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

Abstract

A thin graph-like junction $Ω_{ε} \subset R^{3}$ consists of several thin curvilinear cylinders that are joined through a domain (node) of diameter $O (ε)$ . Here ε is a small parameter characterizing the thickness of the thin cylinders and the node. In $Ω_{ε}$ we consider a semilinear parabolic problem with nonlinear perturbed Robin boundary conditions both on the lateral surfaces of the cylinders and the node boundary.

The purpose is to study the asymptotic behavior of the solution $u_{ε}$ as $ε \to 0$ , i.e. when the thin graph-like junction is shrunk into a graph. The passage to the limit is accompanied by special intensity factor $ε^{α_{0}}$ in the Robin condition on the node boundary. We establish qualitatively different cases in the asymptotic behaviour of the solution depending on the value of parameter $α_{0}$ . For each case we construct the asymptotic approximation for the solution up to the second terms of the asymptotics and prove the asymptotic estimates from which the influence of the local geometric heterogeneity of the node and physical processes inside are observed.

Keywords

Approximation semilinear parabolic problem asymptotic estimate thin graph-like junction

1. Introduction

There are many physical and mathematical articles and books that deal with the study of boundary-value problems on graphs (see e.g. [1,2,10,11,13,18,21,22]). One of the main question discussing in those papers is the choosing of coupling conditions at vertices of a graph. From our papers [7,8,16] it follows that the natural approach to correctly deduce the coupling conditions is the passage to the limit in the corresponding boundary-value problem in a three-dimensional thin graph-like structure when it is shrunk into the graph.

In the present article we continue our research started in [8], where only the first term of the asymptotic was justified for the solution to a semilinear parabolic problem with nonlinear perturbed Robin boundary conditions in a thin graph-like junction $Ω_{ε} \subset R^{3}$ that consists of several thin curvilinear cylinders that are joined through a domain (node) of the diameter $O (ε)$ . Here ε is a small parameter characterizing the thickness of the thin cylinders and the node diameter. If ε tends to zero, then $Ω_{ε}$ is shrunk into a graph (Fig. 1).

Fig. 1.

Transformation of a thin graph-like junction into a graph.

The novelty of this paper is to study the influence of the local geometric heterogeneity of the node and physical processes inside on the asymptotic behaviour of the solution. For this we have performed more detailed asymptotic analysis, namely, we have constructed further terms of the asymptotics and proved the corresponding asymptotic estimates. As a result, we discovered that the second term of the regular asymptotics feels always the node impact and the main our conclusion is as follows. If the local geometric irregularity of the node and the boundary condition on the node boundary don’t affect the view of the limit problem, then the problem for the second term of the regular asymptotics should be considered in parallel with the limit problem to take the node effect into account.

We refer the reader to [8] for the review of different asymptotic approaches to study boundary-value problems in thin graph-like domains. Here we mention some new papers appeared in last time. A big result review for spectral problems is presented in the book [23]. Convergence of the spectrum and the eigenfunctions for the Laplacian in fractals was proved in [24]. Interesting multifarious transmission conditions are obtained in the limit passage for spectral problems on thin periodic honeycomb lattices in [12,20]. Numerical approach to deduce the vertex coupling conditions for the nonlinear Schrödinger equation on two-dimensional thin networks was proposed in [26]. Asymptotic analysis of the uniform Dirichlet problem for the biharmonic equation and the steady incompressible flow of a Bingham fluid in thin T-like shaped plane domains was made in [3,4], respectively.

The paper is organized as follows. After the statement of the problem in Section 2, we briefly recall the main results obtained in [8] and show the need for a more detailed study of this problem. Then we study the asymptotic behaviour of the solution for the cases $α_{0} \in (0, 1)$ , $α_{0} = 0$ , and $α_{0} = 1$ in Sections 3, 4 and 5, respectively. In the Conclusion we analyze obtained results and discuss the other cases.

The results of this paper were reported on the workshop [17].

2. Statement of the problem and its well-posedness

The model thin star-shaped junction $Ω_{ε}$ consists of three thin curvilinear cylinders $\begin{matrix} Ω_{ε}^{(i)} = {x = (x_{1}, x_{2}, x_{3}) \in R^{3} : ε ℓ_{i}^{0} < x_{i} < ℓ_{i}, \sum_{j = 1}^{3} (1 - δ_{i j}) x_{j}^{2} < ε^{2} h_{i}^{2} (x_{i})}, i = 1, 2, 3, \end{matrix}$ that are joined through a domain $Ω_{ε}^{(0)}$ (referred in the sequel “node”). Here ε is a small parameter; $ℓ_{i}^{0} \in (0, \frac{1}{3}), ℓ_{i} ⩾ 1, i = 1, 2, 3$ ; the positive function $h_{i}$ belongs to the space $C^{1} ([0, ℓ_{i}])$ and it is equal to some constants in neighborhoods of the points $x = 0$ and $x_{i} = 1$ $(i = 1, 2, 3)$ ; the symbol $δ_{i j}$ is the Kroneker delta, i.e., $δ_{i i} = 1$ and $δ_{i j} = 0$ if $i \neq j$ .

Fig. 2.

The node $Ω_{ε}^{(0)}$ .

The node $Ω_{ε}^{(0)}$ (see Fig. 2) is formed by the homothetic transformation with coefficient ε from a bounded domain $Ξ^{(0)} \subset R^{3}$ , i.e., $Ω_{ε}^{(0)} = ε Ξ^{(0)}$ . In addition, we assume that its boundary contains the disks $Υ_{ε}^{(i)} (ε ℓ_{i}^{0})$ , $i = 1, 2, 3$ , where $\begin{matrix} Υ_{ε}^{(i)} (s) = {x \in R^{3} : x_{i} = s, \sum_{j = 1}^{3} (1 - δ_{i j}) x_{j}^{2} < ε^{2} h_{i}^{2} (s)}, \end{matrix}$ and denote $Γ_{ε}^{(0)} : = \partial Ω_{ε}^{(0)} ∖ {\overline{Υ_{ε}^{(1)} (ε ℓ_{i}^{0})} \cup \overline{Υ_{ε}^{(2)} (ε ℓ_{i}^{0})} \cup \overline{Υ_{ε}^{(3)} (ε ℓ_{i}^{0})}}$ .

Thus the model thin star-shaped junction $Ω_{ε}$ (see Fig. 3) is the interior of the union $⋃_{i = 0}^{3} \overline{Ω_{ε}^{(i)}}$ and we assume that it has the Lipschitz boundary.

Fig. 3.

The model thin graph-like junction $Ω_{ε}$ .

Remark 2.1.

We can consider thin graph-like junctions with arbitrary finite number of the thin cylinders and with arbitrary their orientation. It was made for a linear parabolic problem in [6]. But to avoid technical and huge calculations we consider this thin junction $Ω_{ε}$ in which the cylinders are placed on the coordinate axes.

In $Ω_{ε}$ , we consider the following semilinear parabolic problem: $\begin{matrix} (2.1) & \{\begin{matrix} \partial_{t} u_{ε} (x, t) - Δ_{x} u_{ε} (x, t) + k (u_{ε} (x, t)) = f (x, t), & (x, t) \in Ω_{ε} \times (0, T), \\ \partial_{ν} u_{ε} (x, t) + ε^{α_{0}} κ_{0} (u_{ε} (x, t)) = ε^{α_{0}} φ_{ε}^{(0)} (x, t), & (x, t) \in Γ_{ε}^{(0)} \times (0, T), \\ \partial_{ν} u_{ε} (x, t) + ε κ_{i} (u_{ε} (x, t)) = ε φ_{ε}^{(i)} (x, t), & (x, t) \in Γ_{ε}^{(i)} \times (0, T), i = 1, 2, 3, \\ u_{ε} (x, t) = 0, & (x, t) \in Υ_{ε}^{(i)} (ℓ_{i}) \times (0, T), i = 1, 2, 3, \\ u_{ε} (x, 0) = 0, & x \in Ω_{ε}, \end{matrix} \end{matrix}$ where $Γ_{ε}^{(i)} = \partial Ω_{ε}^{(i)} \cap {x \in R^{3} : ε ℓ_{i}^{0} < x_{i} < ℓ_{i}}, T > 0, \partial_{t} = \partial / \partial t, \partial_{ν}$ is the outward normal derivative, and the parameter $α_{0} \in [0, 1]$ . For the given functions f, k, ${φ_{ε}^{(i)}, κ_{i}}_{i = 0}^{3}$ we assume the following conditions:

the function f belongs to the space $C_{x}^{1} (\overline{Ω_{a_{0}}} \times [0, T])$ and its restriction on the curvilinear cylinder $Ω_{a_{0}}^{(i)}$ $(i = 1, 2, 3)$ belongs to the space $C_{{\overline{x}}_{i}}^{2} (\overline{Ω_{a_{0}}^{(i)}} \times [0, T])$ (the space of all continuous functions having continuous first and second derivatives with respect to variables $\begin{matrix} {\overline{x}}_{i} = \{\begin{matrix} (x_{2}, x_{3}), & i = 1, \\ (x_{1}, x_{3}), & i = 2, \\ (x_{1}, x_{2}), & i = 3, \end{matrix} \end{matrix}$ in $\overline{Ω_{a_{0}}^{(i)}} \times [0, T])$ , where $a_{0}$ is a fixed positive number such that $Ω_{ε} \subset Ω_{a_{0}}$ for all values of the small parameter $ε \in (0, ε_{0})$ ; in addition $\begin{matrix} (2.2) & f |_{t = 0} = 0; \end{matrix}$

the functions $φ^{(0)} (ξ, t), (ξ, t) \in \overline{Ξ^{(0)}} \times [0, T]$ , and $φ^{(i)} ({\overline{ξ}}_{i}, x_{i}, t), ({\overline{ξ}}_{i}, x_{i}, t) \in {| {\overline{ξ}}_{i} | ⩽ h_{i} (x_{i}), x_{i} \in [0, ℓ_{i}], t \in [0, T]}$ , $i \in {1, 2, 3}$ , are continuous and we will use the following notation: $\begin{matrix} φ_{ε}^{(0)} (x, t) : = φ^{(0)} (\frac{x}{ε}, t), φ_{ε}^{(i)} (x, t) : = φ^{(i)} (\frac{{\overline{x}}_{i}}{ε}, x_{i}, t); \end{matrix}$ in addition $\begin{matrix} (2.3) & φ^{(0)} |_{t = 0} = κ_{0} (0), φ^{(i)} |_{t = 0} = 0, i \in {1, 2, 3}; \end{matrix}$

the function $κ_{0}$ belongs to the space $C^{2} (R)$ with second derivative $κ_{0}^{″}$ from $L^{\infty} (R)$ , the functions k, ${κ_{i}}_{i = 1}^{3}$ , belong to the space $C^{3} (R)$ with third derivatives $k^{‴}, {κ_{i}^{‴}}_{i = 1}^{3}$ from $L^{\infty} (R)$ , and there exists a positive constant $κ_{+}$ such that $\begin{matrix} (2.4) & 0 ⩽ k^{'} (s) ⩽ κ_{+}, 0 ⩽ κ_{i}^{'} (s) ⩽ κ_{+} for s \in R, i = 0, 1, 2, 3; \end{matrix}$ in addition, we impose the zero-absorption condition for k and ${κ_{i}}_{i = 1}^{3}$ , i.e., $\begin{matrix} (2.5) & k (0) = 0, κ_{i} (0) = 0 . \end{matrix}$

Remark 2.2.

The function k and ${κ_{i}}_{i = 0}^{3}$ may additionally depend on x and t (see [8]). However, we have omitted this dependence to avoid cumbersome formulas.

From conditions (2.4) it follows the following inequalities: $\begin{array}{l} (2.6) & \begin{aligned} 0 ⩽ k (s) s ⩽ κ_{+} s^{2}, 0 ⩽ κ_{i} (s) s ⩽ κ_{+} s^{2} for s \in R, i = 1, 2, 3, \\ κ_{0} (0) s ⩽ κ_{0} (s) s ⩽ κ_{+} s^{2} + κ_{0} (0) s for s \in R \end{aligned} \end{array}$ that will be essential used in our analysis. Also we will often use the inequalities $\begin{array}{l} (2.7) & ε \int_{Γ_{ε}^{(i)}} v^{2} d σ_{x} ⩽ C_{2} (ε^{2} \int_{Ω_{ε}^{(i)}} | \nabla_{x} v |^{2} d x + \int_{Ω_{ε}^{(i)}} v^{2} d x), \\ (2.8) & \int_{Ω_{ε}^{(i)}} v^{2} d x ⩽ C_{3} (ε^{2} \int_{Ω_{ε}^{(i)}} | \nabla_{x} v |^{2} d x + ε \int_{Γ_{ε}^{(i)}} v^{2} d σ_{x}), \\ (2.9) & \int_{Γ_{ε}^{(0)}} v^{2} d σ_{x} ⩽ C_{5} (\int_{Ω_{ε}} | \nabla_{x} v |^{2} d x + \int_{Υ_{ε}^{(i)} (ℓ_{i})} v^{2} d {\overline{x}}_{i}) \end{array}$ for all $v \in H^{1} (Ω_{ε})$ and $i \in {1, 2, 3}$ . They were proved in [8,15].

2.1. Well-posedness of the problem (2.1)

Denote by ${⟨ \cdot, \cdot ⟩}_{ε}$ the duality pairing of the dual space $H_{ε}^{*}$ and the Sobolev space $H_{ε} = {u \in H^{1} (Ω_{ε}) : u |_{Υ_{ε}^{(i)} (ℓ_{i})} = 0, i = 1, 2, 3}$ with the scalar product $\begin{matrix} {(u, v)}_{ε} = \int_{Ω_{ε}} \nabla u \cdot \nabla v d x, u, v \in H_{ε} . \end{matrix}$ Due to the uniform Dirichlet condition on $Υ_{ε}^{(i)} (ℓ_{i}), i = 1, 2, 3$ , the norm $‖ \cdot ‖_{ε}$ generated by this product and the standard norm $‖ \cdot ‖_{H^{1} (Ω_{ε})}$ are uniformly equivalent, i.e., there exist constants $C_{1} > 0$ and $ε_{0} > 0$ such that for all $ε \in (0, ε_{0})$ and for all $u \in H_{ε}$ the following estimate hold: $\begin{matrix} (2.10) & ‖ u ‖_{ε} ⩽ ‖ u ‖_{H^{1} (Ω_{ε})} ⩽ C_{1} ‖ u ‖_{ε} . \end{matrix}$

Remark 2.3.
Here and in what follows all constants ${C_{j}}$ and ${c_{j}}$ in inequalities are independent of the parameter ε.

Define a nonlinear operator $A_{ε} (t) : H_{ε} ⟼ H_{ε}^{}$ through the relation $\begin{matrix} {⟨ A_{ε} (t) u, v ⟩}_{ε} = \int_{Ω_{ε}} \nabla u \cdot \nabla v d x + \int_{Ω_{ε}} k (u) v d x + ε^{α_{0}} \int_{Γ_{ε}^{(0)}} κ_{0} (u) v d σ_{x} + \sum_{i = 1}^{3} ε \int_{Γ_{ε}^{(i)}} κ_{i} (u) v d σ_{x} \end{matrix}$ for all $u, v \in H_{ε}$ and the linear functional $F_{ε} (t) \in H_{ε}^{}$ by the formula $\begin{matrix} {⟨ F_{ε} (t), v ⟩}_{ε} = \int_{Ω_{ε}} f v d x + ε^{α_{0}} \int_{Γ_{ε}^{(i)}} φ_{ε}^{(0)} v d σ_{x} + \sum_{i = 1}^{3} ε \int_{Γ_{ε}^{(i)}} φ_{ε}^{(i)} v d σ_{x} \end{matrix}$ for all $v \in H_{ε}$ and a.e. $t \in (0, T)$ .

Recall that a function $u_{ε} \in L^{2} (0, T; H_{ε})$ , with $\partial_{t} u_{ε} \in L^{2} (0, T; H_{ε}^{*})$ , is called a weak solution to the problem (2.1) if it satisfies the identity $\begin{matrix} (2.11) & {⟨ \partial_{t} u_{ε}, v ⟩}_{ε} + {⟨ A_{ε} (t) u_{ε}, v ⟩}_{ε} = {⟨ F_{ε} (t), v ⟩}_{ε} \forall v \in H_{ε}, \end{matrix}$ for a.e. $t \in (0, T)$ , and $u_{ε} |_{t = 0} = 0$ . It is known (see e.g. [25]) that $u_{ε} \in C ([0, T]; L^{2} (Ω_{ε}))$ , and thus the equality $u_{ε} |_{t = 0} = 0$ makes sense.

Let us verify some properties of the operator $A_{ε}$ for a fixed value of ε and for a.e. $t \in (0, T)$ .

With the help of (2.6) and Cauchy’s inequality with $δ > 0 (a b ⩽ δ a^{2} + \frac{b^{2}}{4 δ})$ , we obtain $\begin{matrix} {⟨ A_{ε} (t) v, v ⟩}_{ε} ⩾ \int_{Ω_{ε}} | \nabla v |^{2} d x + ε^{α_{0}} \int_{Γ_{ε}^{(0)}} κ_{0} (0) v d σ_{x} ⩾ ‖ v ‖_{ε}^{2} - δ \int_{Γ_{ε}^{(0)}} v^{2} d σ_{x} - \frac{1}{4 δ} ε^{2 α_{0}} {| κ_{0} (0) |}^{2} {| Γ_{ε}^{(0)} |}_{2} . \end{matrix}$ Here and in what follows $| S |_{n}$ is the n-dimensional Lebesgue measure of a set S. Then using (2.10), (2.9), the assumption C3 and selecting appropriate δ, we find that $\begin{matrix} {⟨ A_{ε} (t) v, v ⟩}_{ε} ⩾ C_{6} ‖ v ‖_{ε}^{2} - C_{7} ε^{2} (1 + ε^{2 α_{0}}) \forall v \in H_{ε} . \end{matrix}$ This inequality means that the operator $A_{ε}$ is coercive.

Taking into account (2.4), we get the strict monotonicity of $A_{ε}$ : $\begin{matrix} {⟨ A_{ε} (t) u_{1} - A_{ε} (t) u_{2}, u_{1} - u_{2} ⟩}_{ε} ⩾ ‖ u_{1} - u_{2} ‖_{ε}^{2} \forall u_{1}, u_{2} \in H_{ε} . \end{matrix}$

The operator $A_{ε}$ is hemicontinuous. Indeed, the real valued function $\begin{matrix} [0, 1] ∋ τ \to {⟨ A_{ε} [u_{1} + τ v], u_{2} ⟩}_{ε} \end{matrix}$ is continuous on $[0, 1]$ for all fixed $u_{1}, u_{2}, v \in H_{ε}$ due to the continuity of the functions $k, {κ_{i}}_{i = 0}^{3}$ and Lebesque’s dominated convergence theorem.

Using (2.6), (2.7), (2.9) and (2.10), we deduce the boundedness of $A_{ε}$ : $\begin{matrix} {⟨ A_{ε u}, v ⟩}_{ε} ⩽ C_{8} (1 + ε^{α_{0}}) (ε + ‖ u ‖_{ε}) ‖ v ‖_{ε} \forall u, v \in H_{ε} . \end{matrix}$

Thus, the existence and uniqueness of the weak solution for every fixed value ε follow directly from Corollary 4.1 (see [25, Chapter 3]).

The aim of the present paper is to
construct the approximation for the solution to the problem (2.1) and prove the corresponding asymptotic estimates (as $ε \to 0$ ) for first terms of the asymptotics from which the influence of the local geometric heterogeneity of the node $Ω_{ε}^{(0)}$ and physical processes inside will be always observed;

study the influence of the parameter $α_{0}$ on the asymptotic behavior of the solution.

