Asymptotic approximation for the solution to a semi-linear parabolic problem in a thin star-shaped junction
aa r X i v : . [ m a t h . A P ] A p r ASYMPTOTIC APPROXIMATION FOR THE SOLUTION TO A SEMI-LINEARPARABOLIC PROBLEM IN A THIN STAR-SHAPED JUNCTION
ARSEN V. KLEVTSOVSKIY AND TARAS A. MEL’NYK
Abstract.
A semi-linear parabolic problem is considered in a thin 3 D star-shaped junction that consistsof several thin curvilinear cylinders that are joined through a domain (node) of diameter O ( ε ) . The purpose is to study the asymptotic behavior of the solution u ε as ε → , i.e. when the star-shapedjunction is transformed in a graph. In addition, the passage to the limit is accompanied by special intensityfactors { ε α i } and { ε β i } in nonlinear perturbed Robin boundary conditions.We establish qualitatively different cases in the asymptotic behaviour of the solution depending on thevalue of the parameters { α i } and { β i } . Using the multi-scale analysis, the asymptotic approximation forthe solution is constructed and justified as the parameter ε → . Namely, in each case we derive the limitproblem ( ε = 0) on the graph with the corresponding Kirchhoff transmission conditions (untypical in somecases) at the vertex, define other terms of the asymptotic approximation and prove appropriate asymptoticestimates that justify these coupling conditions at the vertex and show the impact of the local geometricheterogeneity of the node and physical processes in the node on some properties of the solution. Contents
1. Introduction 11.1. Novelty and method of the study 32. Statement of the problem 42.1. Comments to the statement 63. Existence and uniqueness of the weak solution 73.1. A priori estimates 104. Formal asymptotic approximation. The case α ≥ , α i ≥ , i ∈ { , , } . α < , α i ≥ , i ∈ { , , } α ∈ ( − ,
0) 276.2. The case α = − Introduction
We are interested in the study of evolution phenomena in junctions composed of several thin curvilinearcylinders that are joined through a domain of diameter O ( ε ) (see Fig. 1). Mathematical models those are Key words and phrases.
Approximation, semi-linear parabolic problem, nonlinear perturbed boundary condition, asymptoticestimate, thin star-shaped junction.
MOS subject classification: described by semi-linear parabolic equations that allow to model a variety of biological and physical phe-nomena (reaction and diffusion processes in biology and biochemistry, heat-mass transfer, etc.) in channels,junctions and networks.As we can see from Fig. 1, a thin junction is shrunk into a graph as the small parameter ε, characterizingthickness of the thin cylinders and domain connecting them, tends to zero. Thus, the aim is to find thecorresponding limit problem in this graph and prove the estimate for the difference between the solutions ofthese two problems. ε d=2 ε h ( x ) i i Figure 1.
Transformation of a thin star-shaped junction into a graphA large amount of physical and mathematical articles and books dedicated to different models on graphs,has been published for the last three decades, e.g. [1–13]. The main question arising in problems on graphsis point interactions at nodes of networks, i.e., the type of coupling conditions at vertices of the graph.Also there is increasing interest in the investigation of the influence of a local geometric heterogeneityin vessels on the blood flow. This is both an aneurysm (a pathological extension of an artery like a bulge)and a stenosis (a pathological restriction of an artery). In [14] the authors classified 12 different aneurysmsand proposed a numerical approach for this study. The aneurysm models have been meshed with 800,000– 1,200,000 tetrahedral cells containing three boundary layers. However, as was noted by the authors, thequestion how to model blood flow with sufficient accuracy is still open .Because of those point interactions and local geometric irregularities, the reaction-diffusion processes,heat-mass transfer and flow motions in networks posses many distinguishing features. A natural approachto explain the meaning of point interactions at vertices is the use of the limiting procedure mentioned above.There are several asymptotic approaches to study such problems. As far as we known, the paper [15]was the first paper, where convergence results for linear diffusion processes in a region with narrow tubeswere obtained with the help of the martingal-problem method of proving weak convergence. As a result, thestandard gluing conditions (or so-called ”Kirchhoff” transmission conditions) at the vertices of the graphwere derived. Then this probabilistic approach was generalized in [16].The method of the partial asymptotic domain decomposition was proposed in [17] and then it applied todifferent problems under the following assumptions: the uniform boundary conditions on the lateral surfacesof thin rectilinear cylinders, the right-hand sides depend only on the longitudinal variable in the directionof the corresponding cylinder and they are constant in some neighbourhoods of the nodes and vertices(see [18–22]). It follows from these papers that the main difficulty is the identification of the behaviour ofsolutions in neighbourhoods of the nodes.To overcome this difficulty and to construct the leading terms of the elastic field asymptotics for thesolution of the equations of anisotropic elasticity on junctions of thin three dimensional beams, the following
PPROXIMATION FOR THE SOLUTION TO A PARABOLIC PROBLEM IN A THIN STAR-SHAPED JUNCTION 3 assumptions were made in [23]: the first terms of the volume force f and surface load g on the rods satisfyspecial orthogonality conditions (see (3 . and (3 . f has anidentified form and depends only on the longitudinal variable; similar orthogonality conditions for the right-hand sides on the nodes are satisfied (see (3 .
