis, homogenization, optimal control, oscillating boundary domain, unfolding operator MSC CLASSIFICATION 80M35; 80M40; 49J20; 35B27 Math Meth Appl Sci. 2019;42:6407-6434.wileyonlinelibrary.com/journal/mma

Asymptotic analysis of spectral problems in thick junctions with the branched fractal structure

Mel'nyk

2022

Self Cite

Funding informationThe Grant of the Ministry of Education and Science of Ukraine A spectral problem is considered in a domain Ω 𝜀 that is the union of a domain Ω 0 and a lot of thin trees situated 𝜀-periodically along some manifold on the boundary of Ω 0 . The trees have finite number of branching levels. The perturbed Robin boundary condition 𝜕 𝜈 u 𝜀 + 𝜀 𝛼 i k i,m u 𝜀 = 0 is given on the ith branching layer; {𝛼 i } are real parameters. The asymptotic analysis of this problem is made as 𝜀 → 0, that is, when the number of the thin trees infinitely increases and their thickness vanishes. In particular, the Hausdorff convergence of the spectrum to the spectrum of the corresponding nonstandard homogenized spectral problem is proved, the leading terms of asymptotics are constructed, and the corresponding asymptotic estimates are justified for the eigenvalues and eigenfunctions.