2.2. Comments to the statement

In [8] the problem (2.1) was considered with the following nonlinear Robin boundary conditions: $\begin{matrix} (2.12) & \partial_{ν} u_{ε} + ε^{α_{i}} κ_{i} (u_{ε}, x_{i}, t) = ε^{β_{i}} φ_{ε}^{(i)} (x, t) \end{matrix}$ both on the boundaries of the thin curvilinear cylinders $(i \in {1, 2, 3})$ and on the boundary of the node $(i = 0)$ , where the parameters ${α_{i}}_{i = 0}^{3} \subset R$ , $β_{0} ⩾ 0, β_{i} ⩾ 1, i \in {1, 2, 3}$ . It turned out that the asymptotic behaviour of the solution depends on such values of the parameters ${α_{i}}$ and ${β_{i}}$ : $\begin{array}{l} α_{0} < 0, α_{0} = 0, α_{0} > 0; β_{0} = 0, β_{0} > 0; \\ α_{i} < 1, α_{i} = 1, α_{i} > 1; β_{i} = 1, β_{i} > 1, i \in {1, 2, 3}, \end{array}$ and essentially on the parameter $α_{0}$ characterizing the intensity of processes at the boundary of the node.

If $α_{0} < 0$ , then the limit problem splits in three independent problems with the uniform Dirichlet conditions at the vertex $x = 0$ .

If $α_{0} ⩾ 0, β_{0} ⩾ 0, α_{i}, β_{i} ⩾ 1, i \in {1, 2, 3}$ , then the Kirchhoff transmission conditions for the first term ${\tilde{w}}_{0} : = {w_{0}^{(i)}}_{i = 1}^{3}$ of the asymptotics look as follows $\begin{array}{l} (2.13) & \begin{aligned} w_{0}^{(1)} (0, t) = w_{0}^{(2)} (0, t) = w_{0}^{(3)} (0, t), \\ \sum_{i = 1}^{3} π h_{i}^{2} (0) \frac{\partial w_{0}^{(i)}}{\partial x_{i}} (0, t) - δ_{α_{0}, 0} | Γ_{0} |_{2} κ_{0} (w_{0}^{(1)} (0, t)) = - δ_{β_{0}, 0} \int_{Γ_{0}} φ^{(0)} (ξ, t) d σ_{ξ} . \end{aligned} \end{array}$ In [8] we have also constructed the approximations for different values of the parameters ${α_{i}}$ and ${β_{i}}$ , but the asymptotic estimates were proved only for the first term of the asymptotics, namely $\begin{matrix} (2.14) & \frac{1}{\sqrt{| Ω_{ε} |_{3}}} (max_{t \in [0, T]} {‖ u_{ε} (\cdot, t) - {\tilde{w}}_{0} (\cdot, t) ‖}_{L^{2} (Ω_{ε})} + {‖ u_{ε} (\cdot, t) - {\tilde{w}}_{0} (\cdot, t) ‖}_{L^{2} (0, T; H^{1} (Ω_{ε}))}) ⩽ C_{0} ε^{ϱ}, \end{matrix}$ where ${\tilde{w}}_{0}$ is the solution to the corresponding limit problem, $ϱ \in (0, \frac{1}{2})$ and depends on ${α_{i}}$ and ${β_{i}}$ .

In the case $α_{0} = 0, β_{0} = 0$ the Kroneker deltas in (2.13) are equal to 1. As a result, we have the new type of the gluing conditions at the vertex $x = 0$ of the graph and observe the impact of the Robin boundary condition in the first term of the asymptotics. We cannot see this influence in the limit problem if $α_{0} > 0$ and $β_{0} > 0$ .

At the present time special interest of researchers is focused on various effects observed in vicinities of local irregularities of the geometry (widening or narrowing) of channels, tubes, vessels (e.g., adhesion to the walls, welds, stenosis and aneurysms). Also this is very important in engineering, since physical processes in a neighborhood of the node often directly affect the strength (stability, resistance, power, etc.) of constructions and devices (for more detail see our reviews in [7,16]).

Therefore, in the present paper we more scrupulously study the impact of the node for more realistic cases from physical point of view, namely, $α_{0} \in (0, 1)$ , $α_{0} = 1$ , and $α_{0} = 0$ . For this we propose a new asymptotic scale $\begin{matrix} ε^{0}, ε^{α_{0}}, ε^{1}, ε^{2 α_{0}}, ε^{1 + α_{0}}, ε^{2}, ε^{2 + α_{0}}, ε^{3}, ε^{2 + 2 α_{0}}, \dots, \end{matrix}$ that is quite different from the asymptotic scale in [8].

At first glance it may seem that there is no difference between the nonlinear Robin condition (2.12) and the corresponding linear Neumann condition, since the term $κ_{i} (u_{ε})$ is multiplied by $ε^{α_{i}}$ . However, this is true only if $α_{i} > 1$ . If $α_{i} = 1$ , then as was showed in [8] the new term $\begin{matrix} 2 π h_{i} (x_{i}) κ_{i} (w_{0}^{(i)} (x_{i}, t)) \end{matrix}$ appears in the differential equation of the corresponding limit problem, which takes into account the curvilinearity of the thin cylinder $Ω_{ε}^{(i)}$ through the function $h_{i}$ . Therefore, we take $α_{i} = β_{i} = 1, i \in {1, 2, 3}$ , in the present paper as more interesting case. Also we regard that $α_{0} = β_{0}$ . The case $α_{0} \neq β_{0}$ is discussed in the Conclusion.

Our approach is based on the multi-scale method and method of matching asymptotic expansions and it does not need any additional assumptions that were assumed by other researchers (see the review in [8]). It can take into account various factors (e.g. variable thickness of thin curvilinear cylinders, inhomogeneous boundary conditions, geometric characteristics of nodes, etc.) in formulations of boundary-value problems on graphs and with the help of this approach it is possible to deduce and justify the Kirchhoff transmission conditions (untypical in some cases) at the graph vertex.

3. Asymptotic approximation in the case $α_{0} \in (0, 1)$

Only for the formal construction procedure of the asymptotics we assume that the functions $f, k$ , ${φ_{ε}^{(i)}, κ_{i}}_{i = 0}^{3}$ are smooth enough. We propose ansatzes of the asymptotic approximation for the solution to the problem (2.1) in the following form:

the regular part of the approximation $\begin{matrix} (3.1) & \sum_{n \in A} (ε^{n} w_{n}^{(i)} (x_{i}, t) + ε^{n + 2} u_{n + 2}^{(i)} (\frac{{\overline{x}}_{i}}{ε}, x_{i}, t)) \end{matrix}$ is located inside of each thin cylinder $Ω_{ε}^{(i)} (i = 1, 2, 3)$ and their terms depend both on the corresponding longitudinal variable $x_{i}$ and so-called “fast variables” $\frac{{\overline{x}}_{i}}{ε}$ , where the index set $A = {0, α_{0}, 1, 2 α_{0}, 1 + α_{0}, 2, 1 + 2 α_{0}}$ ;

the boundary-layer part of the approximation $\begin{matrix} (3.2) & ε^{2} Π_{2}^{(i)} (\frac{{\overline{x}}_{i}}{ε}, \frac{ℓ_{i} - x_{i}}{ε}, t) \end{matrix}$ is located in a neighborhood of the base $Υ_{ε}^{(i)} (ℓ_{i})$ of the thin cylinder $Ω_{ε}^{(i)} (i = 1, 2, 3)$ ;

and the inner part of the approximation $\begin{matrix} (3.3) & \sum_{n \in I} ε^{n} N_{n} (\frac{x}{ε}, t) \end{matrix}$ is located in a neighborhood of the node $Ω_{ε}^{(0)}$ , where the index set $I = {0, α_{0}, 1, 2 α_{0}, 1 + α_{0}, 2, 1 + 2 α_{0}, 2 + α_{0}, 3, 2 + 2 α_{0}}$ .

3.1. Regular parts

Substituting the representation (3.1) for each fixed index $i \in {1, 2, 3}$ into the differential equation of the problem (2.1) instead of $u_{ε}$ , using Taylor’s formula for the function f at the point ${\overline{x}}_{i} = (0, 0)$ for the function k at $w_{0}^{(i)}$ , and collecting coefficients at $ε^{n}$ , we obtain differential equations for $u_{n + 2}^{(i)}$ with respect to the variables ${\overline{ξ}}_{i}$ .

Taking into account the view of the outer unit normal to $Γ_{ε}^{(i)}$ $\begin{array}{l} ν^{(i)} (x_{i}, {\overline{ξ}}_{i}) & = \frac{1}{\sqrt{1 + ε^{2} | h_{i}^{'} (x_{i}) |^{2}}} (- ε h_{i}^{'} (x_{i}), {\overline{ν}}_{i} ({\overline{ξ}}_{i})) \\ = \{\begin{matrix} \frac{(- ε h_{1}^{'} (x_{1}), ν_{2}^{(1)} ({\overline{ξ}}_{1}), ν_{3}^{(1)} ({\overline{ξ}}_{1}))}{\sqrt{1 + ε^{2} | h_{1}^{'} (x_{1}) |^{2}}}, & i = 1, \\ \frac{(ν_{1}^{(2)} ({\overline{ξ}}_{2}), - ε h_{2}^{'} (x_{2}), ν_{3}^{(2)} ({\overline{ξ}}_{2}))}{\sqrt{1 + ε^{2} | h_{2}^{'} (x_{2}) |^{2}}}, & i = 2, \\ \frac{(ν_{1}^{(3)} ({\overline{ξ}}_{3}), ν_{2}^{(3)} ({\overline{ξ}}_{3}), - ε h_{3}^{'} (x_{3}))}{\sqrt{1 + ε^{2} | h_{3}^{'} (x_{3}) |^{2}}}, & i = 3, \end{matrix} \end{array}$ where ${\overline{ξ}}_{i} = \frac{{\overline{x}}_{i}}{ε}$ and ${\overline{ν}}_{i} (\frac{{\overline{x}}_{i}}{ε})$ is the outward normal to the disk $Υ_{ε}^{(i)} (x_{i}) : = {{\overline{ξ}}_{i} \in R^{2} : | {\overline{ξ}}_{i} | < h_{i} (x_{i})}$ , then putting (3.1) into the third relation of the problem (2.1), using Taylor’s formula for $κ_{i}$ , and collecting coefficients at $ε^{n + 1}$ , we get the boundary conditions for $u_{n + 2}^{(i)}$ on the boundary of the disk $Υ_{ε}^{(i)} (x_{i})$ .

Thus we obtain boundary-value problems to define ${u_{n + 2}^{(i)}}_{n \in A}$ . The problem for $u_{2}^{(i)}$ is as follows: $\begin{matrix} (3.4) & \{\begin{matrix} - Δ_{{\overline{ξ}}_{i}} u_{2}^{(i)} ({\overline{ξ}}_{i}, x_{i}, t) = - \frac{\partial w_{0}^{(i)}}{\partial t} + \frac{\partial^{2} w_{0}^{(i)}}{\partial {x_{i}}^{2}} - k (w_{0}^{(i)} (x_{i}, t)) + f_{0}^{(i)} (x_{i}, t), & {\overline{ξ}}_{i} \in Υ_{i} (x_{i}), \\ \partial_{ν_{{\overline{ξ}}_{i}}} u_{2}^{(i)} ({\overline{ξ}}_{i}, x_{i}, t) = h_{i}^{'} (x_{i}) \frac{\partial w_{0}^{(i)}}{\partial x_{i}} (x_{i}, t) - κ_{i} (w_{0}^{(i)} (x_{i}, t)) + φ^{(i)} ({\overline{ξ}}_{i}, x_{i}, t), & {\overline{ξ}}_{i} \in \partial Υ_{i} (x_{i}), \\ {⟨ u_{2}^{(i)} (\cdot, x_{i}, t) ⟩}_{Υ_{i} (x_{i})} = 0 . \end{matrix} \end{matrix}$ Here $f_{0}^{(i)} (x_{i}, t) : = f (x, t) |_{{\overline{x}}_{i} = (0, 0)}$ and ${⟨ u (\cdot, x_{i}, t) ⟩}_{Υ_{i} (x_{i})} : = \int_{Υ_{i} (x_{i})} u ({\overline{ξ}}_{i}, x_{i}, t) d {\overline{ξ}}_{i},$ the variables $(x_{i}, t)$ are regarded as parameters from $I_{ε}^{(i)} \times (0, T)$ , where $I_{ε}^{(i)} : = {x : x_{i} \in (ε ℓ_{i}^{0}, ℓ_{i}), \overline{x_{i}} = (0, 0)}$ . We add the third equality in (3.4) for the uniqueness of a solution.

Writing down the necessary and sufficient conditions for the solvability of the problem (3.4), we derive the differential equation $\begin{array}{l} π h_{i}^{2} (x_{i}) \frac{\partial w_{0}^{(i)}}{\partial t} (x_{i}, t) - π \frac{\partial}{\partial x_{i}} (h_{i}^{2} (x_{i}) \frac{\partial w_{0}^{(i)}}{\partial x_{i}} (x_{i}, t)) \\ + π h_{i}^{2} (x_{i}) k (w_{0}^{(i)} (x_{i}, t)) + 2 π h_{i} (x_{i}) κ_{i} (w_{0}^{(i)} (x_{i}, t)) \\ (3.5) & = π h_{i}^{2} (x_{i}) f_{0}^{(i)} (x_{i}, t) + \int_{\partial Υ_{i} (x_{i})} φ^{(i)} ({\overline{ξ}}_{i}, x_{i}, t) d l_{{\overline{ξ}}_{i}}, (x_{i}, t) \in I_{ε}^{(i)} \times (0, T), \end{array}$ to define $w_{0}^{(i)}$ $(i \in {1, 2, 3})$ . Let $w_{0}^{(i)}$ be a solution of the differential equation (3.5) (its existence will be proved in the Section 3.3.1). Then there exists a unique solution to the problem (3.4).

The coefficient $u_{n + 2}^{(i)}$ $(i \in {1, 2, 3}, n \in A ∖ {0})$ should be a solution to the problem $\begin{matrix} (3.6) & \{\begin{matrix} - Δ_{{\overline{ξ}}_{i}} u_{n + 2}^{(i)} \\ = (- \frac{\partial}{\partial t} + \frac{\partial^{2}}{\partial {x_{i}}^{2}}) (w_{n}^{(i)} + δ_{n, 2} u_{2}) - k^{'} (w_{0}^{(i)}) (w_{n}^{(i)} + δ_{n, 2} u_{2}) \\ - k^{″} (w_{0}^{(i)}) (δ_{n, 2 α_{0}} \frac{1}{2} {(w_{α_{0}}^{(i)})}^{2} + δ_{n, 1 + α_{0}} w_{α_{0}}^{(i)} w_{1}^{(i)} + δ_{n, 2} \frac{1}{2} {(w_{1}^{(i)})}^{2} + δ_{n, 1 + 2 α_{0}} w_{α_{0}}^{(i)} w_{1 + α_{0}}^{(i)} \\ + δ_{n, 1 + 2 α_{0}} w_{1}^{(i)} w_{2 α_{0}}^{(i)}) - k^{‴} (w_{0}^{(i)}) δ_{n, 1 + 2 α_{0}} {(w_{α_{0}}^{(i)})}^{2} w_{1}^{(i)} + δ_{n, 1} f_{1}^{(i)} + δ_{n, 2} f_{2}^{(i)}, \\ {\overline{ξ}}_{i} \in Υ_{i} (x_{i}), \\ \partial_{ν_{{\overline{ξ}}_{i}}} u_{n + 2}^{(i)} \\ = h_{i}^{'} \frac{\partial}{\partial x_{i}} (w_{n}^{(i)} + δ_{n, 2} u_{2}) - κ_{i}^{'} (w_{0}^{(i)}) (w_{n}^{(i)} + δ_{n, 2} u_{2}) - κ_{i}^{″} (w_{0}^{(i)}) (δ_{n, 2 α_{0}} \frac{1}{2} {(w_{α_{0}}^{(i)})}^{2} \\ + δ_{n, 1 + α_{0}} w_{α_{0}}^{(i)} w_{1}^{(i)} + δ_{n, 2} \frac{1}{2} {(w_{1}^{(i)})}^{2} + δ_{n, 1 + 2 α_{0}} w_{α_{0}}^{(i)} w_{1 + α_{0}}^{(i)} + δ_{n, 1 + 2 α_{0}} w_{1}^{(i)} w_{2 α_{0}}^{(i)}) \\ - κ_{i}^{‴} (w_{0}^{(i)}) δ_{n, 1 + 2 α_{0}} {(w_{α_{0}}^{(i)})}^{2} w_{1}^{(i)} - δ_{n, 2} \frac{1}{2} | h_{i}^{'} |^{2} κ_{i} (w_{0}^{(i)}) + δ_{n, 2} \frac{1}{2} | h_{i}^{'} |^{2} φ^{(i)}, \\ {\overline{ξ}}_{i} \in \partial Υ_{i} (x_{i}), \\ {⟨ u_{n + 2}^{(i)} (\cdot, x_{i}, t) ⟩}_{Υ_{i} (x_{i})} = 0, \end{matrix} \end{matrix}$ where $\begin{array}{l} (3.7) & \begin{aligned} f_{1}^{(i)} ({\overline{ξ}}_{i}, x_{i}, t) = \sum_{j = 1}^{3} (1 - δ_{i j}) ξ_{j} \frac{\partial}{\partial x_{j}} f (x, t) |_{{\overline{x}}_{i} = (0, 0)}, \\ f_{2}^{(i)} ({\overline{ξ}}_{i}, x_{i}, t) = {(\sum_{j = 1}^{3} (1 - δ_{i j}) ξ_{j} \frac{\partial}{\partial x_{j}})}^{2} f (x, t) |_{{\overline{x}}_{i} = (0, 0)} . \end{aligned} \end{array}$

Writing down the necessary and sufficient conditions for the solvability of the problem (3.6), we get a linear parabolic differential equation for $w_{n}^{(i)}$ (see (3.29) and (3.30)). Due to the uniform Dirichlet conditions on bases of the thin cylinders we should add the following boundary conditions $\begin{matrix} (3.8) & w_{n}^{(i)} (ℓ_{i}, t) = 0, i \in {1, 2, 3}, n \in A, \end{matrix}$ to those differential equations respectively.

3.2. Boundary-layer parts

The regular part of the asymptotics takes into account the inhomogeneity of the right-hand side of the differential equation, the boundary conditions on the lateral surfaces of the thin cylinders ${Ω_{ε}^{(i)}}_{i = 1}^{3}$ in (2.1), and partly the uniform Dirichlet conditions on the bases of the cylinders. To compensate residuals that the regular part leaves on those bases, we should construct the boundary-layer part of the asymptotics. Substituting the term (3.2) into (2.1), using (2.5) and collecting coefficients at $ε^{0}$ , we get the following mixed boundary-value problem: $\begin{matrix} (3.9) & \{\begin{matrix} - Δ_{ξ_{i}^{*}, {\overline{ξ}}_{i}} Π_{2}^{(i)} ({\overline{ξ}}_{i}, ξ_{i}^{*}, t) = 0, & ({\overline{ξ}}_{i}, ξ_{i}^{*}) \in Υ_{i} (ℓ_{i}) \times (0, + \infty), \\ \partial_{ν_{{\overline{ξ}}_{i}}} Π_{2}^{(i)} ({\overline{ξ}}_{i}, ξ_{i}^{*}, t) = 0, & ({\overline{ξ}}_{i}, ξ_{i}^{*}) \in \partial Υ_{i} (ℓ_{i}) \times (0, + \infty), \\ Π_{2}^{(i)} ({\overline{ξ}}_{i}, t, 0) = - u_{2}^{(i)} ({\overline{ξ}}_{i}, ℓ_{i}, t), & {\overline{ξ}}_{i} \in Υ_{i} (ℓ_{i}), \\ Π_{2}^{(i)} ({\overline{ξ}}_{i}, ξ_{i}^{*}, t) \to 0, & ξ_{i}^{*} \to + \infty, {\overline{ξ}}_{i} \in Υ_{i} (ℓ_{i}), \end{matrix} \end{matrix}$ where ${\overline{ξ}}_{i} = \frac{{\overline{x}}_{i}}{ε}, ξ_{i}^{*} = \frac{ℓ_{i} - x_{i}}{ε}$ . Using the method of separation of variables, we determine the solution $\begin{matrix} (3.10) & Π_{2}^{(i)} ({\overline{ξ}}_{i}, ξ_{i}^{*}, t) = \sum_{p = 1}^{+ \infty} b_{p}^{(i)} (t) Θ_{p}^{(i)} ({\overline{ξ}}_{i}) exp (- λ_{p}^{(i)} ξ_{i}^{*}) \end{matrix}$ of problem (3.9), where $\begin{matrix} b_{p}^{(i)} (t) = - \frac{1}{‖ Θ_{p}^{(i)} ‖_{L^{2} (Υ_{i} (ℓ_{i}))}^{2}} \int_{Υ_{i} (ℓ_{i})} u_{2}^{(i)} ({\overline{ξ}}_{i}, ℓ_{i}, t) Θ_{p}^{(i)} ({\overline{ξ}}_{i}) d {\overline{ξ}}_{i}, \end{matrix}$ $b_{0}^{(i)} = 0$ , ${Θ_{p}^{(i)}}_{p = 0}^{+ \infty}$ and ${0 = {(λ_{0}^{(i)})}^{2} < {(λ_{1}^{(i)})}^{2} ⩽ {(λ_{2}^{(i)})}^{2} ⩽ \dots ⩽ {(λ_{p}^{(i)})}^{2} ⩽ \dots}$ are the eigenfunctions and eigenvalues of the spectral problem $\begin{matrix} - Δ_{{\overline{ξ}}_{i}} Θ^{(i)} = {(λ^{(i)})}^{2} Θ^{(i)} in Υ_{i} (ℓ_{i}), \partial_{ν_{{\overline{ξ}}_{i}}} Θ^{(i)} = 0 on \partial Υ_{i} (ℓ_{i}) . \end{matrix}$

Remark 3.1.
From representation (3.10) it follows the following asymptotic relations $\begin{matrix} Π_{2}^{(i)} ({\overline{ξ}}_{i}, ξ_{i}^{}, t) = O (exp (- λ_{1}^{(i)} ξ_{i}^{})) as ξ_{i}^{*} \to + \infty, i = 1, 2, 3 . \end{matrix}$

3.3. Inner part

What conditions should we put for the functions ${w_{n}^{(i)}}$ at the point $x = 0$ ? To give the answer to this question we have to launch the inner asymptotics.