41) ) and the second term is a piecewise constant vector-function(see (3 .
42) ). By these assumptions, the displacement field at each node can be approximated by a rigiddisplacement. As a result, the approximation does not contain boundary layer terms, i.e., the asymptoticexpansion is not complete a priori [23, Remark 3.1]. Similar approach was used for thin two dimensionaljunctions in [24].There is a special interest in spectral problems on thin graph-like structures, since such problems havemany applications. A fairly complete review on this topic has been presented in [25]. The main task isto study the possibility of approximating the spectra of different operators by the spectra of appropriateoperators on the corresponding graph. The convergence of spectra for the Laplacians with different boundaryconditions (Neumann, Dirichlet and Robin) at various levels of generality was proved in [26–33]. In [26] theauthors took into account large protrusions at the vertices; as a result different Kirchhoff conditions areappeared depending on the value of the protrusion. It was demonstrated in [29] that the type of thetransmission conditions depends crucially on the boundary layer phenomenon in the vicinity of the nodes; inaddition the complete asymptotic expansions for the k -th eigenvalue and the eigenfunctions were obtainedthere, uniformly for k, in terms of scattering data on a non-compact limit space. Interesting multifarioustransmission conditions are obtained in the limit passage for spectral problems on thin periodic honeycomblattice [34, 35]. Numerical approach to deduce the vertex coupling conditions for the nonlinear Schr¨odingerequation on two-dimensional thin networks was proposed in [36].1.1. Novelty and method of the study.
In the present paper we continue to develop the asymptoticmethod proposed in our papers [37,38] for linear elliptic problems, which does not need the above mentionedassumptions. In addition, our approach gives the better estimate for the difference between the solution ofthe starting problem and the solution of the corresponding limit problem (compare (1) and (2) in [37]).Here we have adapted this method to semi-linear parabolic problems with nonlinear perturbed Robinboundary conditions ∂ ν u ε + ε α i κ i (cid:0) u ε , x i , t (cid:1) = ε β i ϕ ( i ) ε ( x, t ) (1.1)both on the boundaries of the thin curvilinear cylinders ( i ∈ { , , } ) and on the boundary of the node( i = 0) , which depend on special intensity factors ε α i and ε β i . We study the influence of these factors onthe asymptotic behaviour of the solution as ε → . It turned out that the asymptotic behaviour of the solution depends on the parameters { α i } and { β i } , and essentially on the parameter α that characterizes the intensity of processes at the boundary of the node.It is natural to expect that physical processes on the node boundary provoke crucial changes in the wholeprocess in the thin star-shaped junction, in particular they can reject the traditional Kirchhoff transmissionconditions at the vertex in some cases. We discovery three qualitatively different cases in the asymptoticbehaviour of the solution. If α > , β > , α i , β i ≥ , i ∈ { , , } , then we have classical Kirchhofftransmission conditions. In the case α = 0 , β = 0 , α i , β i ≥ , i ∈ { , , } , new gluing conditions at thevertex x = 0 of the graph look as follows ω (1)0 (0 , t ) = ω (2)0 (0 , t ) = ω (3)0 (0 , t ) ,πh (0) ∂ω (1)0 ∂x (0 , t ) + πh (0) ∂ω (2)0 ∂x (0 , t ) + πh (0) ∂ω (3)0 ∂x (0 , t ) − (cid:12)(cid:12) Γ (cid:12)(cid:12) κ (cid:0) ω (1)0 (0 , t ) (cid:1) = − Z Γ ϕ (0) ( ξ, t ) dσ ξ , where (cid:12)(cid:12) Γ (cid:12)(cid:12) is the Lebesgue measure of the boundary Γ of the node. If α < A.V. KLEVTSOVSKIY AND T.A. MEL’NYK regular part of the asymptotics located inside of each thin cylinder and the inner part of the asymptoticsdiscovered in a neighborhood of the node. The terms of the inner part of the asymptotics are special