Asymptotic Approximation for the Solution to a Semi-linear Parabolic Problem in a Thick Fractal Junction

TL;DR

This paper derives an asymptotic approximation for solutions to a semi-linear parabolic problem in a complex fractal junction, revealing how parameters influence the solution as the structure becomes infinitely fine.

Contribution

It introduces a homogenized model for a semi-linear parabolic problem in a fractal junction and constructs an asymptotic approximation with error estimates.

Findings

Homogenized problem derived for fractal junctions.

Existence and uniqueness proved in multi-sheeted Sobolev spaces.

Asymptotic estimates show parameter influence on solutions.

Abstract

We consider a semi-linear parabolic problem in a model plane thick fractal junction $\Omega_{\varepsilon}$, which is the union of a domain $\Omega_{0}$ and a lot of joined thin trees situated $\varepsilon$-periodically along some interval on the boundary of $\Omega_{0}.$ The trees have finite number of branching levels. The following nonlinear Robin boundary condition $\partial_{\nu}v_{\varepsilon} + \varepsilon^{\alpha_i} \kappa_i(v_{\varepsilon}) = \varepsilon^{\beta_i} g^{(i)}_{\varepsilon}$ is given on the boundaries of the branches from the $i$-th branching layer; $\alpha_i$ and $\beta_i$ are real parameters. The asymptotic analysis of this problem is made as $\varepsilon\to0,$ i.e., when the number of the thin trees infinitely increases and their thickness vanishes. In particular, the corresponding homogenized problem is found and the existence and uniqueness of its solution in an anizotropic Sobolev space of multi-sheeted functions is proved. We construct the asymptotic approximation for the solution $v_\varepsilon$ and prove the corresponding asymptotic estimate in the space $C\big([0,T]; L^2(\Omega_\varepsilon) \big) \cap L^2\big(0, T; H^1(\Omega_\varepsilon)\big)$, which shows the influence of the parameters $\{\alpha_i\}$ and $\{\beta_i\}$ on the asymptotic behavior of the solution.