If we pass to the “fast variables” $ξ = \frac{x}{ε}$ and tend ε to 0, the domain $Ω_{ε}$ is transformed into the unbounded domain Ξ that is the union of the domain $Ξ^{(0)}$ and three semibounded cylinders $\begin{matrix} Ξ^{(i)} = {ξ = (ξ_{1}, ξ_{2}, ξ_{3}) \in R^{3} : ℓ_{i}^{0} < ξ_{i} < + \infty, | {\overline{ξ}}_{i} | < h_{i} (0)}, i = 1, 2, 3, \end{matrix}$ i.e., Ξ is the interior of $⋃_{i = 0}^{3} \overline{Ξ^{(i)}}$ . For parts of the boundary of the domain Ξ we introduce notation $\begin{matrix} Γ_{i} : = {ξ \in R^{3} : ℓ_{i}^{0} < ξ_{i} < + \infty, | {\overline{ξ}}_{i} | = h_{i} (0)}, i = 1, 2, 3, and Γ_{0} : = \partial Ξ ∖ (⋃_{i = 1}^{3} Γ_{i}) . \end{matrix}$

Substituting (3.3) into the problem (2.1), using the Taylor’s formula for the corresponding functions of the original problem and equating coefficients at the same powers of ε, we derive the following boundary-value problems for the coefficients $N_{n}, n \in I$ : $\begin{matrix} (3.11) & \{\begin{matrix} - Δ_{ξ} N_{n} (ξ, t) = F_{n} (ξ, t), & ξ \in Ξ, \\ \partial_{ν_{ξ}} N_{n} (ξ, t) = B_{n}^{(i)} (ξ, t), & ξ \in Γ_{i}, i = 0, 1, 2, 3, \\ N_{n} (ξ, t) \sim w_{n}^{(i)} (0, t) + Ψ_{n}^{(i)} (ξ, t), & ξ_{i} \to + \infty, ξ \in Ξ^{(i)}, i = 1, 2, 3, \end{matrix} \end{matrix}$ where the variable $t \in (0, T)$ is regarded as parameter, $\begin{array}{l} F_{n} \equiv 0, n \in A ∖ {2}, F_{2} (ξ, t) = - \partial_{t} N_{0} - k (N_{0}) + f (0, t), \\ F_{2 + α_{0}} (ξ, t) = - \partial_{t} N_{α_{0}} - k^{'} (N_{0}) N_{α_{0}}, F_{3} (ξ, t) = - \partial_{t} N_{1} - k^{'} (N_{0}) N_{1} + (ξ, \nabla_{x}) f (0), \\ F_{2 + 2 α_{0}} (ξ, t) = - \partial_{t} N_{2 α_{0}} - k^{'} (N_{0}) N_{2 α_{0}} - \frac{1}{2} k^{″} (N_{0}) N_{α_{0}}^{2} . \end{array}$ There are three conditions at the different outlet at infinity in the third line of (3.11). They are obtained by matching the regular asymptotics with the inner one [5], namely the asymptotics of $N_{n}$ as $ξ_{i} \to + \infty$ has to coincide with the corresponding asymptotics of ${w_{n}^{(i)}}$ as $x_{i} = ε ξ_{i} \to + 0$ . As a result, we define $\begin{array}{l} Ψ_{0}^{(i)} \equiv Ψ_{α_{0}}^{(i)} \equiv Ψ_{2 α_{0}}^{(i)} \equiv 0, Ψ_{n}^{(i)} (ξ, t) = ξ_{i} \frac{\partial w_{n - 1}^{(i)}}{\partial x_{i}} (0, t), n \in {1, 1 + α_{0}, 1 + 2 α_{0}}, \\ Ψ_{n}^{(i)} (ξ, t) = \frac{ξ_{i}^{2}}{2} \frac{\partial^{2} w_{n - 2}^{(i)}}{\partial x_{i}^{2}} (0, t) + ξ_{i} \frac{\partial w_{n - 1}^{(i)}}{\partial x_{i}} (0, t) + u_{n}^{(i)} ({\overline{ξ}}_{i}, 0, t), n \in {2, 2 + α_{0}, 2 + 2 α_{0}}, \\ Ψ_{3}^{(i)} (ξ, t) = \frac{ξ_{i}^{3}}{3!} \frac{\partial^{3} w_{0}^{(i)}}{\partial x_{i}^{3}} (0, t) + \frac{ξ_{i}^{2}}{2} \frac{\partial^{2} w_{1}^{(i)}}{\partial x_{i}^{2}} (0, t) + ξ_{i} \frac{\partial w_{2}^{(i)}}{\partial x_{i}} (0, t) + ξ_{i} \frac{\partial u_{2}^{(i)}}{\partial x_{i}} ({\overline{ξ}}_{i}, 0, t) + u_{3}^{(i)} ({\overline{ξ}}_{i}, 0, t) . \end{array}$

A solution of the problem (3.11) is sought in the form $\begin{matrix} (3.12) & N_{n} (ξ, t) = \sum_{i = 1}^{3} χ_{i} (ξ_{i}) Ψ_{n}^{(i)} (ξ, t) + {\tilde{N}}_{n} (ξ, t), \end{matrix}$ where ${χ_{i}}_{i = 1}^{3}$ are smooth cut-off functions such that $0 ⩽ χ_{i} ⩽ 1$ and $χ_{i} (ξ_{i}) = 0$ if $ξ_{i} ⩽ 1 + ℓ_{i}^{0}$ , and $χ_{i} (ξ_{i}) = 1$ if $ξ_{i} ⩾ 2 + ℓ_{i}^{0}$ . Then ${\tilde{N}}_{n}$ has to be a solution to the problem $\begin{matrix} (3.13) & \{\begin{matrix} - Δ_{ξ} {\tilde{N}}_{n} (ξ, t) = {\tilde{F}}_{n} (ξ, t), & ξ \in Ξ, \\ \partial_{ν_{ξ}} {\tilde{N}}_{n} (ξ, t) = {\tilde{B}}_{n}^{(i)} (ξ, t), & ξ \in Γ_{i}, i = 0, 1, 2, 3, \\ {\tilde{N}}_{n} (ξ, t) \to w_{n}^{(i)} (0, t) & as ξ_{i} \to + \infty, ξ \in Ξ^{(i)}, i = 1, 2, 3, \end{matrix} \end{matrix}$ where ${\tilde{F}}_{n} \equiv 0, n \in {0, α_{0}, 2 α_{0}}$ , $\begin{array}{l} {\tilde{F}}_{n} (ξ, t) = \sum_{i = 1}^{3} (ξ_{i} \frac{\partial w_{n - 1}^{(i)}}{\partial x_{i}} (0, t) χ_{i}^{″} (ξ_{i}) + 2 \frac{\partial w_{n - 1}^{(i)}}{\partial x_{i}} (0, t) χ_{i}^{'} (ξ_{i})), n \in {1, 1 + α_{0}, 1 + 2 α_{0}}, \\ {\tilde{F}}_{2} (ξ, t) = \sum_{i = 1}^{3} [(\frac{ξ_{i}^{2}}{2} \frac{\partial^{2} w_{0}^{(i)}}{\partial x_{i}^{2}} (0, t) + ξ_{i} \frac{\partial w_{1}^{(i)}}{\partial x_{i}} (0, t) + u_{2}^{(i)} ({\overline{ξ}}_{i}, 0, t)) χ_{i}^{″} (ξ_{i}) \\ + 2 (ξ_{i} \frac{\partial^{2} w_{0}^{(i)}}{\partial x_{i}^{2}} (0, t) + \frac{\partial w_{1}^{(i)}}{\partial x_{i}} (0, t)) χ_{i}^{'} (ξ_{i})] \\ - \partial_{t} N_{0} - k (N_{0}) + \sum_{i = 1}^{3} χ_{i} (ξ_{i}) (\partial_{t} w_{0}^{(i)} (0, t) + k (w_{0}^{(i)} (0, t))) + (1 - \sum_{i = 1}^{3} χ_{i} (ξ_{i})) f (0, t), \\ B_{n}^{(0)} (ξ, t) \equiv {\tilde{B}}_{n}^{(0)} (ξ, t) \equiv 0, n \in {0, α_{0}, 1, 2 α_{0}, 2, 3}, \\ B_{1 + α_{0}}^{(0)} (ξ, t) = {\tilde{B}}_{1 + α_{0}}^{(0)} (ξ, t) = - κ_{0} (N_{0}) + φ^{(0)} (ξ, t), \\ B_{1 + 2 α_{0}}^{(0)} (ξ, t) = {\tilde{B}}_{1 + 2 α_{0}}^{(0)} (ξ, t) = - κ_{0}^{'} (N_{0}) N_{α_{0}}, B_{2 + α_{0}}^{(0)} (ξ, t) = {\tilde{B}}_{2 + α_{0}}^{(0)} (ξ, t) = - κ_{0}^{'} (N_{0}) N_{1}, \\ B_{2 + 2 α_{0}}^{(0)} (ξ, t) = {\tilde{B}}_{2 + 2 α_{0}}^{(0)} (ξ, t) = - κ_{0}^{'} (N_{0}) N_{1 + α_{0}} - κ_{0}^{″} (N_{0}) N_{α_{0}} N_{1}, \end{array}$ and $\begin{array}{l} B_{n}^{(i)} \equiv {\tilde{B}}_{n}^{(i)} \equiv 0, n \in {0, α_{0}, 1, 2 α_{0}, 1 + α_{0}, 1 + 2 α_{0}}, \\ B_{2}^{(i)} (ξ, t) = - κ_{i} (N_{0}) + φ^{(i)} ({\overline{ξ}}_{i}, 0, t), \\ {\tilde{B}}_{2}^{(i)} (ξ, t) = - (κ_{i} (N_{0}) - κ_{i} (w_{0}^{(i)} (0, t)) χ_{i} (ξ_{i})) + φ^{(i)} ({\overline{ξ}}_{i}, 0, t) (1 - χ_{i} (ξ_{i})), \\ B_{2 + α_{0}}^{(i)} (ξ, t) = {\tilde{B}}_{2 + α_{0}}^{(i)} (ξ, t) = - κ_{i}^{'} (N_{0}) N_{α_{0}}, \\ B_{3}^{(i)} (ξ, t) = {\tilde{B}}_{3}^{(i)} (ξ, t) - κ_{i}^{'} (N_{0}) N_{1} - ξ_{i} \frac{\partial φ^{(i)}}{\partial x_{i}} ({\overline{ξ}}_{i}, 0, t), \\ B_{2 + 2 α_{0}}^{(i)} (ξ, t) = {\tilde{B}}_{2 + 2 α_{0}}^{(i)} (ξ, t) = - κ_{i}^{'} (N_{0}) N_{2 α_{0}} - \frac{1}{2} κ_{i}^{″} (N_{0}) N_{α_{0}}^{2}, i = 1, 2, 3 . \end{array}$

The existence of a solution to the problem (3.13) in the corresponding energetic space can be obtained from general results about the asymptotic behavior of solutions to elliptic problems in domains with different exits to infinity (see e.g. [9,19]). We will use approach proposed in [14,19].

Let $C_{0, ξ}^{\infty} (\overline{Ξ})$ be a space of functions infinitely differentiable in $\overline{Ξ}$ and finite with respect to ξ, i.e., $\begin{matrix} \forall v \in C_{0, ξ}^{\infty} (\overline{Ξ}) \exists R > 0 \forall ξ \in \overline{Ξ} ξ_{i} ⩾ R, i = 1, 2, 3 : v (ξ) = 0 . \end{matrix}$ We now define a space $H : = \overline{(C_{0, ξ}^{\infty} (\overline{Ξ}), ‖ \cdot ‖_{H})}$ , where $\begin{matrix} ‖ v ‖_{H} = \sqrt{\int_{Ξ} {| \nabla v (ξ) |}^{2} d ξ + \int_{Ξ} {| v (ξ) |}^{2} {| ρ (ξ) |}^{2} d ξ}, \end{matrix}$ and the weight function $ρ \in C^{\infty} (R^{3}), 0 ⩽ ρ ⩽ 1$ and $\begin{matrix} ρ (ξ) = \{\begin{matrix} 1, & if ξ \in Ξ^{(0)}, \\ | ξ_{i} |^{- 1}, & if ξ_{i} ⩾ ℓ_{i}^{0} + 1, ξ \in Ξ^{(i)}, i = 1, 2, 3 . \end{matrix} \end{matrix}$

Definition 3.1.
A function ${\tilde{N}}_{n} \in H$ is called a weak solution of the problem (3.13) if the identity $\begin{matrix} \int_{Ξ} \nabla {\tilde{N}}_{n} \cdot \nabla v d ξ = \int_{Ξ} {\tilde{F}}_{n} v d ξ + \sum_{i = 0}^{3} \int_{Γ_{i}} {\tilde{B}}_{n}^{(i)} v d σ_{ξ} \end{matrix}$ holds for all $v \in H$ .
Proposition 3.1.
Let $ρ^{- 1} {\tilde{F}}_{n} (\cdot, t) \in L^{2} (Ξ), {\tilde{B}}_{n}^{(0)} (\cdot, t) \in L^{2} (Γ_{0}), ρ^{- 1} {\tilde{B}}_{n}^{(i)} (\cdot, t) \in L^{2} (Γ_{i}), i = 1, 2, 3$ , for a.e. $t \in (0, T)$ . Then there exist a weak solution of problem ( 3.13 ) if and only if $\begin{matrix} (3.14) & \int_{Ξ} {\tilde{F}}_{n} d ξ + \sum_{i = 0}^{3} \int_{Γ_{i}} {\tilde{B}}_{n}^{(i)} d σ_{ξ} = 0 . \end{matrix}$ It is defined up to an additive constant. The additive constant can be chosen to guarantee the existence and uniqueness of a weak solution of problem ( 3.13 ) with the following differentiable asymptotics: $\begin{matrix} (3.15) & {\hat{N}}_{n} (ξ, t) = \{\begin{matrix} O (exp (- γ_{1} ξ_{1})) & as ξ_{1} \to + \infty, ξ \in Ξ^{(1)}, \\ δ_{n}^{(2)} (t) + O (exp (- γ_{2} ξ_{2})) & as ξ_{2} \to + \infty, ξ \in Ξ^{(2)}, \\ δ_{n}^{(3)} (t) + O (exp (- γ_{3} ξ_{3})) & as ξ_{3} \to + \infty, ξ \in Ξ^{(3)}, \end{matrix} \end{matrix}$ where $γ_{i}, i = 1, 2, 3$ are positive constants.

The values $δ_{n}^{(2)}$ and $δ_{n}^{(3)}$ in (3.15) are defined as follows: $\begin{matrix} (3.16) & δ_{n}^{(i)} (t) = \int_{Ξ} N_{i} (ξ) {\tilde{F}}_{n} (ξ, t) d ξ + \sum_{j = 0}^{3} \int_{Γ_{j}} N_{i} (ξ) {\tilde{B}}_{n}^{(j)} (ξ, t) d σ_{ξ}, i = 2, 3, n \in I, \end{matrix}$ where $N_{2}$ and $N_{3}$ are special solutions to the homogeneous problem $\begin{matrix} (3.17) & - Δ_{ξ} N = 0 in Ξ, \partial_{ν} N = 0 on \partial Ξ . \end{matrix}$ Proposition 3.2.
The problem ( 3.17 ) has two linearly independent solutions $N_{2}$ and $N_{3}$ that do not belong to the space $H$ and they have the following differentiable asymptotics: $\begin{array}{l} N_{2} (ξ) = \{\begin{matrix} - \frac{ξ_{1}}{π h_{1}^{2} (0)} + O (exp (- γ_{1} ξ_{1})) & as ξ_{1} \to + \infty, ξ \in Ξ^{(1)}, \\ \frac{ξ_{2}}{π h_{2}^{2} (0)} + C_{2}^{(2)} + O (exp (- γ_{2} ξ_{2})) & as ξ_{2} \to + \infty, ξ \in Ξ^{(2)}, \\ C_{2}^{(3)} + O (exp (- γ_{3} ξ_{3})) & as ξ_{3} \to + \infty, ξ \in Ξ^{(3)}, \end{matrix} \\ N_{3} (ξ) = \{\begin{matrix} - \frac{ξ_{1}}{π h_{1}^{2} (0)} + O (exp (- γ_{1} ξ_{1})) & as ξ_{1} \to + \infty, ξ \in Ξ^{(1)}, \\ C_{3}^{(2)} + O (exp (- γ_{2} ξ_{2})) & as ξ_{2} \to + \infty, ξ \in Ξ^{(2)}, \\ \frac{ξ_{3}}{π h_{3}^{2} (0)} + C_{3}^{(3)} + O (exp (- γ_{3} ξ_{3})) & as ξ_{3} \to + \infty, ξ \in Ξ^{(3)} . \end{matrix} \end{array}$

Any other solution to the homogeneous problem, which has polynomial growth at infinity, can be presented as a linear combination $c_{1} + c_{2} N_{2} + c_{3} N_{3}$ .
Remark 3.2.
To obtain formulas (3.16) it is necessary to substitute the functions ${\hat{N}}_{n}, N_{2}$ and ${\hat{N}}_{n}, N_{3}$ in the second Green–Ostrogradsky formula $\begin{matrix} \int_{Ξ_{R}} (\hat{N} Δ_{ξ} N - N Δ_{ξ} \hat{N}) d ξ = \int_{\partial Ξ_{R}} (\hat{N} \partial_{ν_{ξ}} N - N \partial_{ν_{ξ}} \hat{N}) d σ_{ξ} \end{matrix}$ respectively, and then pass to the limit as $R \to + \infty$ . Here $Ξ_{R} = Ξ \cap {ξ : | ξ_{i} | < R, i = 1, 2, 3}$ .

3.3.1. Limit problem

The problem (3.11) at $n = 0$ is as follows: $\begin{matrix} \{\begin{matrix} - Δ_{ξ} N_{0} (ξ, t) = 0, & ξ \in Ξ, \\ \partial_{ν_{ξ}} N_{0} (ξ, t) = 0, & ξ \in \partial Ξ, \\ N_{0} (ξ, t) ⟶ w_{0}^{(i)} (0, t), & ξ_{i} \to + \infty, ξ \in Ξ^{(i)}, i = 1, 2, 3 . \end{matrix} \end{matrix}$ It is easy to verify that $δ_{0}^{(2)} = δ_{0}^{(3)} = 0$ . Only a constant is a solution to this problem in $H$ . We can satisfy the limits if and only if $\begin{matrix} (3.18) & w_{0}^{(1)} (0, t) = w_{0}^{(2)} (0, t) = w_{0}^{(3)} (0, t) . \end{matrix}$ Thereby, $N_{0} = {\tilde{N}}_{0} \equiv w_{0}^{(1)} (0, t)$ .