solutionsof boundary-value problems in an unbounded domain with different outlets at infinity. It turns out theyhave polynomial growth at infinity. Matching these parts, we derive the limit problem ( ε = 0) in the graphand the corresponding coupling conditions at the vertex.Also we have proved energetic estimates in each case which allow to identify more precisely the impactof the local geometric heterogeneity of the node and physical processes in the node on some properties ofthe solution. It should be stressed that the error estimates and convergence rate are very important bothfor justification of adequacy of one- or two-dimensional models that aim at description of actual three-dimensional thin bodies and for the study of boundary effects and effects of local (internal) inhomogeneitiesin applied problems. In addition, those estimates justify transmission conditions of Kirchhoff type for metricgraphs.Thus, our approach makes it possible to take into account various factors (e.g. variable thickness of thincurvilinear cylinders, inhomogeneous nonlinear boundary conditions, geometric characteristics of nodes, etc.)in statements of boundary-value problems on graphs.The rest of this paper is organized as follows. The statement of the problem and features of the investi-gation are presented in Section 2. In Section 3, the existence and uniqueness of the weak solution is provedfor every fixed value ε. Also a priori estimates and auxiliary inequalities are deduced there. In Section 4we formally construct the leading terms both of the regular part of the asymptotics and the inner one inthe case α ≥ , α i ≥ , i ∈ { , , } . Then using the constructed terms we build the approximation andprove the corresponding asymptotic estimates in Section 5. Section 6 shows us what will happen in the case α < , α i ≥ , i ∈ { , , } . The main novelty is that the limit problem splits into three independentproblems with the uniform Dirichlet condition at the vertex. In addition, the view of asymptotic ansatzesare very sensitive to the parameter α . Here we construct the approximation and prove the correspondingestimates for more typical and realistic subcases α ∈ ( − ,
0) and α = −
1; general case is only discussed.In Section 7, we analyze obtained results and discuss research perspectives.2.
Statement of the problem
The model thin star-shaped junction Ω ε consists of three thin curvilinear cylindersΩ ( i ) ε = ( x = ( x , x , x ) ∈ R : εℓ < x i < ℓ i , X j =1 (1 − δ ij ) x j < ε h i ( x i ) ) , i = 1 , , , that are joined through a domain Ω (0) ε (referred in the sequel ”node”). Here ε is a small parameter; ℓ ∈ (0 , ) , ℓ i ≥ , i = 1 , ,
3; the positive function h i belongs to the space C ([0 , ℓ i ]) and it is equalto some constants in neighborhoods of the points x = 0 and x i = 1 ( i = 1 , ,
3) ; the symbol δ ij is theKroneker delta, i.e., δ ii = 1 and δ ij = 0 if i = j. x x x ε l ε l ε l Figure 2.
The node Ω (0) ε PPROXIMATION FOR THE SOLUTION TO A PARABOLIC PROBLEM IN A THIN STAR-SHAPED JUNCTION 5
The node Ω (0) ε (see Fig. 2) is formed by the homothetic transformation with coefficient ε from a boundeddomain Ξ (0) ⊂ R , i.e., Ω (0) ε = ε Ξ (0) . In addition, we assume that its boundary contains the disksΥ ( i ) ε ( εℓ ) = ( x ∈ R : x i = εℓ , X j =1 (1 − δ ij ) x j < ε h i ( εℓ ) ) , i = 1 , , , and denote Γ (0) ε := ∂ Ω (0) ε \ n Υ (1) ε ( εℓ ) ∪ Υ (2) ε ( εℓ ) ∪ Υ (3) ε ( εℓ ) o . Thus the model thin star-shaped junction Ω ε (see Fig. 3) is the interior of the union S i =0 Ω ( i ) ε and weassume that it has the Lipschitz boundary. x x x d=2 ε h ( x ) l Figure 3.
The model thin star-shaped junction Ω ε Remark 2.1.
We can consider more general thin star-shaped junctions with arbitrary orientation of thincylinders (their number can be also arbitrary). But to avoid technical and huge calculations and to demon-strate the main steps of the proposed asymptotic approach we consider the case when the cylinders are placedon the coordinate axes.