The solvability condition (3.14) for the problem (3.13) at $n = 1$ reads as follows: $\begin{matrix} (3.19) & π h_{1}^{2} (0) \frac{\partial w_{0}^{(1)}}{\partial x_{1}} (0, t) + π h_{2}^{2} (0) \frac{\partial w_{0}^{(2)}}{\partial x_{2}} (0, t) + π h_{3}^{2} (0) \frac{\partial w_{0}^{(3)}}{\partial x_{3}} (0, t) = 0 . \end{matrix}$

Thus, taking into account (3.5) and (3.8), we obtain for ${w_{0}^{(i)}}_{i = 1}^{3}$ the following semilinear problem: $\begin{array}{l} (3.20) & \{\begin{matrix} π h_{i}^{2} (x_{i}) \frac{\partial w_{0}^{(i)}}{\partial t} (x_{i}, t) - π \frac{\partial}{\partial x_{i}} (h_{i}^{2} (x_{i}) \frac{\partial w_{0}^{(i)}}{\partial x_{i}} (x_{i}, t)) + π h_{i}^{2} (x_{i}) k (w_{0}^{(i)} (x_{i}, t)) \\ + 2 π h_{i} (x_{i}) κ_{i} (w_{0}^{(i)} (x_{i}, t)) = {\hat{F}}_{0}^{(i)} (x_{i}, t), (x_{i}, t) \in I_{i} \times (0, T), i = 1, 2, 3, \\ w_{0}^{(i)} (ℓ_{i}, t) = 0, t \in (0, T), i = 1, 2, 3, \\ w_{0}^{(1)} (0, t) = w_{0}^{(2)} (0, t) = w_{0}^{(3)} (0, t), t \in (0, T), \\ \sum_{i = 1}^{3} π h_{i}^{2} (0) \frac{\partial w_{0}^{(i)}}{\partial x_{i}} (0, t) = 0, t \in (0, T), \\ w_{0}^{(i)} (x_{i}, 0) = 0, x_{i} \in I_{i}, i = 1, 2, 3, \end{matrix} \end{array}$ where $I_{i} : = {x : x_{i} \in (0, ℓ_{i}), {\overline{x}}_{i} = (0, 0)}$ and $\begin{matrix} (3.21) & {\hat{F}}_{0}^{(i)} (x_{i}, t) : = π h_{i}^{2} (x_{i}) f (x, t) |_{{\overline{x}}_{i} = (0, 0)} + \int_{\partial Υ_{i} (x_{i})} φ^{(i)} ({\overline{ξ}}_{i}, x_{i}, t) d l_{{\overline{ξ}}_{i}}, x \in I_{i} . \end{matrix}$ The problem (3.20) is called the limit problem for problem (2.1).

For functions $\begin{matrix} \tilde{ϕ} (x) = \{\begin{matrix} ϕ^{(1)} (x_{1}), & if x_{1} \in I_{1}, \\ ϕ^{(2)} (x_{2}), & if x_{2} \in I_{2}, \\ ϕ^{(3)} (x_{3}), & if x_{3} \in I_{3}, \end{matrix} \end{matrix}$ defined on the graph $I : = \overline{I_{1}} \cup \overline{I_{2}} \cup \overline{I_{3}}$ , we introduce the Sobolev space $\begin{matrix} H_{0} : = {\tilde{ϕ} : ϕ^{(i)} \in H^{1} (I_{i}), ϕ^{(i)} (ℓ_{i}) = 0, i = 1, 2, 3, and ϕ^{(1)} (0) = ϕ^{(2)} (0) = ϕ^{(3)} (0)} \end{matrix}$ with the scalar product $\begin{matrix} {(\tilde{ϕ}, \tilde{ψ})}_{0} : = \sum_{i = 1}^{3} π \int_{0}^{ℓ_{i}} h_{i}^{2} (x_{i}) \frac{d ϕ^{(i)}}{d x_{i}} \frac{d ψ^{(i)}}{d x_{i}} d x_{i}, \tilde{ϕ}, \tilde{ψ} \in H_{0} . \end{matrix}$

Definition 3.2.
A function $\tilde{w} \in L^{2} (0, T; H_{0})$ , with $\partial_{t} \tilde{w} \in L^{2} (0, T; H_{0}^{})$ , is called a weak solution to the problem (3.20) if it satisfies the identity $\begin{matrix} {⟨ \partial_{t} \tilde{w}, \tilde{ψ} ⟩}_{0} + {⟨ A_{0} (t) \tilde{w}, \tilde{ψ} ⟩}_{0} = {⟨ F_{0} (t), \tilde{ψ} ⟩}_{0}, \end{matrix}$ for any function $\tilde{ψ} \in H_{0}$ and a.e. $t \in (0, T)$ , and $\tilde{w} |_{t = 0} = 0$ .

Here the nonlinear operator $A_{0} (t) : H_{0} \mapsto H_{0}^{}$ is defined through the relation $\begin{matrix} {⟨ A_{0} (t) ϕ^{(i)}, ψ^{(i)} ⟩}_{0} = {(\tilde{ϕ}, \tilde{ψ})}_{0} + \sum_{i = 1}^{3} (π \int_{0}^{ℓ_{i}} h_{i}^{2} k (ϕ^{(i)}) ψ^{(i)} d x_{i} + 2 π \int_{0}^{ℓ_{i}} h_{i} κ_{i} (ϕ^{(i)}) ψ^{(i)} d x_{i}) \end{matrix}$ for all $\tilde{ϕ}, \tilde{ψ} \in H_{0}$ , and the linear functional $F_{0} (t) \in H_{0}^{}$ is defined by the formula $\begin{matrix} {⟨ F_{0} (t), \tilde{ψ} ⟩}_{0} = \sum_{i = 1}^{3} \int_{0}^{ℓ_{i}} {\hat{F}}_{0}^{(i)} ψ^{(i)} d x_{i} \forall \tilde{ψ} \in H_{0}, \end{matrix}$ where ${⟨ \cdot, \cdot ⟩}_{0}$ is the duality pairing of the dual space $H_{0}^{}$ and $H_{0}$ .

Using (2.4) and (2.6), we can prove that the operator $A_{0}$ is bounded, strictly monotone, hemicontinuous and coercive. As a result, the existence and uniqueness of the weak solution to the problem (3.20) follow directly from Corollary 4.1 (see [25, Chapter 3]).
3.3.2. Problems for ${{\tilde{w}}_{n}}$

Similarly as before, we find that $δ_{n}^{(2)} = δ_{n}^{(3)} = 0$ , $N_{α_{0}} = {\tilde{N}}_{α_{0}} = w_{α_{0}}^{(1)} (0, t)$ , $N_{2 α_{0}} = {\tilde{N}}_{2 α_{0}} = w_{2 α_{0}}^{(1)} (0, t)$ , and $\begin{matrix} (3.22) & w_{n}^{(1)} (0, t) = w_{n}^{(2)} (0, t) = w_{n}^{(3)} (0, t) \forall n \in {α_{0}, 2 α_{0}} . \end{matrix}$

Taking into account the third relation in problems (3.4), (3.6) and relation (3.5), from the solvability condition (3.14) at $n \in I ∖ {0, α_{0}, 2 α_{0}}$ it follows the following transmission condition for ${w_{n}^{(i)}}_{i = 1}^{3}$ : $\begin{matrix} (3.23) & \sum_{i = 1}^{3} π h_{i}^{2} (0) \frac{\partial w_{n}^{(i)}}{\partial x_{i}} (0, t) = d_{n}^{*} (t), n \in A, \end{matrix}$ where $\begin{array}{l} (3.24) & d_{α_{0}}^{*} (t) = & | Γ_{0} |_{2} κ_{0} (w_{0}^{(1)} (0, t)) - \int_{Γ_{0}} φ^{(0)} (ξ, t) d σ_{ξ}, \\ d_{1}^{*} (t) = & ({| Ξ^{(0)} |}_{3} - \sum_{i = 1}^{3} ℓ_{i}^{0} π h_{i}^{2} (0)) (\partial_{t} w_{0}^{(1)} (0, t) + k (w_{0}^{(1)} (0, t)) - f (0, t)) \\ (3.25) & - \sum_{i = 1}^{3} ℓ_{i}^{0} (2 π h_{i} (0) κ_{i} (w_{0}^{(1)} (0, t)) - \int_{\partial Υ_{i} (0)} φ^{(i)} ({\overline{ξ}}_{i}, 0, t) d l_{{\overline{ξ}}_{i}}), \\ (3.26) & d_{2 α_{0}}^{*} (t) = & | Γ_{0} |_{2} κ_{0}^{'} (w_{0}^{(1)} (0, t)) w_{α_{0}}^{(1)} (0, t) . \end{array}$

Hence, if the functions ${w_{n}^{(i)}}_{i = 1}^{3}$ at $n \in A$ satisfy (3.23), then there exist a weak solution ${\tilde{N}}_{n + 1}$ of the problem (3.13). According to Proposition 3.1, it can be chosen in a unique way to guarantee the asymptotics (3.15).

It remains to satisfy the stabilization conditions in the problem (3.13) at $n \in {1, 1 + α_{0}, 2, 1 + 2 α_{0}}$ . For this, we represent the solution in the form: ${\tilde{N}}_{n} (ξ, t) = w_{n}^{(1)} (0, t) + {\hat{N}}_{n} (ξ, t), ξ \in Ξ$ . Taking into account the asymptotics (3.15), we have to put $\begin{matrix} (3.27) & w_{n}^{(1)} (0, t) = w_{n}^{(2)} (0, t) - δ_{n}^{(2)} (t) = w_{n}^{(3)} (0, t) - δ_{n}^{(3)} (t) . \end{matrix}$ As a result, we get the solution to the problem (3.11) with the following asymptotics: $\begin{matrix} (3.28) & N_{n} (ξ, t) = w_{n}^{(i)} (0, t) + Ψ_{n}^{(i)} (ξ, t) + O (exp (- γ_{i} ξ_{i})) as ξ_{i} \to + \infty, i = 1, 2, 3 . \end{matrix}$

Let us denote by $G_{n} (ξ, t) : = w_{n}^{(i)} (0, t) + Ψ_{n}^{(i)} (ξ, t), (ξ, t) \in Ξ^{(i)} \times (0, T), i = 1, 2, 3$ .

Remark 3.3.
Due to (3.28), functions ${N_{n} - G_{n}}$ are exponentially vanishing as $ξ_{i} \to + \infty$ , $i = 1, 2, 3$ .

Relations (3.22), (3.27) and (3.23) are transmission conditions for ${w_{n}^{(i)}}_{i = 1}^{3}$ at $x = 0$ . Thus, the next terms ${\tilde{w}}_{n}$ , $n \in A ∖ {0}$ of the regular asymptotics are determined from the following problems: $\begin{array}{l} (3.29) & \{\begin{matrix} π h_{i}^{2} (x_{i}) \frac{\partial w_{α_{0}}^{(i)}}{\partial t} (x_{i}, t) - π \frac{\partial}{\partial x_{i}} (h_{i}^{2} (x_{i}) \frac{\partial w_{α_{0}}^{(i)}}{\partial x_{i}} (x_{i}, t)) + π h_{i}^{2} (x_{i}) k^{'} (w_{0}^{(i)} (x_{i}, t)) w_{α_{0}}^{(i)} (x_{i}, t) \\ + 2 π h_{i} (x_{i}) κ_{i}^{'} (w_{0}^{(i)} (x_{i}, t)) w_{α_{0}}^{(i)} (x_{i}, t) = 0, (x_{i}, t) \in I_{i} \times (0, T), i = 1, 2, 3, \\ w_{α_{0}}^{(i)} (ℓ_{i}, t) = 0, t \in (0, T), i = 1, 2, 3, \\ w_{α_{0}}^{(1)} (0, t) = w_{α_{0}}^{(2)} (0, t) = w_{α_{0}}^{(3)} (0, t), t \in (0, T), \\ \sum_{i = 1}^{3} π h_{i}^{2} (0) \frac{\partial w_{α_{0}}^{(i)}}{\partial x_{i}} (0, t) = | Γ_{0} |_{2} κ_{0} (w_{0}^{(1)} (0, t)) - \int_{Γ_{0}} φ^{(0)} (ξ, t) d σ_{ξ}, t \in (0, T), \\ w_{α_{0}}^{(i)} (x_{i}, 0) = 0, x_{i} \in I_{i}, i = 1, 2, 3; \end{matrix} \\ (3.30) & \{\begin{matrix} π h_{i}^{2} (x_{i}) \frac{\partial w_{n}^{(i)}}{\partial t} (x_{i}, t) - π \frac{\partial}{\partial x_{i}} (h_{i}^{2} (x_{i}) \frac{\partial w_{n}^{(i)}}{\partial x_{i}} (x_{i}, t)) + π h_{i}^{2} (x_{i}) k^{'} (w_{0}^{(i)} (x_{i}, t)) w_{n}^{(i)} (x_{i}, t) \\ + 2 π h_{i} (x_{i}) κ_{i}^{'} (w_{0}^{(i)} (x_{i}, t)) w_{n}^{(i)} (x_{i}, t) = {\hat{F}}_{n}^{(i)} (x_{i}, t), (x_{i}, t) \in I_{i} \times (0, T), \\ i = 1, 2, 3, \\ w_{n}^{(i)} (ℓ_{i}, t) = 0, i \in {1, 2, 3}, t \in (0, T), \\ w_{n}^{(1)} (0, t) = w_{n}^{(2)} (0, t) - δ_{n}^{(2)} (t) = w_{n}^{(3)} (0, t) - δ_{n}^{(3)} (t), t \in (0, T), \\ \sum_{i = 1}^{3} π h_{i}^{2} (0) \frac{\partial w_{n}^{(i)}}{\partial x_{i}} (0, t) = d_{n}^{} (t), t \in (0, T), \\ w_{n}^{(i)} (x_{i}, 0) = 0, i \in {1, 2, 3}, x_{i} \in I_{i}, \end{matrix} \end{array}$ for $n \in {1, 2 α_{0}, 1 + α_{0}, 2, 1 + 2 α_{0}}$ . In the case $n = 1$ we have $\begin{matrix} (3.31) & {\hat{F}}_{1}^{(i)} (x_{i}, t) = \int_{Υ_{i} (x_{i})} f_{1}^{(i)} ({\overline{ξ}}_{i}, x_{i}, t) d {\overline{ξ}}_{i} (x_{i}, t) \in I_{i} \times (0, T), i = 1, 2, 3, \end{matrix}$ and the values $d_{1}^{}$ and $δ_{1}^{(2)}$ , $δ_{1}^{(3)}$ (see Remark 3.2) are uniquely determined by (3.25) and formula $\begin{matrix} (3.32) & δ_{1}^{(i)} (t) = \int_{Ξ} N_{i} (ξ) \sum_{j = 1}^{3} (ξ_{j} \frac{\partial w_{0}^{(j)}}{\partial x_{j}} (0, t) χ_{j}^{″} (ξ_{j}) + 2 \frac{\partial w_{0}^{(j)}}{\partial x_{j}} (0, t) χ_{j}^{'} (ξ_{j})) d ξ, i = 2, 3 . \end{matrix}$

With the help of the substitutions $\begin{matrix} ϕ_{n}^{(1)} = w_{n}^{(1)}, ϕ_{n}^{(i)} (x_{i}, t) = w_{n}^{(i)} (x_{i}, t) - δ_{n}^{(i)} (t) \frac{ℓ_{i} - x_{i}}{ℓ_{i}}, i \in {2, 3}, \end{matrix}$ we reduce (3.30) to the respective linear parabolic problem in the space $L^{2} (0, T; H_{0})$ . Then the existence and uniqueness of the solution follow from the classical theory of linear parabolic problems.
3.4. Justification

With the help of ${w_{n}^{(i)}}_{i = 1}^{3}$ , ${N_{n}}$ , $n \in A$ and smooth cut-off functions defined by formula $\begin{matrix} (3.33) & χ_{ℓ_{i}^{0}}^{(i)} (ζ_{i}) = \{\begin{matrix} 1, & if ζ_{i} ⩾ 3 ℓ_{i}^{0}, \\ 0, & if ζ_{i} ⩽ 2 ℓ_{i}^{0}, \end{matrix} χ_{δ}^{(i)} (x_{i}) = \{\begin{matrix} 1, & if x_{i} ⩾ ℓ_{i} - δ, \\ 0, & if x_{i} ⩽ ℓ_{i} - 2 δ, \end{matrix} \end{matrix}$ we construct the following asymptotic approximation: $\begin{array}{l} U_{ε} (x, t) = & \sum_{i = 1}^{3} χ_{ℓ_{i}^{0}}^{(i)} (\frac{x_{i}}{ε^{a}}) (\sum_{n \in {0, α_{0}, 1, 2 α_{0}, 1 + α_{0}, 1 + 2 α_{0}}} ε^{n} w_{n}^{(i)} (x_{i}, t) \\ + ε^{2} (w_{2}^{(i)} (x_{i}, t) + u_{2}^{(i)} (\frac{{\overline{x}}_{i}}{ε}, x_{i}, t) + χ_{δ}^{(i)} Π_{2}^{(i)} (\frac{{\overline{x}}_{i}}{ε}, \frac{ℓ_{i} - x_{i}}{ε}, t))) \\ + (1 - \sum_{i = 1}^{3} χ_{ℓ_{i}^{0}}^{(i)} (\frac{x_{i}}{ε^{a}})) (w_{0}^{(1)} (0, t) + ε^{α_{0}} w_{α_{0}}^{(1)} (0, t) + ε^{2 α_{0}} w_{2 α_{0}}^{(1)} (0, t) \\ (3.34) & + \sum_{n \in {1, 1 + α_{0}, 2, 1 + 2 α_{0}}} ε^{n} N_{n} (\frac{x}{ε}, t)), x \in Ω_{ε} \times (0, T), \end{array}$ where $a$ is a fixed number from the interval $(\frac{2}{3}, 1)$ , and δ is a sufficiently small fixed positive number.

Lemma 3.1.
Let conditions C1, C2, C3 be satisfied. Then the difference between the solution $u_{ε}$ to the problem ( 2.1 ) and the sum ( 3.34 ) admits the following estimate:

$\exists C_{0} > 0$ $\exists ε_{0} > 0$ $\forall ε \in (0, ε_{0})$ : $\begin{matrix} (3.35) & max_{t \in [0, T]} {‖ U_{ε} (\cdot, t) - u_{ε} (\cdot, t) ‖}_{L^{2} (Ω_{ε})} + ‖ U_{ε} - u_{ε} ‖_{L^{2} (0, T; H^{1} (Ω_{ε}))} ⩽ C_{0} ε^{1 + min (3 α_{0}, α_{0} + \frac{a}{2}, \frac{3}{2} a)} . \end{matrix}$
Proof.
Substituting the sum (3.34) in (2.1) instead of $u_{ε}$ and taking (2.2), (2.3), (3.4), (3.6), (3.9) and (3.11) into account, we find $\begin{matrix} (3.36) & \{\begin{matrix} \partial_{t} U_{ε} - Δ_{x} U_{ε} + k (U_{ε}) - f = {\hat{R}}_{ε} in Ω_{ε} \times (0, T), \\ \partial_{ν} U_{ε} + ε^{α_{0}} κ_{0} (U_{ε}) - ε^{α_{0}} φ_{ε}^{(0)} = {\overset{˘}{R}}_{ε}^{(0)} on Γ_{ε}^{(0)} \times (0, T), \\ \partial_{ν} U_{ε} + ε κ_{i} (U_{ε}) - ε φ_{ε}^{(i)} = {\overset{˘}{R}}_{ε}^{(i)} on Γ_{ε}^{(i)} \times (0, T), i = 1, 2, 3, \\ U_{ε} = 0 on Υ_{ε}^{(i)} (ℓ_{i}) \times (0, T), i = 1, 2, 3, \\ U_{ε} |_{t = 0} = 0 in Ω_{ε}, \end{matrix} \end{matrix}$ where $\begin{array}{l} {\hat{R}}_{ε} (x, t) = & - \sum_{i = 1}^{3} ((ε^{- 2 a} \frac{d^{2} χ_{ℓ_{i}^{0}}^{(i)}}{d ζ_{i}^{2}} (ζ_{i}) + 2 ε^{- a} \frac{d χ_{ℓ_{i}^{0}}^{(i)}}{d ζ_{i}} (ζ_{i}) \frac{\partial}{\partial x_{i}}) |_{ζ_{i} = \frac{x_{i}}{ε^{a}}} (w_{0}^{(i)} (x_{i}, t) - w_{0}^{(i)} (0, t) \\ - x_{i} \frac{\partial w_{0}^{(i)}}{\partial x_{i}} (0, t) - \frac{x_{i}^{2}}{2} \frac{\partial^{2} w_{0}^{(i)}}{\partial x_{i}^{2}} (0, t) + ε^{α_{0}} (w_{α_{0}}^{(i)} (x_{i}, t) - w_{α_{0}}^{(i)} (0, t) - x_{i} \frac{\partial w_{α_{0}}^{(i)}}{\partial x_{i}} (0, t)) \\ + ε (w_{1}^{(i)} (x_{i}, t) - w_{1}^{(i)} (0, t) - x_{i} \frac{\partial w_{1}^{(i)}}{\partial x_{i}} (0, t)) \\ + ε^{2 α_{0}} (w_{2 α_{0}}^{(i)} (x_{i}, t) - w_{2 α_{0}}^{(i)} (0, t) - x_{i} \frac{\partial w_{2 α_{0}}^{(i)}}{\partial x_{i}} (0, t)) \\ + ε^{1 + α_{0}} (w_{1 + α_{0}}^{(i)} (x_{i}, t) - w_{1 + α_{0}}^{(i)} (0, t)) \\ + ε^{2} (w_{2}^{(i)} (x_{i}, t) + u_{2}^{(i)} (\frac{x}{ε}, x_{i}, t) - w_{2}^{(i)} (0, t) - u_{2}^{(i)} (\frac{x}{ε}, 0, t)) \\ + ε^{1 + 2 α_{0}} (w_{1 + 2 α_{0}}^{(i)} (x_{i}, t) - w_{1 + 2 α_{0}}^{(i)} (0, t)) \\ - \sum_{n \in {1, 1 + α_{0}, 2, 1 + 2 α_{0}}} ε^{n} (N_{n} (\frac{x}{ε}, t) - G_{n} (\frac{x}{ε}, t))) \\ + χ_{ℓ_{i}^{0}}^{(i)} (\frac{x_{i}}{ε^{a}}) (k (w_{0}^{(i)} (x_{i})) - f_{0}^{(i)} (x_{i}, t) \\ + ε^{α_{0}} (k^{'} (w_{0}^{(i)} (x_{i})) w_{α_{0}}^{(i)} (x_{i}) - Δ_{{\overline{ξ}}_{i}} u_{2 + α_{0}} ({\overline{ξ}}_{i}, x_{i}, t)) \\ + ε (k^{'} (w_{0}^{(i)} (x_{i})) w_{1}^{(i)} (x_{i}) - f_{1}^{(i)} ({\overline{ξ}}_{i}, x_{i}, t) - Δ_{{\overline{ξ}}_{i}} u_{3} ({\overline{ξ}}_{i}, x_{i}, t)) \\ + ε^{2 α_{0}} (k^{'} (w_{0}^{(i)} (x_{i})) w_{2 α_{0}}^{(i)} (x_{i}) - \frac{1}{2} k^{″} (w_{0}^{(i)} (x_{i})) {(w_{α_{0}}^{(i)} (x_{i}))}^{2} - Δ_{{\overline{ξ}}_{i}} u_{2 + 2 α_{0}} ({\overline{ξ}}_{i}, x_{i}, t)) \\ + ε^{1 + α_{0}} (k^{'} (w_{0}^{(i)} (x_{i})) w_{1 + α_{0}}^{(i)} - k^{″} (w_{0}^{(i)} (x_{i})) w_{α_{0}}^{(i)} (x_{i}) w_{1}^{(i)} (x_{i}) - Δ_{{\overline{ξ}}_{i}} u_{3 + α_{0}} ({\overline{ξ}}_{i}, x_{i}, t)) \\ + ε^{2} (k^{'} (w_{0}^{(i)} (x_{i})) (w_{2}^{(i)} (x_{i}) + u_{2} ({\overline{ξ}}_{i}, x_{i}, t)) \\ - \frac{1}{2} k^{″} (w_{0}^{(i)} (x_{i})) {(w_{1}^{(i)} (x_{i}))}^{2} - f_{2}^{(i)} ({\overline{ξ}}_{i}, x_{i}, t) \\ - Δ_{{\overline{ξ}}_{i}} u_{4} ({\overline{ξ}}_{i}, x_{i}, t)) + ε^{1 + 2 α_{0}} (k^{'} (w_{0}^{(i)} (x_{i})) w_{1 + 2 α_{0}}^{(i)} (x_{i}) - k^{″} (w_{0}^{(i)} (x_{i})) w_{1}^{(i)} (x_{i}) w_{2 α_{0}}^{(i)} (x_{i}) \\ - k^{″} (w_{0}^{(i)} (x_{i})) w_{α_{0}}^{(i)} (x_{i}) w_{1 + α_{0}}^{(i)} (x_{i}) \\ - k^{‴} (w_{0}^{(i)} (x_{i})) {(w_{α_{0}}^{(i)} (x_{i}))}^{2} w_{1}^{(i)} (x_{i}) - Δ_{{\overline{ξ}}_{i}} u_{3 + 2 α_{0}} ({\overline{ξ}}_{i}, x_{i}, t))) |_{\overline{ξ} = \frac{{\overline{x}}_{i}}{ε}} \\ + (- ε^{2} χ_{δ}^{(i)} (x_{i}) \frac{\partial}{\partial t} + ε^{2} \frac{d^{2} χ_{δ}^{(i)}}{d x_{i}^{2}} (x_{i}) - ε \frac{d χ_{δ}^{(i)}}{d x_{i}} (x_{i}) \frac{\partial}{\partial ξ_{i}^{}}) Π_{2}^{(i)} (\frac{{\overline{x}}_{i}}{ε}, ξ_{i}^{}, t) |_{ξ_{i}^{} = \frac{ℓ_{i} - x_{i}}{ε}}) \\ + (1 - \sum_{i = 1}^{3} χ_{ℓ_{i}^{0}}^{(i)} (\frac{x_{i}}{ε^{a}})) (f (0, t) - k (w_{0}^{(i)} (0)) + \sum_{n \in A ∖ {0}} ε^{n} \frac{\partial N_{n}}{\partial t} (\frac{x}{ε}, t)) \\ + k (U_{ε} (x, t)) - f (x, t), \end{array}$ and $\begin{array}{l} {\overset{˘}{R}}_{ε}^{(0)} (x, t) = & ε^{α_{0}} κ_{0} (U_{ε} (x, t)) - ε^{α_{0}} κ_{0} (w_{0}^{(1)} (0, t)) - ε^{2 α_{0}} κ_{0}^{'} (w_{0}^{(1)} (0, t)) w_{α_{0}}^{(1)} (0, t), \\ {\overset{˘}{R}}_{ε}^{(i)} (x, t) = & - \frac{ε}{\sqrt{1 + ε^{2} | h_{i}^{'} (x_{i}) |^{2}}} χ_{ℓ_{i}^{0}}^{(i)} (\frac{x_{i}}{ε^{a}}) (κ_{i} (w_{0}^{(i)} (x_{i})) - φ_{ε}^{(i)} (x, t) \\ + ε^{α_{0}} (κ_{i}^{'} (w_{0}^{(i)} (x_{i})) w_{α_{0}}^{(i)} (x_{i}) - \partial_{ν_{{\overline{ξ}}_{i}}} u_{2 + α_{0}} ({\overline{ξ}}_{i}, x_{i}, t)) \\ + ε (κ_{i}^{'} (w_{0}^{(i)} (x_{i})) w_{1}^{(i)} (x_{i}) - \partial_{ν_{{\overline{ξ}}_{i}}} u_{3} ({\overline{ξ}}_{i}, x_{i}, t)) \\ + ε^{2 α_{0}} (κ_{i}^{'} (w_{0}^{(i)} (x_{i})) w_{2 α_{0}}^{(i)} (x_{i}) - \frac{1}{2} κ_{i}^{″} (w_{0}^{(i)} (x_{i})) {(w_{α_{0}}^{(i)} (x_{i}))}^{2} - \partial_{ν_{{\overline{ξ}}_{i}}} u_{2 + 2 α_{0}} ({\overline{ξ}}_{i}, x_{i}, t)) \\ + ε^{1 + α_{0}} (κ_{i}^{'} (w_{0}^{(i)} (x_{i})) w_{1 + α_{0}}^{(i)} - κ_{i}^{″} (w_{0}^{(i)} (x_{i})) w_{α_{0}}^{(i)} (x_{i}) w_{1}^{(i)} (x_{i}) - \partial_{ν_{{\overline{ξ}}_{i}}} u_{3 + α_{0}} ({\overline{ξ}}_{i}, x_{i}, t)) \\ + ε^{2} (κ_{i}^{'} (w_{0}^{(i)} (x_{i})) (w_{2}^{(i)} (x_{i}) + u_{2} ({\overline{ξ}}_{i}, x_{i}, t)) \\ - \frac{1}{2} κ_{i}^{″} (w_{0}^{(i)} (x_{i})) {(w_{1}^{(i)} (x_{i}))}^{2} + \frac{1}{2} {| h_{i}^{'} (x_{i}) |}^{2} κ_{i} (w_{0}^{(i)} (x_{i})) \\ - \frac{1}{2} {| h_{i}^{'} (x_{i}) |}^{2} φ_{ε}^{(i)} (x, t) - \partial_{ν_{{\overline{ξ}}_{i}}} u_{4} ({\overline{ξ}}_{i}, x_{i}, t)) + ε^{1 + 2 α_{0}} (κ_{i}^{'} (w_{0}^{(i)} (x_{i})) w_{1 + 2 α_{0}}^{(i)} (x_{i}) \\ - κ_{i}^{″} (w_{0}^{(i)} (x_{i})) w_{1}^{(i)} (x_{i}) w_{2 α_{0}}^{(i)} (x_{i}) - κ_{i}^{″} (w_{0}^{(i)} (x_{i})) w_{α_{0}}^{(i)} (x_{i}) w_{1 + α_{0}}^{(i)} (x_{i}) \\ - κ_{i}^{‴} (w_{0}^{(i)} (x_{i})) {(w_{α_{0}}^{(i)} (x_{i}))}^{2} w_{1}^{(i)} (x_{i}) - \partial_{ν_{{\overline{ξ}}_{i}}} u_{3 + 2 α_{0}} ({\overline{ξ}}_{i}, x_{i}, t))) |_{\overline{ξ} = \frac{{\overline{x}}_{i}}{ε}} \\ + (1 - χ_{ℓ_{i}^{0}}^{(i)} (\frac{x_{i}}{ε^{a}})) ε (- κ_{i} (w_{0}^{(i)} (x_{i})) + φ^{(i)} (\frac{{\overline{x}}_{i}}{ε}, 0, t)) \\ + ε κ_{i} (U_{ε} (x, t)) - ε φ_{ε}^{(i)} (x, t), i = 1, 2, 3 . \end{array}$

From (3.36) we derive the following relation: $\begin{matrix} (3.37) & {⟨ \partial_{t} U_{ε}, v ⟩}_{ε} + {⟨ A_{ε} (t) U_{ε}, v ⟩}_{ε} - {⟨ F_{ε} (t), v ⟩}_{ε} = R_{ε} (v) \end{matrix}$ for all $v \in L^{2} (0, T; H_{ε})$ and a.e. $t \in (0, T)$ . Here $\begin{matrix} R_{ε} (v) = \int_{Ω_{ε}} {\hat{R}}_{ε} v d x + \sum_{i = 0}^{3} \int_{Γ_{ε}^{(i)}} {\overset{˘}{R}}_{ε}^{(i)} v d σ_{x} . \end{matrix}$

Let us denote $\begin{array}{l} Ω_{ε, a}^{(0)} : = Ω_{ε} \cap {x : x_{i} < 2 ℓ_{i}^{0} ε^{a}, i = 1, 2, 3}, {\hat{Ω}}_{ε, a}^{(i)} : = Ω_{ε}^{(i)} \cap {x : x_{i} \in {\hat{I}}_{ε, a}^{(i)}}, \\ Ω_{ε, a}^{(i)} : = Ω_{ε}^{(i)} \cap {x : x_{i} \in I_{ε, a}^{(i)}}, \\ {\dot{Γ}}_{ε, a}^{(i)} : = Γ_{ε}^{(i)} \cap {x : x_{i} < 2 ℓ_{i}^{0} ε^{a}}, {\hat{Γ}}_{ε, a}^{(i)} : = Γ_{ε}^{(i)} \cap {x : x_{i} \in {\hat{I}}_{ε, a}^{(i)}}, \\ Γ_{ε, a}^{(i)} : = Γ_{ε}^{(i)} \cap {x : x_{i} \in I_{ε, a}^{(i)}}, \\ {\hat{I}}_{ε, a}^{(i)} : = (2 ℓ_{i}^{0} ε^{a}, 3 ℓ_{i}^{0} ε^{a}), I_{ε, a}^{(i)} : = (3 ℓ_{i}^{0} ε^{a}, ℓ_{i}), i \in {1, 2, 3} . \end{array}$