In Ω ε , we consider the following semi-linear parabolic problem: ∂ t u ε ( x, t ) − ∆ x u ε ( x, t ) + k (cid:0) u ε ( x, t ) (cid:1) = f ( x, t ) , ( x, t ) ∈ Ω ε × (0 , T ) ,∂ ν u ε ( x, t ) + ε α κ (cid:0) u ε ( x, t ) (cid:1) = ε β ϕ (0) ε ( x, t ) , ( x, t ) ∈ Γ (0) ε × (0 , T ) ,∂ ν u ε ( x, t ) + ε α i κ i (cid:0) u ε ( x, t ) , x i , t (cid:1) = ε β i ϕ ( i ) ε ( x, t ) , ( x, t ) ∈ Γ ( i ) ε × (0 , T ) , i = 1 , , ,u ε ( x, t ) = 0 , ( x, t ) ∈ Υ ( i ) ε ( ℓ i ) × (0 , T ) , i = 1 , , ,u ε ( x,
0) = 0 , x ∈ Ω ε , (2.1)where Γ ( i ) ε = ∂ Ω ( i ) ε ∩ { x ∈ R : εℓ < x i < ℓ i } , T > , ∂ t = ∂/∂t, ∂ ν is the outward normal derivative,and the parameters { α i } i =0 ⊂ R , β ≥ , β i ≥ , i = 1 , . For the given functions f, k, { ϕ ( i ) ε , κ i } i =0 weassume the following conditions: C1. the function f belongs to the space C (cid:0) Ω a × [0 , T ] (cid:1) and its restriction on the curvilinear cylinderΩ ( i ) a ( i = 1 , ,
3) belong to the space C x i (cid:16) Ω ( i ) a × [0 , T ] (cid:17) (the space of all continuous functionshaving continuous derivatives with respect to variables x i in Ω ( i ) a × [0 , T ]) , where a is a fixed A.V. KLEVTSOVSKIY AND T.A. MEL’NYK positive number such that Ω ε ⊂ Ω a for all values of the small parameter ε ∈ (0 , ε ) and x i = ( x , x ) , i = 1 , ( x , x ) , i = 2 , ( x , x ) , i = 3; C2. the functions ϕ (0) ε ( x, t ) := ϕ (0) (cid:16) xε , t (cid:17) and ϕ ( i ) ε ( x, t ) := ϕ ( i ) (cid:18) x i ε , x i , t (cid:19) , i = 1 , , , belong to thespaces C (cid:16) Ω ( i ) a × [0 , T ] (cid:17) , i ∈ { , , , } , respectively; C3. the functions { κ i ( s, x i , t ) } i =1 , ( s, x i , t ) ∈ R × [0 , ℓ i ] × [0 , T ] are continuous in their domains ofdefinition and have the partial derivatives with respect to s, k ∈ C ( R ) , κ ∈ C ( R ) , and thereexists a positive constant k + such that0 ≤ k ′ ( s ) ≤ k + , ≤ κ ′ ( s ) ≤ k + , ≤ ∂ s κ i ( s, x i , t ) ≤ k + for s ∈ R (2.2)uniformly with respect to x i ∈ [0 , ℓ i ] and t ∈ [0 , T ] , respectively;(a) if α < , then in addition, the function κ is a C -function with bounded derivatives, thereexists a constant k − such that 0 < k − ≤ κ ′ ( s ) for all s ∈ R and κ (0) = 0 (so-calledcondition of zero-absorption).Denote by H ∗ ε the dual space to the Sobolev space H ε = (cid:8) u ∈ H (Ω ε ) : u | Υ ( i ) ε ( ℓ i ) = 0 , i = 1 , , (cid:9) . Recall that a function u ε ∈ L (0 , T ; H ε ) , with ∂ t u ε ∈ L (0 , T ; H ∗ ε ) , is called a weak solution to theproblem (2.1) if it satisfies the integral identity Z Ω ε ∂ t u ε v dx + Z Ω ε ∇ u ε · ∇ v dx + Z Ω ε k ( u ε ) v dx + ε α Z Γ (0) ε κ ( u ε ) v dσ x + X i =1 ε α i Z Γ ( i ) ε κ i ( u ε , x i , t ) v dσ x = Z Ω ε f v dx + X i =0 ε β i Z Γ ( i ) ε ϕ ( i ) ε v dσ x (2.3)for any function v ∈ H ε and a.e. t ∈ (0 , T ) , and u ε | t =0 = 0 . It is known (see e.g. [47]) that u ε ∈ C (cid:0) [0 , T ]; L (Ω ε ) (cid:1) , and thus the equality u ε | t =0 = 0 makes sense.The aim of the present paper is to • construct the asymptotic approximation for the solution to the problem (2.1) as the parameter ε → • derive the corresponding limit problem ( ε = 0); • prove the corresponding asymptotic estimates from which the influence of the local geometric het-erogeneity of the node Ω (0) ε and physical processes inside will be observed; • study the influence of the parameters { α i , β i } i =0 on the asymptotic behavior of the solution.2.1. Comments to the statement.