With the help of Green–Ostrogradsky formula $\begin{matrix} \int_{Υ_{i}} Δ_{{\overline{ξ}}_{i}} u v d {\overline{ξ}}_{i} = \int_{\partial Υ_{i}} \partial_{ν_{{\overline{ξ}}_{i}}} u v d σ_{{\overline{ξ}}_{i}} - \int_{Υ_{i}} \nabla_{{\overline{ξ}}_{i}} u \nabla_{{\overline{ξ}}_{i}} v d {\overline{ξ}}_{i}, \end{matrix}$ we rewrite $R_{ε}$ in the form $R_{ε} (v) = \sum_{j = 1}^{12} R_{ε, j} (v)$ , where $\begin{array}{l} R_{ε, 1} (v) = & \int_{Ω_{ε}} (k (U_{ε} (x, t)) - \sum_{i = 1}^{3} χ_{ℓ_{i}^{0}}^{(i)} (\frac{x_{i}}{ε^{a}}) (k (w_{0}^{(i)} (x_{i}, t)) ε^{α_{0}} + ε^{α_{0}} k^{'} (w_{0}^{(i)} (x_{i})) w_{α_{0}}^{(i)} (x_{i}) \\ + ε k^{'} (w_{0}^{(i)} (x_{i}, t)) w_{1}^{(i)} (x_{i}, t) + ε^{2 α_{0}} (k^{'} (w_{0}^{(i)} (x_{i})) w_{2 α_{0}}^{(i)} (x_{i}) + \frac{1}{2} k^{″} (w_{0}^{(i)} (x_{i})) {(w_{α_{0}}^{(i)} (x_{i}))}^{2}) \\ + ε^{1 + α_{0}} (k^{'} (w_{0}^{(i)} (x_{i})) w_{1 + α_{0}}^{(i)} - k^{″} (w_{0}^{(i)} (x_{i})) w_{α_{0}}^{(i)} (x_{i}) w_{1}^{(i)} (x_{i})) \\ + ε^{2} (k^{'} (w_{0}^{(i)} (x_{i})) (w_{2}^{(i)} (x_{i}) + u_{2} ({\overline{ξ}}_{i}, x_{i}, t)) - \frac{1}{2} k^{″} (w_{0}^{(i)} (x_{i})) {(w_{1}^{(i)} (x_{i}))}^{2}) \\ + ε^{1 + 2 α_{0}} (k^{'} (w_{0}^{(i)} (x_{i})) w_{1 + 2 α_{0}}^{(i)} (x_{i}) - k^{″} (w_{0}^{(i)} (x_{i})) w_{1}^{(i)} (x_{i}) w_{2 α_{0}}^{(i)} (x_{i}) \\ - k^{″} (w_{0}^{(i)} (x_{i})) w_{α_{0}}^{(i)} (x_{i}) w_{1 + α_{0}}^{(i)} (x_{i}) - k^{‴} (w_{0}^{(i)} (x_{i})) {(w_{α_{0}}^{(i)} (x_{i}))}^{2} w_{1}^{(i)} (x_{i}))) \\ - (1 - \sum_{i = 1}^{3} χ_{ℓ_{i}^{0}}^{(i)} (\frac{x_{i}}{ε^{a}})) k (w_{0}^{(i)} (0))) v (x) d x, \\ R_{ε, 2} (v) = & - \int_{Ω_{ε}} (f (x, t) - \sum_{i = 1}^{3} χ_{ℓ_{i}^{0}}^{(i)} (\frac{x_{i}}{ε^{a}}) (f_{0}^{(i)} (x_{i}, t) + ε f_{1}^{(i)} (\frac{{\overline{x}}_{i}}{ε}, x_{i}, t) + ε^{2} f_{2}^{(i)} (\frac{{\overline{x}}_{i}}{ε}, x_{i}, t)) \\ - (1 - \sum_{i = 1}^{3} χ_{ℓ_{i}^{0}}^{(i)} (\frac{x_{i}}{ε^{a}})) f (0, t)) v (x) d x, \\ R_{ε, 3} (v) = & ε^{α_{0}} \int_{Γ_{ε}^{(0)}} (κ_{0} (U_{ε} (x, t)) - κ_{0} (w_{0}^{(1)} (0, t)) - ε^{α_{0}} κ_{0}^{'} (w_{0}^{(1)} (0, t)) w_{α_{0}}^{(1)} (0, t)) v (x) d σ_{x}, \\ R_{ε, 4} (v) = & \sum_{i = 1}^{3} ε \int_{Γ_{ε}^{(i)}} (κ_{i} (U_{ε} (x, t)) - χ_{ℓ_{i}^{0}}^{(i)} (\frac{x_{i}}{ε^{a}}) (κ_{i} (w_{0}^{(i)} (x_{i}, t)) ε^{α_{0}} + ε^{α_{0}} κ_{i}^{'} (w_{0}^{(i)} (x_{i})) w_{α_{0}}^{(i)} (x_{i}) \\ + ε κ_{i}^{'} (w_{0}^{(i)} (x_{i}, t)) w_{1}^{(i)} (x_{i}, t) + ε^{2 α_{0}} (κ_{i}^{'} (w_{0}^{(i)} (x_{i})) w_{2 α_{0}}^{(i)} (x_{i}) + \frac{1}{2} κ_{i}^{″} (w_{0}^{(i)} (x_{i})) {(w_{α_{0}}^{(i)} (x_{i}))}^{2}) \\ + ε^{1 + α_{0}} (κ_{i}^{'} (w_{0}^{(i)} (x_{i})) w_{1 + α_{0}}^{(i)} - κ_{i}^{″} (w_{0}^{(i)} (x_{i})) w_{α_{0}}^{(i)} (x_{i}) w_{1}^{(i)} (x_{i})) \\ + ε^{2} (κ_{i}^{'} (w_{0}^{(i)} (x_{i})) (w_{2}^{(i)} (x_{i}) + u_{2} ({\overline{ξ}}_{i}, x_{i}, t)) - \frac{1}{2} κ_{i}^{″} (w_{0}^{(i)} (x_{i})) {(w_{1}^{(i)} (x_{i}))}^{2}) \\ + ε^{1 + 2 α_{0}} (κ_{i}^{'} (w_{0}^{(i)} (x_{i})) w_{1 + 2 α_{0}}^{(i)} (x_{i}) - κ_{i}^{″} (w_{0}^{(i)} (x_{i})) w_{1}^{(i)} (x_{i}) w_{2 α_{0}}^{(i)} (x_{i}) \\ - κ_{i}^{″} (w_{0}^{(i)} (x_{i})) w_{α_{0}}^{(i)} (x_{i}) w_{1 + α_{0}}^{(i)} (x_{i}) - κ_{i}^{‴} (w_{0}^{(i)} (x_{i})) {(w_{α_{0}}^{(i)} (x_{i}))}^{2} w_{1}^{(i)} (x_{i}))) \\ - (1 - χ_{ℓ_{i}^{0}}^{(i)} (\frac{x_{i}}{ε^{a}})) κ_{i} (w_{0}^{(i)} (0))) v (x) d σ_{x}, \\ R_{ε, 5} (v) = & - \sum_{i = 1}^{3} ε \int_{Γ_{ε}^{(i)}} χ_{ℓ_{i}^{0}}^{(i)} (\frac{x_{i}}{ε^{a}}) \frac{φ_{ε}^{(i)} (x, t)}{\sqrt{1 + ε^{2} | h_{i}^{'} (x_{i}) |^{2}}} \\ \times (\sqrt{1 + ε^{2} {| h_{i}^{'} (x_{i}) |}^{2}} - 1 - \frac{1}{2} ε^{2} {| h_{i}^{'} (x_{i}) |}^{2}) v (x) d σ_{x}, \\ R_{ε, 6} (v) = & - \sum_{i = 1}^{3} ε \int_{Γ_{ε}^{(i)}} (1 - χ_{ℓ_{i}^{0}}^{(i)} (\frac{x_{i}}{ε^{a}})) (φ^{(i)} (\frac{{\overline{x}}_{i}}{ε}, x_{i}, t) - φ^{(i)} (\frac{{\overline{x}}_{i}}{ε}, 0, t)) v (x) d σ_{x}, \\ R_{ε, 7} (v) = & \sum_{i = 1}^{3} ε \int_{Γ_{ε}^{(i)}} \frac{1}{\sqrt{1 + ε^{2} | h_{i}^{'} (x_{i}) |^{2}}} χ_{ℓ_{i}^{0}}^{(i)} (\frac{x_{i}}{ε^{a}}) ((ε^{α_{0}} (κ_{i}^{'} (w_{0}^{(i)} (x_{i})) w_{α_{0}}^{(i)} (x_{i}) \\ - \partial_{ν_{{\overline{ξ}}_{i}}} u_{2 + α_{0}} ({\overline{ξ}}_{i}, x_{i}, t)) + ε (κ_{i}^{'} (w_{0}^{(i)} (x_{i})) w_{1}^{(i)} (x_{i}) - \partial_{ν_{{\overline{ξ}}_{i}}} u_{3} ({\overline{ξ}}_{i}, x_{i}, t)) \\ + ε^{2 α_{0}} (κ_{i}^{'} (w_{0}^{(i)} (x_{i})) w_{2 α_{0}}^{(i)} (x_{i}) \\ - \frac{1}{2} κ_{i}^{″} (w_{0}^{(i)} (x_{i})) {(w_{α_{0}}^{(i)} (x_{i}))}^{2} - \partial_{ν_{{\overline{ξ}}_{i}}} u_{2 + 2 α_{0}} ({\overline{ξ}}_{i}, x_{i}, t)) \\ + ε^{1 + α_{0}} (κ_{i}^{'} (w_{0}^{(i)} (x_{i})) w_{1 + α_{0}}^{(i)} - κ_{i}^{″} (w_{0}^{(i)} (x_{i})) w_{α_{0}}^{(i)} (x_{i}) w_{1}^{(i)} (x_{i}) - \partial_{ν_{{\overline{ξ}}_{i}}} u_{3 + α_{0}} ({\overline{ξ}}_{i}, x_{i}, t)) \\ + ε^{2} (κ_{i}^{'} (w_{0}^{(i)} (x_{i})) (w_{2}^{(i)} (x_{i}) + u_{2} ({\overline{ξ}}_{i}, x_{i}, t)) \\ - \frac{1}{2} κ_{i}^{″} (w_{0}^{(i)} (x_{i})) {(w_{1}^{(i)} (x_{i}))}^{2} - \partial_{ν_{{\overline{ξ}}_{i}}} u_{4} ({\overline{ξ}}_{i}, x_{i}, t)) \\ + ε^{1 + 2 α_{0}} (κ_{i}^{'} (w_{0}^{(i)} (x_{i})) w_{1 + 2 α_{0}}^{(i)} (x_{i}) \\ - κ_{i}^{″} (w_{0}^{(i)} (x_{i})) w_{1}^{(i)} (x_{i}) w_{2 α_{0}}^{(i)} (x_{i}) - κ_{i}^{″} (w_{0}^{(i)} (x_{i})) w_{α_{0}}^{(i)} (x_{i}) w_{1 + α_{0}}^{(i)} (x_{i}) \\ - κ_{i}^{‴} (w_{0}^{(i)} (x_{i})) {(w_{α_{0}}^{(i)} (x_{i}))}^{2} w_{1}^{(i)} (x_{i}) \\ - \partial_{ν_{{\overline{ξ}}_{i}}} u_{3 + 2 α_{0}} ({\overline{ξ}}_{i}, x_{i}, t))) |_{\overline{ξ} = \frac{{\overline{x}}_{i}}{ε}} (\sqrt{1 + ε^{2} {| h_{i}^{'} (x_{i}) |}^{2}} - 1) \\ + κ_{i} (w_{0}^{(i)} (x_{i})) (\sqrt{1 + ε^{2} {| h_{i}^{'} (x_{i}) |}^{2}} - 1 - \frac{1}{2} ε^{2} {| h_{i}^{'} (x_{i}) |}^{2})) v (x) d σ_{x}, \\ R_{ε, 8} (v) = & \int_{Ω_{ε}} (1 - \sum_{i = 1}^{3} χ_{ℓ_{i}^{0}}^{(i)} (\frac{x_{i}}{ε^{a}})) \frac{\partial}{\partial t} (ε^{α_{0}} w_{α_{0}}^{(i)} (0, t) + ε^{2 α_{0}} w_{2 α_{0}}^{(i)} (0, t) \\ + \sum_{n \in {1, 1 + α_{0}, 2, 1 + 2 α_{0}}} ε^{n} \partial N_{n} (\frac{x}{ε}, t)) v (x) d x, \\ R_{ε, 9} (v) = & - \sum_{n \in A ∖ {0}} ε^{2 + n} \sum_{i = 1}^{3} \int_{I_{ε}^{(i)}} \int_{Υ_{i} (x_{i})} χ_{ℓ_{i}^{0}}^{(i)} (\frac{x_{i}}{ε^{a}}) \nabla_{{\overline{ξ}}_{i}} u_{n + 2}^{(i)} ({\overline{ξ}}_{i}, x_{i}, t) \cdot \nabla_{{\overline{ξ}}_{i}} v (x, t) d {\overline{ξ}}_{i} d x_{i}, \\ R_{ε, 10} (v) = & - \sum_{i = 1}^{3} \int_{{\hat{Ω}}_{ε, a}^{(i)}} (ε^{- 2 a} \frac{d^{2} χ_{ℓ_{i}^{0}}^{(i)}}{d ζ_{i}^{2}} (ζ_{i}) + 2 ε^{- a} \frac{d χ_{ℓ_{i}^{0}}^{(i)}}{d ζ_{i}} (ζ_{i}) \frac{\partial}{\partial x_{i}}) |_{ζ_{i} = \frac{x_{i}}{ε^{a}}} (w_{0}^{(i)} (x_{i}, t) - w_{0}^{(i)} (0, t) \\ - x_{i} \frac{\partial w_{0}^{(i)}}{\partial x_{i}} (0, t) - \frac{x_{i}^{2}}{2} \frac{\partial^{2} w_{0}^{(i)}}{\partial x_{i}^{2}} (0, t) + ε^{α_{0}} (w_{α_{0}}^{(i)} (x_{i}, t) - w_{α_{0}}^{(i)} (0, t) - x_{i} \frac{\partial w_{α_{0}}^{(i)}}{\partial x_{i}} (0, t)) \\ + ε (w_{1}^{(i)} (x_{i}, t) - w_{1}^{(i)} (0, t) - x_{i} \frac{\partial w_{1}^{(i)}}{\partial x_{i}} (0, t)) + ε^{2 α_{0}} (w_{2 α_{0}}^{(i)} (x_{i}, t) - w_{2 α_{0}}^{(i)} (0, t) \\ - x_{i} \frac{\partial w_{2 α_{0}}^{(i)}}{\partial x_{i}} (0, t)) + ε^{1 + α_{0}} (w_{1 + α_{0}}^{(i)} (x_{i}, t) - w_{1 + α_{0}}^{(i)} (0, t)) \\ + ε^{2} (w_{2}^{(i)} (x_{i}, t) + u_{2}^{(i)} (\frac{x}{ε}, x_{i}, t) \\ - w_{2}^{(i)} (0, t) - u_{2}^{(i)} (\frac{x}{ε}, 0, t)) + ε^{1 + 2 α_{0}} (w_{1 + 2 α_{0}}^{(i)} (x_{i}, t) - w_{1 + 2 α_{0}}^{(i)} (0, t))) v (x) d x, \\ R_{ε, 11} (v) = & - \sum_{i = 1}^{3} \int_{{\hat{Ω}}_{ε, a}^{(i)}} (ε^{- 2 a} \frac{d^{2} χ_{ℓ_{i}^{0}}^{(i)}}{d ζ_{i}^{2}} (ζ_{i}) + 2 ε^{- a} \frac{d χ_{ℓ_{i}^{0}}^{(i)}}{d ζ_{i}} (ζ_{i}) \frac{\partial}{\partial ξ_{i}}) |_{ζ_{i} = \frac{x_{i}}{ε^{a}}} \\ \times \sum_{n \in {1, 1 + α_{0}, 2, 1 + 2 α_{0}}} ε^{n} (N_{n} (ξ, t) - G_{n} (ξ, t)) |_{ξ = \frac{x}{ε}} v (x) d x, \\ R_{ε, 12} (v) = & - \sum_{i = 1}^{3} \int_{Ω_{ε}^{(i)}} (- ε^{2} χ_{δ}^{(i)} (x_{i}) \frac{\partial}{\partial t} + ε^{2} \frac{d^{2} χ_{δ}^{(i)}}{d x_{i}^{2}} (x_{i}) - ε \frac{d χ_{δ}^{(i)}}{d x_{i}} (x_{i}) \frac{\partial}{\partial ξ_{i}^{}}) \\ \times Π_{2}^{(i)} (\frac{{\overline{x}}_{i}}{ε}, ξ_{i}^{}, t) |_{ξ_{i}^{} = \frac{ℓ_{i} - x_{i}}{ε}} v (x) d x . \end{array}$ Let us estimate the value $R_{ε}$ . Using (2.7), (2.9) and (2.6), we deduce the following estimates: $\begin{array}{l} | R_{ε, 1} (v) | ⩽ {\overset{ˇ}{C}}_{0} (ε^{2 α_{0}} \sum_{i = 1}^{3} \sqrt{{| Ω_{ε, a}^{(i)} |}_{3}} + ε^{min (α_{0}, a)} \sum_{i = 1}^{3} \sqrt{{| {\hat{Ω}}_{ε, a}^{(i)} |}_{3}} + ε^{α_{0}} \sqrt{{| Ω_{ε, a}^{(0)} |}_{3}}) ‖ v ‖_{L^{2} (Ω_{ε})} \\ (3.38) & ⩽ \overset{ˇ}{C} ε^{1 + min (3 α_{0}, α_{0} + \frac{a}{2}, \frac{3}{2} a)} ‖ v ‖_{H^{1} (Ω_{ε})}, \\ | R_{ε, 2} (v) | ⩽ {\overset{ˇ}{C}}_{0} (ε^{2} \sum_{i = 1}^{3} \sqrt{{| Ω_{ε, a}^{(i)} |}_{3}} + ε^{a} \sum_{i = 1}^{3} \sqrt{{| {\hat{Ω}}_{ε, a}^{(i)} |}_{3}} + ε^{a} \sqrt{{| Ω_{ε, a}^{(0)} |}_{3}}) ‖ v ‖_{L^{2} (Ω_{ε})} \\ (3.39) & ⩽ \overset{ˇ}{C} ε^{1 + \frac{3}{2} a} ‖ v ‖_{H^{1} (Ω_{ε})}, \\ (3.40) & | R_{ε, 3} (v) | ⩽ {\overset{ˇ}{C}}_{0} ε^{α_{0} + min (1, 2 α_{0})} \sqrt{{| Γ_{ε}^{(0)} |}_{2}} ‖ v ‖_{L^{2} (Γ_{ε}^{(0)})} ⩽ {\overset{ˇ}{C}}_{0} ε^{1 + α_{0} + min (1, 2 α_{0})} ‖ v ‖_{H^{1} (Ω_{ε})}, \\ | R_{ε, 4} (v) | ⩽ {\overset{ˇ}{C}}_{0} \sum_{i = 1}^{3} ε (ε^{2 α_{0}} \sqrt{{| Γ_{ε, a}^{(i)} |}_{2}} + ε^{min (α_{0}, a)} \sqrt{{| {\hat{Γ}}_{ε, a}^{(i)} |}_{2}} + ε^{α_{0}} \sqrt{{| {\dot{Γ}}_{ε, a}^{(i)} |}_{2}}) ‖ v ‖_{L^{2} (Γ_{ε}^{(i)})} \\ (3.41) & ⩽ \overset{ˇ}{C} ε^{1 + min (3 α_{0}, α_{0} + \frac{a}{2}, \frac{3}{2} a)} ‖ v ‖_{H^{1} (Ω_{ε})}, \\ (3.42) & | R_{ε, 5} (v) | ⩽ {\overset{ˇ}{C}}_{0} \sum_{i = 1}^{3} ε^{5} \sqrt{{| Γ_{ε}^{(i)} |}_{2}} ‖ v ‖_{L^{2} (Γ_{ε}^{(i)})} ⩽ \overset{ˇ}{C} ε^{5} ‖ v ‖_{H^{1} (Ω_{ε})}, \\ (3.43) & | R_{ε, 6} (v) | ⩽ {\overset{ˇ}{C}}_{0} ε^{1 + a} \sum_{i = 1}^{3} (\sqrt{{| {\hat{Γ}}_{ε, a}^{(i)} |}_{2} + {| {\dot{Γ}}_{ε, a}^{(i)} |}_{2}}) ‖ v ‖_{L^{2} (Γ_{ε}^{(i)})} ⩽ \overset{ˇ}{C} ε^{1 + \frac{3}{2} a} ‖ v ‖_{H^{1} (Ω_{ε})}, \\ (3.44) & | R_{ε, 7} (v) | ⩽ {\overset{ˇ}{C}}_{0} ε (ε^{2 + α_{0}} + ε^{4}) \sum_{i = 1}^{3} \sqrt{{| Γ_{ε}^{(i)} |}_{2}} ‖ v ‖_{L^{2} (Γ_{ε}^{(i)})} ⩽ \overset{ˇ}{C} ε^{3 + α_{0}} ‖ v ‖_{H^{1} (Ω_{ε})}, \\ (3.45) & | R_{ε, 8} (v) | ⩽ {\overset{ˇ}{C}}_{0} ε^{α_{0}} \sqrt{{| Ω_{ε, a}^{(0)} |}_{3} + \sum_{i = 1}^{3} {| {\hat{Ω}}_{ε, a}^{(i)} |}_{3}} ‖ v ‖_{L^{2} (Ω_{ε})} ⩽ \overset{ˇ}{C} ε^{1 + α_{0} + \frac{a}{2}} ‖ v ‖_{H^{1} (Ω_{ε})}, \\ (3.46) & | R_{ε, 9} (v) | ⩽ \overset{ˇ}{C} ε^{2 + α_{0}} ‖ \nabla_{x} v ‖_{L^{2} (Ω_{ε})}, \\ (3.47) & | R_{ε, 10} (v) | ⩽ {\overset{ˇ}{C}}_{0} (ε^{a} + ε^{α_{0}}) \sum_{i = 1}^{3} \sqrt{{| {\hat{Ω}}_{ε, a}^{(i)} |}_{3}} ‖ v ‖_{L^{2} (Ω_{ε, a}^{(i)})} ⩽ \overset{ˇ}{C} ε^{1 + \frac{a}{2} + min (α_{0}, a)} ‖ v ‖_{H^{1} (Ω_{ε})} . \end{array}$ Due to the exponential decreasing of functions ${N_{n} - G_{n}}$ (see Remark 3.3), we arrive that $\begin{array}{l} | R_{ε, 11} (v) | & ⩽ {\overset{ˇ}{C}}_{0} ε^{- 1 - a} \sum_{i = 1}^{3} \sqrt{{| {\hat{Ω}}_{ε, a}^{(i)} |}_{3}} exp (- \frac{2 ℓ_{i}^{0}}{ε^{1 - a}} γ_{i}) ‖ v ‖_{L^{2} (Ω_{ε})} \\ (3.48) & ⩽ \overset{ˇ}{C} ε^{- \frac{a}{2}} exp (- \frac{2 ℓ_{i}^{0}}{ε^{1 - a}} min_{i = 1, 2, 3} γ_{i}) ‖ v ‖_{H^{1} (Ω_{ε})} . \end{array}$ Taking into account Remark 3.1, we get $\begin{matrix} (3.49) & | R_{ε, 12} (v) | ⩽ \overset{ˇ}{C} ε δ^{\frac{1}{2}} (ε^{2} + exp (- \frac{ε}{δ} min_{i = 1, 2, 3} λ_{1}^{(i)})) ‖ v ‖_{L^{2} (Ω_{ε})} . \end{matrix}$ Subtracting the identity (2.11) from (3.37) and integrating over $t \in (0, τ)$ , where $τ \in (0, T]$ , we obtain $\begin{matrix} (3.50) & \int_{0}^{τ} ({⟨ \partial_{t} U_{ε} - \partial_{t} u_{ε}, v ⟩}_{ε} + {⟨ A_{ε} (t) U_{ε} - A_{ε} (t) u_{ε}, v ⟩}_{ε}) d t = \int_{0}^{τ} R_{ε} (v) d t \forall v \in L^{2} (0, T; H_{ε}) . \end{matrix}$ Now set $v = U_{ε} - u_{ε}$ in (3.50). Then, taking into account that $A_{ε}$ is strictly monotone and (3.38)–(3.49), we arrive to the inequality $\begin{matrix} {‖ U_{ε} (\cdot, τ) - u_{ε} (\cdot, τ) ‖}_{L^{2} (Ω_{ε})}^{2} + ‖ U_{ε} - u_{ε} ‖_{L^{2} (0, τ; H_{ε})}^{2} ⩽ C ε^{1 + min (3 α_{0}, α_{0} + \frac{a}{2}, \frac{3}{2} a)} ‖ U_{ε} - u_{ε} ‖_{L^{2} (0, τ; H^{1} (Ω_{ε}))}, \end{matrix}$ whence thanks to (2.10) it follows (3.35). □
Theorem 3.1 (The case $α_{0} \in (0, 1)$ ).

Let conditions C1, C2, C3 be satisfied. Then the sum $\begin{array}{l} U_{ε}^{(α_{0})} (x, t) = & \sum_{i = 1}^{3} χ_{ℓ_{i}^{0}}^{(i)} (\frac{x_{i}}{ε^{a}}) (w_{0}^{(i)} (x_{i}, t) + ε^{α_{0}} w_{α_{0}}^{(i)} (x_{i}, t)) \\ + (1 - \sum_{i = 1}^{3} χ_{ℓ_{i}^{0}}^{(i)} (\frac{x_{i}}{ε^{a}})) (w_{0}^{(1)} (0, t) + ε^{α_{0}} w_{α_{0}}^{(1)} (0, t)), x \in Ω_{ε} \times (0, T) \end{array}$ is the asymptotic approximation for the solution $u_{ε}$ to the boundary-value problem ( 2.1 ), i.e., $\exists C_{1} > 0$ $\exists ε_{0} > 0$ $\forall ε \in (0, ε_{0})$ : $\begin{array}{l} (3.51) & \frac{1}{\sqrt{| Ω_{ε} |_{3}}} {‖ u_{ε} - U_{ε}^{(α_{0})} ‖}_{L^{2} (0, T; H^{1} (Ω_{ε}))} ⩽ C_{1} ε^{min (2 α_{0}, \frac{a}{2})}, \\ (3.52) & \frac{1}{\sqrt{| Ω_{ε} |_{3}}} max_{t \in [0, T]} {‖ u_{ε} (\cdot, t) - U_{ε}^{(α_{0})} (\cdot, t) ‖}_{L^{2} (Ω_{ε})} ⩽ C_{1} ε^{min (2 α_{0}, α_{0} + \frac{a}{2}, 1)}, \end{array}$ where $a$ is a fixed number from the interval $(\frac{2}{3}, 1)$ .

Proof.
Denote by $χ_{ℓ_{i}^{0}, a, ε}^{(i)} (\cdot) : = χ_{ℓ_{i}^{0}}^{(i)} (\frac{\cdot}{ε^{a}})$ (the function $χ_{ℓ_{i}^{0}}^{(i)}$ is determined in (3.33)) and $\begin{matrix} ‖ v ‖_{Ω}^{} : = max_{t \in [0, T]} {‖ v (\cdot, t) ‖}_{L^{2} (Ω)} + ‖ v ‖_{L^{2} (0, T; H^{1} (Ω))} . \end{matrix}$ Using the smoothness of the functions ${w_{1}^{(i)}}_{i = 1}^{3}$ and the exponential decay of the functions ${N_{n} - G_{n}}$ , ${Π_{2}^{(i)}}_{i = 1}^{3}$ at infinity, we deduce the inequality (3.51) from estimate (3.35), namely $\begin{array}{l} {‖ u_{ε} - U_{ε}^{(α_{0})} ‖}_{Ω_{ε}}^{} \\ ⩽ ‖ u_{ε} - U_{ε} ‖_{Ω_{ε}}^{} \\ + ‖ \sum_{n \in {1, 2 α_{0}, 1 + α_{0}, 1 + 2 α_{0}}} ε^{n} (\sum_{i = 1}^{3} χ_{ℓ_{i}^{0}, a, ε}^{(i)} w_{n}^{(i)} + (1 - \sum_{i = 1}^{3} χ_{ℓ_{i}^{0}, a, ε}^{(i)}) N_{n}) \\ + ε^{2} (\sum_{i = 1}^{3} (χ_{ℓ_{i}^{0}, a, ε}^{(i)} (u_{2}^{(i)} + w_{2}^{(i)}) + χ_{δ}^{(i)} Π_{2}^{(i)}) + (1 - \sum_{i = 1}^{3} χ_{ℓ_{i}^{0}, a, ε}^{(i)}) N_{2}) ‖_{Ω_{ε}}^{} \\ ⩽ C_{0} ε^{1 + min (3 α_{0}, α_{0} + \frac{a}{2}, \frac{3}{2} a)} + \sum_{i = 1}^{3} ‖ \sum_{n \in {1, 2 α_{0}, 1 + α_{0}, 1 + 2 α_{0}}} ε^{n} (χ_{ℓ_{i}^{0}, a, ε}^{(i)} w_{n}^{(i)} + (1 - χ_{ℓ_{i}^{0}, a, ε}^{(i)}) N_{n}) \\ + ε^{2} (χ_{ℓ_{i}^{0}, a, ε}^{(i)} (u_{2}^{(i)} + w_{2}^{(i)}) + (1 - χ_{ℓ_{i}^{0}, a, ε}^{(i)}) N_{2} + χ_{δ}^{(i)} Π_{2}^{(i)}) ‖_{Ω_{ε}^{(i)}}^{} \\ + ε^{2 α_{0}} {‖ w_{2 α_{0}}^{(1)} (0, \cdot) ‖}_{Ω_{ε}^{(0)}}^{} + \sum_{n \in {1, 1 + α_{0}, 1 + 2 α_{0}}} ε^{n} ‖ N_{n} ‖_{Ω_{ε}^{(0)}}^{} ⩽ C_{0} ε^{1 + min (3 α_{0}, α_{0} + \frac{a}{2}, \frac{3}{2} a)} \\ + \sum_{i = 1}^{3} ({‖ (1 - χ_{ℓ_{i}^{0}, a, ε}^{(i)}) (x_{i} \frac{\partial w_{0}^{(i)}}{\partial x_{i}} (0, \cdot) + \frac{x_{i}^{2}}{2} \frac{\partial^{2} w_{0}^{(i)}}{\partial x_{i}^{2}} (0, \cdot)) ‖}_{Ω_{ε}^{(i)}}^{} \\ + ε^{α_{0}} {‖ (1 - χ_{ℓ_{i}^{0}, a, ε}^{(i)}) x_{i} \frac{\partial w_{α_{0}}^{(i)}}{\partial x_{i}} (0, \cdot) ‖}_{Ω_{ε}^{(i)}}^{} \\ + \sum_{n \in {1, 2 α_{0}}} ε^{n} {‖ (1 - χ_{ℓ_{i}^{0}, a, ε}^{(i)}) (w_{n}^{(i)} (0, \cdot) + x_{i} \frac{\partial w_{n}^{(i)}}{\partial x_{i}} (0, \cdot) - w_{n}^{(i)}) ‖}_{Ω_{ε}^{(i)}}^{} \\ + ε^{1 + α_{0}} {‖ (1 - χ_{ℓ_{i}^{0}, a, ε}^{(i)}) (w_{1 + α_{0}}^{(i)} (0, \cdot) - w_{1 + α_{0}}^{(i)}) ‖}_{Ω_{ε}^{(i)}}^{} \\ + ε^{1 + 2 α_{0}} {‖ (1 - χ_{ℓ_{i}^{0}, a, ε}^{(i)}) (w_{1 + 2 α_{0}}^{(i)} (0, \cdot) - w_{1 + 2 α_{0}}^{(i)}) ‖}_{Ω_{ε}^{(i)}}^{} \\ + ε^{2} {‖ (1 - χ_{ℓ_{i}^{0}, a, ε}^{(i)}) (u_{2}^{(i)} (\cdot, 0, \cdot) + w_{2}^{(i)} (0, \cdot) - u_{2}^{(i)} - w_{2}^{(i)}) ‖}_{Ω_{ε}^{(i)}}^{} \\ + \sum_{n \in {1, 1 + α_{0}, 2, 1 + 2 α_{0}}} ε^{n} {‖ (1 - χ_{ℓ_{i}^{0}, a, ε}^{(i)}) (N_{n} - G_{n}) ‖}_{Ω_{ε}^{(i)}}^{} + ε^{2} {‖ χ_{δ}^{(i)} Π_{2}^{(i)} ‖}_{Ω_{ε}^{(i)}}^{} \\ + \sum_{n \in {1, 2 α_{0}, 1 + α_{0}, 1 + 2 α_{0}}} ε^{n} {‖ w_{n}^{(i)} ‖}_{Ω_{ε}^{(i)}}^{} + ε^{2} {‖ u_{2}^{(i)} + w_{2}^{(i)} ‖}_{Ω_{ε}^{(i)}}^{}) + ε^{\frac{3}{2} + 2 α_{0}} {‖ w_{2 α_{0}}^{(1)} (0, \cdot) ‖}_{Ξ^{(0)}}^{} \\ + \sum_{n \in {1, 1 + α_{0}, 2, 1 + 2 α_{0}}} (ε^{\frac{3}{2} + n} ‖ N_{n} ‖_{C (0, T; L^{2} (Ξ^{(0)}))} + ε^{\frac{1}{2} + n} ‖ N_{n} ‖_{L (0, T; H^{1} (Ξ^{(0)}))}) ⩽ {\overset{˘}{C}}_{1} ε^{1 + min (2 α_{0}, \frac{a}{2})} . \end{array}$ The main contribution to the right hand-side of this inequality is brought with the following summands: $(1 - χ_{ℓ_{i}^{0}, a, ε}^{(i)}) x_{i} \frac{\partial w_{0}^{(i)}}{\partial x_{i}} (0, \cdot)$ and $ε^{2 α_{0}} w_{2 α_{0}}^{(i)}$ whose norms are estimated by the values ${\overset{˘}{c}}_{1} ε^{1 + \frac{a}{2}}$ and ${\overset{˘}{c}}_{1} ε^{1 + 2 α_{0}}$ , respectively.

In order to get the estimate (3.52), we need to replace the norm $‖ \cdot ‖_{Ω_{ε}}^{}$ to the norm in the space $C ([0, T]; L^{2} (Ω))$ at the beginning of the proof. □
Corollary 3.1.
In each thin cylinder* $Ω_{ε, a}^{(i)} : = Ω_{ε}^{(i)} \cap {x \in R^{3} : x_{i} \in I_{ε, a}^{(i)} : = (3 ℓ_{i}^{0} ε^{a}, ℓ_{i})} (i = 1, 2, 3)$ , the following estimate holds: $\begin{array}{l} \frac{1}{\sqrt{| Ω_{ε, a}^{(i)} |_{3}}} (max_{t \in [0, T]} {‖ u_{ε} (\cdot, t) - w_{0}^{(i)} (\cdot, t) - ε^{α_{0}} w_{α_{0}}^{(i)} (\cdot, t) ‖}_{L^{2} (Ω_{ε, a}^{(i)})} \\ (3.53) & + {‖ u_{ε} - w_{0}^{(i)} - ε^{α_{0}} w_{α_{0}}^{(i)} ‖}_{L^{2} (0, T; H^{1} (Ω_{ε, a}^{(i)}))}) ⩽ {\tilde{C}}_{1} ε^{min (2 α_{0}, α_{0} + \frac{a}{2}, 1)}, \end{array}$ where ${w_{0}^{(i)}}_{i = 1}^{3}$ is the solution to the limit problem ( 3.20 ), ${w_{α_{0}}^{(i)}}_{i = 1}^{3}$ is the solution to the problem ( 3.29 ).

In the neighborhood $Ω_{ε, a}^{(0)} : = Ω_{ε} \cap {x : x_{i} < 2 ℓ_{i}^{0} ε^{a}, i = 1, 2, 3}$ of the node $Ω_{ε}^{(0)}$ , we get $\begin{matrix} (3.54) & \frac{1}{\sqrt{| Ω_{ε, a}^{(0)} |_{3}}} ‖ \nabla_{x} u_{ε} - \nabla_{ξ} N_{1} ‖_{L^{2} (Ω_{ε, a}^{(0)} \times (0, T))} ⩽ {\tilde{C}}_{2} ε^{min (3 α_{0}, α_{0} + \frac{a}{2}, \frac{3}{2} a) - \frac{a}{2}}, \end{matrix}$ where $N_{1}$ is the solution to the problem $\begin{matrix} (3.55) & \{\begin{matrix} - Δ_{ξ} N_{1} (ξ, t) = 0, ξ \in Ξ, \\ \partial_{ν_{ξ}} N_{1} (ξ, t) = 0, ξ \in \partial Ξ, \\ N_{1} (ξ, t) \sim w_{1}^{(i)} (0, t) + ξ_{i} \frac{\partial w_{0}^{(i)}}{\partial x_{i}} (0, t), ξ_{i} \to + \infty, ξ \in Ξ^{(i)}, i = 1, 2, 3 . \end{matrix} \end{matrix}$
Proof.
With the help of estimate (3.35), we derive $\begin{array}{l} {‖ u_{ε} - w_{0}^{(i)} - ε^{α_{0}} w_{α_{0}}^{(i)} ‖}_{Ω_{ε, a}^{(i)}}^{} \\ ⩽ ‖ u_{ε} - U_{ε} ‖_{Ω_{ε}}^{} \\ + \sum_{n \in {1, 2 α_{0}, 1 + α_{0}, 1 + 2 α_{0}}} ε^{n} {‖ w_{n}^{(i)} ‖}_{Ω_{ε, a}^{(i)}}^{} + ε^{2} {‖ w_{2}^{(i)} + u_{2}^{(i)} + χ_{δ}^{(i)} Π_{2}^{(i)} ‖}_{Ω_{ε, a}^{(i)}}^{} ⩽ {\tilde{c}}_{1} ε^{1 + min (2 α_{0}, α_{0} + \frac{a}{2}, 1)}, \end{array}$ whence we get (3.53). From inequality $\begin{array}{l} ‖ \nabla_{x} u_{ε} - \nabla_{ξ} N_{1} ‖_{L^{2} (Ω_{ε, a}^{(0)} \times (0, T))} \\ ⩽ {‖ u_{ε} - w_{0}^{(1)} (0, \cdot) - ε^{α_{0}} w_{α_{0}}^{(1)} (0, \cdot) - ε N_{1} ‖}_{Ω_{ε, a}^{(0)}}^{} \\ ⩽ ‖ u_{ε} - U_{ε} ‖_{Ω_{ε}}^{} + ε^{2 α_{0}} {‖ w_{2 α_{0}}^{(1)} (0, \cdot) ‖}_{Ω_{ε, a}^{(0)}}^{} + \sum_{n \in {1 + α_{0}, 2, 1 + 2 α_{0}}} ‖ N_{n} ‖_{Ω_{ε, a}^{(0)}}^{} ⩽ {\tilde{c}}_{2} ε^{1 + min (3 α_{0}, α_{0} + \frac{a}{2}, \frac{3}{2} a)} \end{array}$ it follows the estimates (3.54). □

Using the Cauchy–Buniakovskii–Schwarz inequality and the continuous embedding of the space $H^{1} (I_{ε, a}^{(i)})$ in $C ([3 ℓ_{i}^{0} ε^{a}, ℓ_{i}])$ , it follows from (3.53) the following corollary. Corollary 3.2.
If for some $i \in {1, 2, 3}$ the function $h_{i} (x_{i}) \equiv c o n s t$ , then $\begin{array}{l} max_{t \in [0, T]} {‖ (E_{ε}^{(i)} u_{ε}) (\cdot, t) - w_{0}^{(i)} (\cdot, t) - ε^{α_{0}} w_{α_{0}}^{(i)} (\cdot, t) ‖}_{L^{2} (I_{ε, a}^{(i)})} \\ (3.56) & + {‖ E_{ε}^{(i)} u_{ε} - w_{0}^{(i)} - ε^{α_{0}} w_{α_{0}}^{(i)} ‖}_{L^{2} (0, T; C ([3 ℓ_{i}^{0} ε^{a}, ℓ_{i}]))} ⩽ C_{2} ε^{min (2 α_{0}, α_{0} + \frac{a}{2}, 1)}, \end{array}$ where $\begin{matrix} (E_{ε}^{(i)} u_{ε}) (x_{i}, t) = \frac{1}{π ε^{2} h_{i}^{2}} \int_{Υ_{ε}^{(i)} (0)} u_{ε} (x, t) d {\overline{x}}_{i} . \end{matrix}$

4. Asymptotic approximation in the case $α_{0} = 1$

In this case we propose ansatzes of the asymptotic approximation for the solution to the problem (2.1) in the following form:

the regular part of the approximation $\begin{array}{l} w_{0}^{(i)} (x_{i}, t) + ε w_{1}^{(i)} (x_{i}, t) + ε^{2} w_{2}^{(i)} (x_{i}, t) + ε^{2} u_{2}^{(i)} (\frac{{\overline{x}}_{i}}{ε}, x_{i}, t) \\ (4.1) & + ε^{3} u_{3}^{(i)} (\frac{{\overline{x}}_{i}}{ε}, x_{i}, t) + ε^{4} u_{4}^{(i)} (\frac{{\overline{x}}_{i}}{ε}, x_{i}, t) \end{array}$ is located inside of each thin cylinder $Ω_{ε}^{(i)}$ $(i = 1, 2, 3)$ ;

the boundary-layer part of the approximation $\begin{matrix} (4.2) & ε^{2} Π_{2}^{(i)} (\frac{{\overline{x}}_{i}}{ε}, \frac{ℓ_{i} - x_{i}}{ε}, t) \end{matrix}$ is located in a neighborhood of the base $Υ_{ε}^{(i)} (ℓ_{i})$ of the thin cylinder $Ω_{ε}^{(i)} (i = 1, 2, 3)$ ;

and the inner part of the approximation $\begin{matrix} (4.3) & N_{0} (\frac{x}{ε}, t) + ε N_{1} (\frac{x}{ε}, t) + ε^{2} N_{2} (\frac{x}{ε}, t) + ε^{3} N_{3} (\frac{x}{ε}, t) \end{matrix}$ is located in a neighborhood of the node $Ω_{ε}^{(0)}$ .

Similarly as was done in Section 3.1, we obtain the linear inhomogeneous Neumann boundary-value problems to define coefficients $u_{n + 2}^{(i)}, n \in {0, 1, 2}$ , of the regular part (4.1), namely problem (3.4) and problems (3.6) at $n \in {1, 2}$ for each $i \in {1, 2, 3}$ with $f_{1}$ and $f_{2}$ defined in (3.7)). Writing down the necessary and sufficient conditions for the solvability of those problems, we derive the corresponding differential equations for ${w_{n}^{(i)}}_{i = 1}^{3}, n \in {0, 1, 2}$ .

The coefficient $Π_{2}^{(i)}$ is defined in the Section 3.2 $(i \in {1, 2, 3})$ .

The coefficients ${N_{n}}_{n = 0}^{3}$ of the inner part (4.3) are solutions to the problems (3.11) (see also problems (3.13)) at $n \in {0, 1, 2, 3}$ with the same functions in the right-hand sides except of ${B_{n}^{(0)}}_{n = 0}^{3}$ : $\begin{array}{l} B_{n}^{(0)} (ξ, t) \equiv {\tilde{B}}_{n}^{(0)} (ξ, t) \equiv 0, n \in {0, 1}, \\ B_{2}^{(0)} (ξ, t) = {\tilde{B}}_{2}^{(0)} (ξ, t) = - κ_{0} (N_{0}) + φ^{(0)} (ξ, t), B_{3}^{(0)} (ξ, t) = {\tilde{B}}_{3}^{(0)} (ξ, t) = - κ_{0}^{'} (N_{0}) N_{1} . \end{array}$

Similarly as in the Section 3.3.1 we obtain (3.18) and $N_{0} \equiv {\tilde{N}}_{0} \equiv w_{0}^{(1)} (0, t)$ . From the solvability condition (3.14) for the problem (3.13) at $n = 1$ , we get Kirchhoff transmission condition (3.19). Thus, taking into account (3.5) and (3.8), we arrive at the same limit problem (3.20) to define ${w_{0}^{(i)}}_{i = 1}^{3}$ .

Taking into account the third equality both in the problem (3.4) and in (3.6), from the solvability condition (3.14) at $n \in {2, 3}$ we deduce the following Kirchhoff transmission condition for ${w_{n}^{(i)}}_{i = 1}^{3}$ : $\begin{matrix} \sum_{i = 1}^{3} π h_{i}^{2} (0) \frac{\partial w_{n}^{(i)}}{\partial x_{i}} (0, t) = d_{n}^{*} (t), n \in {1, 2}, \end{matrix}$ where $\begin{array}{l} d_{1}^{*} (t) = & ({| Ξ^{(0)} |}_{3} - \sum_{i = 1}^{3} ℓ_{i}^{0} π h_{i}^{2} (0)) (\partial_{t} w_{0}^{(1)} (0, t) + k (w_{0}^{(1)} (0, t)) - f (0, t)) + | Γ_{0} |_{2} κ_{0} (w_{0}^{(1)} (0, t)) \\ (4.4) & - \int_{Γ_{0}} φ^{(0)} (ξ, t) d σ_{ξ} - \sum_{i = 1}^{3} ℓ_{i}^{0} (2 π h_{i} (0) κ_{i} (w_{0}^{(1)} (0, t)) - \int_{\partial Υ_{i} (0)} φ^{(i)} ({\overline{ξ}}_{i}, 0, t) d l_{{\overline{ξ}}_{i}}) . \end{array}$

Repeating the previous reasoning, we get a solution $N_{n}$ of the problem (3.11) with the asymptotics (3.28) (see also Remark 3.3). The next term ${\tilde{w}}_{1}$ of the regular asymptotics are determined from the problem (3.30) at $n = 1$ , where the right-hand side ${\hat{F}}_{1}^{(i)}$ , the values $δ_{1}^{(2)}$ , $δ_{1}^{(3)}$ and $d_{1}^{*}$ are uniquely determined by formulas (3.31), (3.32) and (4.4), respectively. By the same way we can determine the term ${\tilde{w}}_{2}$ from the problem (3.30) at $n = 2$ .