To our knowledge, the first works on the study of boundary-valueproblems for reaction-diffusion equations were papers by Kolmogorov, Petrovskii, Piskunov [40] and Fisher[41]. Standard assumptions for reaction terms of semilinear equations are as follows: • ∃
C > ∀ s , s ∈ R : | k ( s ) − k ( s ) | ≤ C | s − s | ; • ∃ C > C ≥ ∀ s ∈ R : k ( s ) s ≥ C s − C . This is sufficient for the existence and uniqueness of the weak solution. However, many physical processes,especially in chemistry and medicine, have monotonous nature. Therefore, it is naturally to impose specialmonotonous conditions for nonlinear terms. In our case we propose simple conditions (2.2) which are easyto verify. For instance, the functions k ( s ) = λs + cos s ( λ ≥ k ( s ) = λs νs ( λ, ν > PPROXIMATION FOR THE SOLUTION TO A PARABOLIC PROBLEM IN A THIN STAR-SHAPED JUNCTION 7 satisfy this condition. The last one corresponds to the Michaelis-Menten hypothesis in biochemical reactionsand to the Langmuir kinetics adsorption models (see [42, 43]).From conditions (2.2) it follows the following inequalities: k (0) s ≤ k ( s ) s ≤ k + s + k (0) s, κ (0) s ≤ κ ( s ) s ≤ k + s + κ (0) s,κ i (0 , x i , t ) s ≤ κ i ( s, x i , t ) s ≤ k + s + κ i (0 , x i , t ) s, for s ∈ R (2.4)uniformly with respect to ( x i , t ) ∈ [0 , ℓ i ] × [0 , T ] , respectively; i = 1 , , . For the case C3 (a) we have k − s ≤ κ ( s ) s ≤ k + s ∀ s ∈ R . (2.5)Doubtless both the function k and κ may also depend on x and t. However, we have omitted thisdependence to avoid cumbersome formulas, leaving it only for the functions { κ i } i =1 . As will be seen from further calculations in the case when some parameter α i > , the condition (2.2)for the corresponding function κ i can be weakened. In this case it is sufficient that κ i is continuous andthere exist constants c > , c ≥ s , s , s ∈ R , x i ∈ [0 , ℓ i ] , t ∈ [0 , T ] : (cid:0) κ i ( s , x i , t ) − κ i ( s , x i , t ) (cid:1) ( s − s ) ≥ , (cid:12)(cid:12) κ i ( s, x i , t ) (cid:12)(cid:12) ≤ c (cid:0) | s | (cid:1) , κ i ( s, x i , t ) s ≥ − c . It should be noted here that the asymptotic behaviour of solutions to the reaction-diffusion equation indifferent kind of thin domains with the uniform Neumann conditions was studied in [44,45]. The convergencetheorems were proved under the following assumptions for the reaction term k : in [44] it is a C -functionwith bounded derivatives and lim inf | s |→ + ∞ k ( s ) s >
0; (2.6)in [45] it is a C -function, | k ′ ( s ) | ≤ C (1 + | s | q ) , where q ∈ (0 , + ∞ ) , and the dissipative condition (2.6) issatisfied. It is easy to see that from (2.5) it follows (2.6).In a typical interpretation the solution to the problem (2.1) denotes the density of some quantity (temper-ature, chemical concentration, the potential of a vector-field, etc.) within the thin star-shaped junction Ω ε . The nonlinear Robin boundary conditions considered in this problem mean that there is some interactionbetween the surrounding density and the density just inside Ω ε . It is evident from the results we havepresented that these conditions (essentially the condition at the boundary of the node) have a substantialinfluence on the asymptotic behavior of the solution. To study this influence, we introduce special intensityfactors ε α i , i ∈ { , , , } . Since in this paper we are more interested in the study of the boundary inter-actions at the node, we take the parameter α from R and the other ones from [1 , + ∞ ) . The case when α i < i ∈ { , , } ) is only discussed in Sec. 7.3. Existence and uniqueness of the weak solution
In order to obtain the operator statement for the problem (2.1) we introduce the new norm k · k ε in H ε ,which is generated by the scalar product( u, v ) ε = Z Ω ε ∇ u · ∇ v dx, u, v ∈ H ε . Due to the uniform Dirichlet condition on Υ ( i ) ε ( ℓ i ) , i = 1 , , , the norm k · k ε and the ordinary norm k · k H (Ω ε ) are uniformly equivalent, i.e. , there exist constants C > ε > ε ∈ (0 , ε ) and for all u ∈ H ε the following estimate hold: k u k ε ≤ k u k H (Ω ε ) ≤ C k u k ε . (3.1) Remark 3.1.