With the help of ${w_{n}^{(i)}}_{i = 1}^{3}$ , ${N_{n}}$ , $n \in {0, 1, 2}$ and smooth cut-off functions ${χ_{ℓ_{i}^{0}}^{(i)}, χ_{δ}^{(i)}}_{i = 1}^{3}$ (see (3.33)) we construct the following approximation: $\begin{array}{l} U_{ε} (x, t) = & \sum_{i = 1}^{3} χ_{ℓ_{i}^{0}}^{(i)} (\frac{x_{i}}{ε^{a}}) (w_{0}^{(i)} (x_{i}, t) + ε w_{1}^{(i)} (x_{i}, t) \\ + ε^{2} (w_{2}^{(i)} (x_{i}, t) + u_{2}^{(i)} (\frac{{\overline{x}}_{i}}{ε}, x_{i}, t) + χ_{δ}^{(i)} Π_{2}^{(i)} (\frac{{\overline{x}}_{i}}{ε}, \frac{ℓ_{i} - x_{i}}{ε}, t))) \\ + (1 - \sum_{i = 1}^{3} χ_{ℓ_{i}^{0}}^{(i)} (\frac{x_{i}}{ε^{a}})) (w_{0}^{(1)} (0, t) + ε N_{1} (\frac{x}{ε}, t) + ε^{2} N_{2} (\frac{x}{ε}, t)), \\ (4.5) & x \in Ω_{ε} \times (0, T), \end{array}$ where $a$ is a fixed number from the interval $(\frac{2}{3}, 1)$ , and δ is a sufficiently small fixed positive number.

Lemma 4.1.
Let conditions C1, C2, C3 be satisfied. The difference between the solution $u_{ε}$ of problem ( 2.1 ) and the sum ( 4.5 ) admits the following estimate: $\begin{matrix} (4.6) & max_{t \in [0, T]} {‖ U_{ε} (\cdot, t) - u_{ε} (\cdot, t) ‖}_{L^{2} (Ω_{ε})} + ‖ U_{ε} - u_{ε} ‖_{L^{2} (0, T; H^{1} (Ω_{ε}))} ⩽ C_{0} ε^{1 + \frac{3}{2} a} . \end{matrix}$

The proof is carrying out in the same reasoning as Lemma 3.1. Then with the help of the estimate (4.6) similarly to the proof of Theorem 3.1 we prove the following theorem and corollaries. Theorem 4.1 (The case $α_{0} = 1$ ).

Let conditions C1, C2, C3 be satisfied. Then the sum $\begin{array}{l} U_{ε}^{(1)} (x, t) = & \sum_{i = 1}^{3} χ_{ℓ_{i}^{0}}^{(i)} (\frac{x_{i}}{ε^{a}}) (w_{0}^{(i)} (x_{i}, t) + ε w_{1}^{(i)} (x_{i}, t)) \\ (4.7) & + (1 - \sum_{i = 1}^{3} χ_{ℓ_{i}^{0}}^{(i)} (\frac{x_{i}}{ε^{a}})) (w_{0}^{(1)} (0, t) + ε N_{1} (\frac{x}{ε}, t)), x \in Ω_{ε} \times (0, T), \end{array}$ is the asymptotic approximation for the solution $u_{ε}$ to the boundary-value problem ( 2.1 ), i.e., $\exists C_{1} > 0$ $\exists ε_{0} > 0$ $\forall ε \in (0, ε_{0})$ : $\begin{array}{l} (4.8) & \frac{1}{\sqrt{| Ω_{ε} |_{3}}} max_{t \in [0, T]} {‖ u_{ε} (\cdot, t) - U_{ε}^{(1)} (\cdot, t) ‖}_{L^{2} (Ω_{ε})} ⩽ C_{1} ε^{\frac{3}{2} a}, \\ (4.9) & \frac{1}{\sqrt{| Ω_{ε} |_{3}}} {‖ u_{ε} - U_{ε}^{(1)} ‖}_{L^{2} (0, T; H^{1} (Ω_{ε}))} ⩽ C_{1} ε, \end{array}$ where $a$ is a fixed number from the interval $(\frac{2}{3}, 1)$ .

Corollary 4.1 ( $α_{0} = 1$ ).

In each cylinder $Ω_{ε, a}^{(i)} : = Ω_{ε}^{(i)} \cap {x \in R^{3} : x_{i} \in I_{ε, a}^{(i)} : = (3 ℓ_{i}^{0} ε^{a}, ℓ_{i})}$ the following estimate holds: $\begin{matrix} (4.10) & \frac{1}{\sqrt{| Ω_{ε, a}^{(i)} |_{3}}} max_{t \in [0, T]} {‖ u_{ε} (\cdot, t) - w_{0}^{(i)} (\cdot, t) - ε w_{1}^{(i)} (\cdot, t) ‖}_{L^{2} (Ω_{ε, a}^{(i)})} ⩽ {\tilde{C}}_{1} ε^{\frac{3}{2} a}, i \in {1, 2, 3}, \end{matrix}$ where ${w_{0}^{(i)}}_{i = 1}^{3}$ is the solution of the limit problem ( 3.20 ).

In the neighborhood $Ω_{ε, a}^{(0)} : = Ω_{ε} \cap {x : x_{i} < 2 ℓ_{i}^{0} ε^{a}, i = 1, 2, 3}$ of the node $Ω_{ε}^{(0)}$ , we get the estimate $\begin{matrix} (4.11) & \frac{1}{\sqrt{| Ω_{ε, a}^{(0)} |_{3}}} ‖ \nabla_{x} u_{ε} - \nabla_{ξ} N_{1} ‖_{L^{2} (Ω_{ε, a}^{(0)} \times (0, T))} ⩽ {\tilde{C}}_{2} ε^{a}, \end{matrix}$ where $N_{1}$ is the solution to the problem ( 3.55 ).

If for some $i \in {1, 2, 3}$ the function $h_{i} (x_{i}) \equiv c o n s t$ , then $\begin{matrix} (4.12) & max_{t \in [0, T]} {‖ (E_{ε}^{(i)} u_{ε}) (\cdot, t) - w_{0}^{(i)} (\cdot, t) - ε w_{1}^{(i)} (\cdot, t) ‖}_{L^{2} (I_{ε, a}^{(i)})} ⩽ C_{2} ε^{\frac{3}{2} a}, i \in {1, 2, 3} . \end{matrix}$

5. Asymptotic approximation in the case $α_{0} = 0$

This case is very similar to the previous one. As in the Section 4 we seek the ansatzes of the asymptotics in the form (4.1), (4.2) and (4.3). For coefficients $u_{2}^{(i)}, u_{3}^{(i)}, u_{4}^{(i)}$ we obtain the same problems.

The coefficients ${N_{n}}_{n = 0}^{3}$ of the inner part (4.3) are solutions to the problems (3.11) (see also problems (3.13)) at $n \in {0, 1, 2, 3}$ with the same functions in the right-hand sides except of ${B_{n}^{(0)}}_{n = 0}^{3}$ : $\begin{array}{l} B_{0}^{(0)} (ξ, t) \equiv {\tilde{B}}_{0}^{(0)} (ξ, t) \equiv 0, B_{1}^{(0)} (ξ, t) = {\tilde{B}}_{1}^{(0)} (ξ, t) = - κ_{0} (N_{0}) + φ^{(0)} (ξ, t), \\ B_{2}^{(0)} (ξ, t) = {\tilde{B}}_{2}^{(0)} (ξ, t) = - κ_{0}^{'} (N_{0}) N_{1}, B_{3}^{(0)} (ξ, t) = {\tilde{B}}_{3}^{(0)} (ξ, t) = - κ_{0}^{'} (N_{0}) N_{2} - \frac{1}{2} κ_{0}^{″} (N_{0}) N_{1}^{2} . \end{array}$

Similarly as in the Section 3.3.1 we obtain (3.18) and $N_{0} \equiv {\tilde{N}}_{0} \equiv w_{0}^{(1)} (0, t)$ .

But now, the solvability condition (3.14) for the problem (3.13) at $n = 1$ reads as follows: $\begin{matrix} (5.1) & \sum_{i = 1}^{3} π h_{i}^{2} (0) \frac{\partial w_{0}^{(i)}}{\partial x_{i}} (0, t) - | Γ_{0} |_{2} κ_{0} (w_{0}^{(1)} (0, t)) = - \int_{Γ_{0}} φ^{(0)} (ξ, t) d σ_{ξ}, t \in (0, T) . \end{matrix}$ As a result, to define ${w_{0}^{(i)}}_{i = 1}^{3}$ we get the following semilinear problem: $\begin{matrix} (5.2) & \{\begin{matrix} π h_{i}^{2} (x_{i}) \frac{\partial w_{0}^{(i)}}{\partial t} (x_{i}, t) - π \frac{\partial}{\partial x_{i}} (h_{i}^{2} (x_{i}) \frac{\partial w_{0}^{(i)}}{\partial x_{i}} (x_{i}, t)) + π h_{i}^{2} (x_{i}) k (w_{0}^{(i)} (x_{i}, t)) \\ + 2 π h_{i} (x_{i}) κ_{i} (w_{0}^{(i)} (x_{i}, t)) = {\hat{F}}_{0}^{(i)} (x_{i}, t), (x_{i}, t) \in I_{i} \times (0, T), i = 1, 2, 3, \\ w_{0}^{(i)} (ℓ_{i}, t) = 0, i \in {1, 2, 3}, t \in (0, T), \\ w_{0}^{(1)} (0, t) = w_{0}^{(2)} (0, t) = w_{0}^{(3)} (0, t), t \in (0, T), \\ \sum_{i = 1}^{3} π h_{i}^{2} (0) \frac{\partial w_{0}^{(i)}}{\partial x_{i}} (0, t) - | Γ_{0} |_{2} κ_{0} (w_{0}^{(1)} (0, t)) = - \int_{Γ_{0}} φ^{(0)} (ξ, t) d σ_{ξ}, t \in (0, T), \\ w_{0}^{(i)} (x_{i}, 0) = 0, i \in {1, 2, 3}, x_{i} \in I_{i}, \end{matrix} \end{matrix}$ where $I_{i} : = {x : x_{i} \in (0, ℓ_{i}), {\overline{x}}_{i} = (0, 0)}$ and ${\hat{F}}_{0}^{(i)}$ are defined in (3.21). The existence and uniqueness of this problem is proved in [8, §4.2.1].

Repeating the previous reasoning, we get a solution $N_{n}$ of the problem (3.11) with the asymptotics (3.28). The next terms ${\tilde{w}}_{n}$ , $n \in {1, 2}$ of the regular asymptotics are determined from the problems (3.30) at $n = 1, 2$ . At $n = 1$ the right-hand side ${\hat{F}}_{1}^{(i)}$ are defined in (3.31), and the values $d_{1}^{*}$ and $δ_{1}^{(2)}$ , $δ_{1}^{(3)}$ are uniquely determined by formulas $\begin{array}{l} d_{1}^{*} (t) = & ({| Ξ^{(0)} |}_{3} - \sum_{i = 1}^{3} ℓ_{i}^{0} π h_{i}^{2} (0)) (\partial_{t} w_{0}^{(1)} (0, t) + k (w_{0}^{(1)} (0, t)) - f (0, t)) \\ + \int_{Γ_{0}} κ_{0}^{'} (w_{0}^{(1)} (0, t)) N_{1} (ξ, t) d ξ \\ (5.3) & - \sum_{i = 1}^{3} ℓ_{i}^{0} (2 π h_{i} (0) κ_{i} (w_{0}^{(1)} (0, t)) - \int_{\partial Υ_{i} (0)} φ^{(i)} ({\overline{ξ}}_{i}, 0, t) d l_{{\overline{ξ}}_{i}}), \\ δ_{1}^{(i)} (t) = & \int_{Ξ} N_{i} (ξ) \sum_{j = 1}^{3} (ξ_{j} \frac{\partial w_{0}^{(j)}}{\partial x_{j}} (0, t) χ_{j}^{″} (ξ_{j}) + 2 \frac{\partial w_{0}^{(j)}}{\partial x_{j}} (0, t) χ_{j}^{'} (ξ_{j})) d ξ \\ (5.4) & + \int_{Γ_{0}} N_{i} (ξ) (- κ_{0} (w_{0}^{(i)} (0, t)) + φ^{(0)} (ξ, t)) d σ_{ξ}, i = 2, 3 . \end{array}$

With the help of ${w_{n}^{(i)}}_{i = 1}^{3}$ , ${N_{n}}_{n = 0}^{3}$ and smooth cut-off functions ${χ_{ℓ_{i}^{0}}^{(i)}, χ_{δ}^{(i)}}_{i = 1}^{3}$ (see (3.33)), the asymptotic approximation (4.5) is constructed, where $a$ is a fixed number from the interval $(\frac{2}{3}, 1)$ , and δ is a sufficiently small fixed positive number. Then, similar as before we prove the statement.

Theorem 5.1 ( $α_{0} = 0$ ).

Let conditions C1, C2, C3 be satisfied. Then the difference between the solution $u_{ε}$ to the problem ( 2.1 ) and the sum ( 4.5 ) admits the estimate ( 4.6 ). In addition, the sum ( 4.7 ) is the asymptotic approximation for the solution $u_{ε}$ to the problem ( 2.1 ) and the estimates ( 4.9 ) and ( 4.8 ) hold.

Also the Corollary 4.1 holds, but now $N_{1}$ is a solution to the problem $\begin{matrix} (5.5) & \{\begin{matrix} - Δ_{ξ} N_{1} (ξ, t) = 0, ξ \in Ξ, \\ \partial_{ν_{ξ}} N_{1} (ξ, t) = - κ_{0} (w_{0}^{(1)} (0, t)) + φ^{(0)} (ξ, t) ξ \in Γ_{0}, \\ \partial_{ν_{ξ}} N_{1} (ξ, t) = 0, ξ \in Γ_{i}, i = 1, 2, 3, \\ N_{1} (ξ, t) \sim w_{1}^{(i)} (0, t) + ξ_{i} \frac{\partial w_{0}^{(i)}}{\partial x_{i}} (0, t), ξ_{i} \to + \infty, ξ \in Ξ^{(i)}, i = 1, 2, 3 . \end{matrix} \end{matrix}$

6. Conclusion

Our asymptotic approach is a very general and unified algorithm for finding the asymptotic approximation for solutions to boundary-value problems in thin graph-like junctions. Results presented in the paper are significant improvement and extension of the results obtained previously in our paper [8]. Especially, the asymptotic estimates in Theorems 3.1, 4.1 and 5.1, which show the influence of the local geometric heterogeneity of the node and physical processes inside on the asymptotic behaviour of the solution, justify our main conclusion made in the Introduction.

This impact is also observed through the values $d_{1}^{*}$ , $δ_{1}^{(2)}$ , $δ_{1}^{(3)}$ defined with formulas (4.4) and (3.32) for $α_{0} = 1$ , with (5.3) and (5.4) for $α_{0} = 0$ , and through the value $d_{α_{0}}^{*}$ in (3.24) for $α_{0} \in (0, 1)$ .

The estimates obtained in Theorems 4.1 and 5.1 make it possible to improve estimates for the difference between the solution and the zero approximation $\begin{matrix} (6.1) & U_{ε}^{(0)} (x, t) = \sum_{i = 1}^{3} χ_{ℓ_{i}^{0}}^{(i)} (\frac{x_{i}}{ε^{a}}) w_{0}^{(i)} (x_{i}, t) + (1 - \sum_{i = 1}^{3} χ_{ℓ_{i}^{0}}^{(i)} (\frac{x_{i}}{ε^{a}})) w_{0}^{(1)} (0, t), x \in Ω_{ε} \times (0, T) \end{matrix}$ in comparison with the results of our paper [8, Section 5]. Here $a$ is a fixed number from $(\frac{2}{3}, 1)$ .

Corollary 6.1 ( $α_{0} \in {0, 1}$ ).

The difference between the solution $u_{ε}$ of problem ( 2.1 ) and the sum ( 6.1 ) admits the following asymptotic estimates: $\begin{array}{l} (6.2) & \frac{1}{\sqrt{| Ω_{ε} |_{3}}} max_{t \in [0, T]} {‖ u_{ε} (\cdot, t) - U_{ε}^{(0)} (\cdot, t) ‖}_{L^{2} (Ω_{ε})} ⩽ {\tilde{C}}_{0} ε, \\ (6.3) & \frac{1}{\sqrt{| Ω_{ε} |_{3}}} {‖ u_{ε} - U_{ε}^{(0)} ‖}_{L^{2} (0, T; H^{1} (Ω_{ε}))} ⩽ {\tilde{C}}_{0} ε^{\frac{1}{2} a} . \end{array}$

In each thin cylinder $Ω_{ε, a}^{(i)} : = Ω_{ε}^{(i)} \cap {x \in R^{3} : x_{i} \in I_{ε, a}^{(i)} : = (3 ℓ_{i}^{0} ε^{a}, ℓ_{i})} (i = 1, 2, 3)$ , the following estimate holds: $\begin{matrix} (6.4) & \frac{1}{\sqrt{| Ω_{ε, a}^{(i)} |_{3}}} (max_{t \in [0, T]} {‖ u_{ε} (\cdot, t) - w_{0}^{(i)} (\cdot, t) ‖}_{L^{2} (Ω_{ε, a}^{(i)})} + {‖ u_{ε} - w_{0}^{(i)} ‖}_{L^{2} (0, T; H^{1} (Ω_{ε, a}^{(i)}))}) ⩽ {\tilde{C}}_{1} ε, \end{matrix}$ where ${w_{0}^{(i)}}_{i = 1}^{3}$ is the solution of the limit problem ( 5.2 ).

If for some $i \in {1, 2, 3}$ the function $h_{i} (x_{i}) \equiv c o n s t$ , then $\begin{matrix} (6.5) & max_{t \in [0, T]} {‖ (E_{ε}^{(i)} u_{ε}) (\cdot, t) - w_{0}^{(i)} (\cdot, t) ‖}_{L^{2} (I_{ε, a}^{(i)})} + {‖ E_{ε}^{(i)} u_{ε} - w_{0}^{(i)} ‖}_{L^{2} (0, T; C ([3 ℓ_{i}^{0} ε^{a}, ℓ_{i}]))} ⩽ C_{2} ε . \end{matrix}$

For $α_{0} \in (0, 1)$ , when the limit problem is independent of $α_{0}$ but the interaction intensity at the node boundary still remains sensitive, this difference is estimated with the same order of ε as in [8], namely $O (ε^{min {α_{0}, \frac{a}{2}}})$ . To obtain better asymptotic estimates in this case we should construct next term of the asymptotics (see Theorem 3.1). Based on those facts we can conclude that the nonlinear Robin condition can be neglected for $α_{0} ⩾ 1$ and instead of it the uniform Neumann condition on the node boundary can be considered.

To construct the asymptotic approximation for the solution $u_{ε}$ in the case $α_{0} \neq β_{0}$ and $β_{0} \in (0, 1)$ in (2.12) at $i = 0$ , we should consider the following asymptotic scale $\begin{matrix} ε^{0}, ε^{α_{0}}, ε^{β_{0}}, ε^{1}, ε^{2 α_{0}}, ε^{2 β_{0}}, ε^{1 + α_{0}}, ε^{1 + β_{0}}, ε^{2}, ε^{2 + α_{0}}, ε^{2 + β_{0}}, ε^{3}, ε^{2 + 2 α_{0}}, ε^{2 + 2 β_{0}}, \dots \end{matrix}$ and repeat all steps made in the Section 3 with the corresponding changes.

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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

Abstract

Keywords

1. Introduction

3. Asymptotic approximation in the case α 0 ∈ ( 0 , 1 )

3.1. Regular parts

3.2. Boundary-layer parts

Remark 3.1. From representation (3.10) it follows the following asymptotic relations Π 2 ( i ) ( ξ ‾ i , ξ i ∗ , t ) = O ( exp ( − λ 1 ( i ) ξ i ∗ ) ) as ξ i ∗ → + ∞ , i = 1 , 2 , 3 . 3.3. Inner part

Corollary 4.1 ( α 0 = 1 ).

5. Asymptotic approximation in the case α 0 = 0

Theorem 5.1 ( α 0 = 0 ).

6. Conclusion

Corollary 6.1 ( α 0 ∈ { 0 , 1 } ).

References

3. Asymptotic approximation in the case $α_{0} \in (0, 1)$

Remark 3.1.
From representation (3.10) it follows the following asymptotic relations $\begin{matrix} Π_{2}^{(i)} ({\overline{ξ}}_{i}, ξ_{i}^{}, t) = O (exp (- λ_{1}^{(i)} ξ_{i}^{})) as ξ_{i}^{*} \to + \infty, i = 1, 2, 3 . \end{matrix}$

3.3. Inner part

Corollary 4.1 ( $α_{0} = 1$ ).

5. Asymptotic approximation in the case $α_{0} = 0$

Theorem 5.1 ( $α_{0} = 0$ ).

Corollary 6.1 ( $α_{0} \in {0, 1}$ ).