Here and in what follows all constants { C j } and { c j } in inequalities are independent of theparameter ε. A.V. KLEVTSOVSKIY AND T.A. MEL’NYK
Further we will often use the inequalities ε Z Γ ( i ) ε v dσ x ≤ C ε Z Ω ( i ) ε |∇ x v | dx + Z Ω ( i ) ε v dx ! , (3.2) Z Ω ( i ) ε v dx ≤ C ε Z Ω ( i ) ε |∇ x v | dx + ε Z Γ ( i ) ε v dσ x ! ∀ v ∈ H (Ω ( i ) ε ) , ( i ∈ { , , } ) (3.3)proved in [46]. Let us prove similar inequalities for the node ( i = 0) . Proposition 3.1.
Let Q be a bounded domain in R with the smooth boundary ∂Q. Then there exists apositive constant C > that is independent of ε such that for any function v from the space H ( Q ε ) thefollowing inequalities hold: ε Z ∂Q ε v dσ x ≤ C ε Z Q ε |∇ x v | dx + Z Q ε v dx ! and Z Q ε v dx ≤ C ε Z Q ε |∇ x v | dx + ε Z ∂Q ε v dσ x ! , (3.4) where Q ε := ε Q is the homothetic transformation with the coefficient ε of Q. Proof.
Let r ( s ) = (cid:0) r ( s ) , r ( s ) , r ( s ) (cid:1) , s ∈ S ⊂ R , be a smooth parametrisation of ∂Q. Then r ε := εr = (cid:0) εr , εr , εr (cid:1) is the parametrization of ∂Q ε . Denote by ρ ( r ) := √ EG − F , where E = P i =1 (cid:16) ∂r i ∂ s (cid:17) , G = P i =1 (cid:16) ∂r i ∂ s (cid:17) , F = P i =1 ∂r i ∂ s ∂r i ∂ s . Then ρ ( r ε ) = ε ρ ( r ) . Using definition of the surface integral, we get Z ∂Q ε v ( x ) dσ x = Z S v (cid:0) r ε ( s ) (cid:1) ρ (cid:0) r ε ( s ) (cid:1) d s = ε Z S v (cid:0) εr ( s ) (cid:1) ρ (cid:0) r ( s ) (cid:1) d s = ε Z ∂Q v ε ( ξ ) dσ ξ (3.5)for all v ∈ H ( Q ε ) , where v ε ( ξ ) := v ( εξ ) , ξ = ( ξ , ξ , ξ ) , and x = εξ. Taking into account the boundedness of the trace operator, i.e., ∃ c > k v ε k L ( ∂Q ) ≤ c k v ε k H ( Q ) , where constant c does not depend on v ε , and the equality ε (cid:18)Z Q |∇ ξ v ε | dξ + Z Q v ε dξ (cid:19) = ε Z Q ε |∇ x v | dx + Z Q ε v dx, we obtain the first inequality in (3.4). By the same arguments we can prove the second one. (cid:3) It is easy to prove the inequality Z Ω (0) ε v dx ≤ C ε Z Ω ε |∇ x v | dx + Z Υ ( i ) ε ( ℓ i ) v dx i ! and then with the help of the first inequality in (3.4) the following one: Z Γ (0) ε v dσ x ≤ C Z Ω ε |∇ x v | dx + Z Υ ( i ) ε ( ℓ i ) v dx i ! (3.6)for all v ∈ H (Ω ε ) and i ∈ { , , } . Define a nonlinear operator A ε ( t ) : H ε −→ H ∗ ε through the relation (cid:10) A ε ( t ) u, v (cid:11) ε = Z Ω ε ∇ u · ∇ v dx + Z Ω ε k ( u ) v dx + ε α Z Γ (0) ε κ ( u ) v dσ x + X i =1 ε α i Z Γ ( i ) ε κ i ( u, x i , t ) v dσ x ∀ u, v ∈ H ε , PPROXIMATION FOR THE SOLUTION TO A PARABOLIC PROBLEM IN A THIN STAR-SHAPED JUNCTION 9 and the linear functional F ε ( t ) ∈ H ∗ ε by the formula (cid:10) F ε ( t ) , v (cid:11) ε = Z Ω ε f v dx + X i =0 ε β i Z Γ ( i ) ε ϕ ( i ) ε v dσ x ∀ v ∈ H ε , for a.e. t ∈ (0 , T ) , where h· , ·i ε is the duality pairing of H ∗ ε and H ε .Then the integral identity (2.3) can be rewritten as follows (cid:10) ∂ t u ε , v (cid:11) ε + (cid:10) A ε ( t ) u ε , v (cid:11) ε = (cid:10) F ε ( t ) , v (cid:11) ε ∀ v ∈ H ε , (3.7)for a.e. t ∈ (0 , T ) , and u ε | t =0 = 0 . To prove the well-posedness result, we verify some properties of the operator A ε for a fixed value of ε. (1) With the help of (2.4) and Cauchy’s inequality with δ > ab ≤ δa + b δ ) , we obtain (cid:10) A ε ( t ) v, v (cid:11) ε ≥ Z Ω ε |∇ v | dx + Z Ω ε k (0) v dx + ε α Z Γ (0) ε κ (0) v dσ x + X i =1 ε α i Z Γ ( i ) ε κ i (0 , x i , t ) v dσ x ≥ k v k ε − δ Z Ω ε v dx + Z Γ (0) ε v dσ x + ε X i =1 Z Γ ( i ) ε v dσ x ! − δ | k (0) | | Ω ε | + ε α | κ (0) | | Γ (0) ε | + X i =1 ε α i − max [0 ,ℓ i ] × [0 ,T ] | κ i (0 , x i , t ) | | Γ ( i ) ε | ! . (3.8)Here and in what follows | S | n is the n -dimensional Lebesgue measure of a set S. Then using (3.1),(3.2), (3.6) and recalling the assumption C3 (a), we can select appropriate δ such that (cid:10) A ε ( t ) v, v (cid:11) ε ≥ C k v k ε − C ε (cid:18) ε α + X i =1 ε α i − (cid:19) ∀ v ∈ H ε . This inequality means that the operator A ε is coercive for a.e. t ∈ (0 , T ) . (2) Let us show that it is strongly monotone for a.e. t ∈ (0 , T ) . Taking into account (2.2), we get (cid:10) A ε ( t ) u − A ε ( t ) u , u − u (cid:11) ε ≥ k u − u k ε ∀ u , u ∈ H ε . (3) The operator A ε is hemicontinuous for a.e. t ∈ (0 , T ) . Indeed, the real valued function[0 , ∋ τ → (cid:10) A ε [ u + τ v ] , u (cid:11) ε is continuous on [0 ,
1] for all fixed u , u , v ∈ H ε due to the continuity of the functions k, { κ i } i =0 and Lebesque’s dominated convergence theorem.(4) Let us prove that operator A ε is bounded. Using Cauchy-Bunyakovsky integral inequality, (3.1)and (2.4), we deduce the following inequality: (cid:10) A ε u, v (cid:11) ε ≤ Z Ω ε ∇ u · ∇ v dx + Z Ω ε (cid:0) k + | u | + | k (0) | (cid:1) | v | dx + ε α Z Γ (0) ε (cid:0) k + | u | + | κ (0) | (cid:1) | v | dσ x + X i =1 ε α i Z Γ ( i ) ε (cid:0) k + | u | + | κ i (0 , x i , t ) | (cid:1) | v | dσ x ≤ k u k ε k v k ε + k + k u k L (Ω ε ) k v k L (Ω ε ) + k + 3 X i =0 ε α i k u k L (Γ ( i ) ε ) k v k L (Γ ( i ) ε ) + | k (0) | p | Ω ε | k v k L (Ω ε ) + ε α | κ (0) | q(cid:12)(cid:12) Γ (0) ε (cid:12)(cid:12) k v k L (Γ (0) ε ) + X i =1 ε α i max [0 ,ℓ i ] × [0 ,T ] | κ i (0 , x i , t ) | q(cid:12)(cid:12) Γ ( i ) ε (cid:12)(cid:12) k v k L (Γ ( i ) ε ) . (3.9) Now with the help of (3.2) and (3.6), we obtain (cid:10) A ε u, v (cid:11) ε ≤ C (cid:18) ε α + X i =1 ε α i − (cid:19)(cid:16) ε + k u k ε (cid:17) k v k ε ∀ u, v ∈ H ε and a.e. t ∈ (0 , T ) . Thus, the existence and uniqueness of the weak solution for every fixed value ε follow directly fromCorollary 4.1 (see [47, Chapter 3]).3.1. A priori estimates.
Taking into account (3.8), (3.2) and (3.6), we derive from (3.7) that12 Z Ω ε u ε ( x, τ ) dx + c Z τ k u k ε dt − c ε τ (cid:18) | k (0) | + ε α | κ (0) | + X i =1 ε α i − max [0 ,ℓ i ] × [0 ,T ] | κ i (0 , x i , t ) | (cid:19) ≤ δ Z τ k u k ε dt + c δ Z τ k f k L (Ω ε ) dt + ε β c δ Z τ k ϕ (0) ε k L (Γ (0) ε ) dt + X i =1 ε β i − c δ Z τ k ϕ ( i ) ε k L (Γ ( i ) ε ) dt for any τ ∈ (0 , T ] . Selecting appropriate δ > C1 – C3 ( a ) into account, weobtain the uniform estimate max t ∈ [0 ,T ] k u ε ( · , t ) k L (Ω ε ) + k u ε k L (0 ,T ; H ε ) ≤ C (cid:18) √ T ε (cid:16) | k (0) | + ε α | κ (0) | + X i =1 ε α i − max [0 ,ℓ i ] × [0 ,T ] | κ i (0 , x i , t ) | (cid:17) + k f k L (Ω ε × (0 ,T )) + ε β k ϕ (0) ε k L (Γ (0) ε × (0 ,T )) + X i =1 ε β i − k ϕ ( i ) ε k L (Γ ( i ) ε × (0 ,T )) (cid:19) ≤ C ε (3.10)for all values of the parameters { α i } i =0 and β ≥ , β i ≥ , i ∈ { , , } . Now let us consider the case C3 (a) ( α < . From the integral identity (2.3) and inequalities (2.4),(2.5), (3.2), (3.1), (3.6) and (3.10) it follows ε α Z Γ (0) ε × (0 ,T ) u ε dσ x dt ≤ C (cid:18) √ T ε (cid:16) | k (0) | + X i =1 ε α i − max [0 ,ℓ i ] × [0 ,T ] | κ i (0 , x i , t ) | (cid:17) k f k L (Ω ε × (0 ,T )) + ε β k ϕ (0) ε k L (Γ (0) ε × (0 ,T )) + X i =1 ε β i − k ϕ ( i ) ε k L (Γ ( i ) ε × (0 ,T )) (cid:17) k u ε k L (0 ,T ; H ε ) ≤ C ε . Now with the help of (3.4) we get Z Ω (0) ε × (0 ,T ) u ε dxdt ≤ C ε Z Ω (0) ε × (0 ,T ) |∇ x u ε | dxdt + ε − α ε α Z Γ (0) ε × (0 ,T ) u ε dσ x dt ! ≤ C ε ϑ , (3.11)where ϑ := min { , − α } . This means that1 ε Z Ω (0) ε × (0 ,T ) u ε dxdt ≤ C ε min { , − α } −→ ε → . (3.12)4. Formal asymptotic approximation. The case α ≥ , α i ≥ , i ∈ { , , } . In this section we assume that the functions f, k, { ϕ ( i ) ε , κ i } i =0 are smooth enough. Following theapproach of [37], we propose ansatzes of the asymptotic approximation for the solution to the problem (2.1)in the following form: PPROXIMATION FOR THE SOLUTION TO A PARABOLIC PROBLEM IN A THIN STAR-SHAPED JUNCTION 11 (1) the regular parts of the approximation ω ( i )0 ( x i , t ) + εω ( i )1 ( x i , t ) + ε u ( i )2 (cid:18) x i ε , x i , t (cid:19) + ε u ( i )3 (cid:18) x i ε , x i , t (cid:19) (4.1)is located inside of each thin cylinder Ω ( i ) ε and their terms depend both on the correspondinglongitudinal variable x i and so-called “fast variables” x i ε ( i = 1 , , N (cid:16) xε , t (cid:17) + εN (cid:16) xε , t (cid:17) + ε N (cid:16) xε , t (cid:17) (4.2)is located in a neighborhood of the node Ω (0) ε .4.1. Regular parts.
Substituting the representation (4.1) for each fixed index i ∈ { , , } into the differ-ential equation of the problem (2.1), using Taylor’s formula for the function f at the point x i = (0 ,
0) forthe function k at ω ( i )0 , and collecting coefficients at ε , we obtain − ∆ ξ i u ( i )2 ( ξ i , x i , t ) = − ∂ω ( i )0 ∂t ( x i , t ) + ∂ ω ( i )0 ∂x i ( x i , t ) − k (cid:16) ω ( i )0 ( x i , t ) (cid:17) + f ( i )0 ( x i , t ) , (4.3)where ξ i = x i ε and f ( i )0 ( x i , t ) := f ( x, t ) | x i =(0 , . It is easy to calculate the outer unit normal to Γ ( i ) ε : ν ( i ) ( x i , ξ i ) = 1 p ε | h ′ i ( x i ) | (cid:0) − εh ′ i ( x i ) , ν i ( ξ i ) (cid:1) = (cid:0) − εh ′ ( x ) , ν (1)2 ( ξ ) , ν (1)3 ( ξ ) (cid:1)p ε | h ′ ( x ) | , i = 1 , (cid:0) ν (2)1 ( ξ ) , − εh ′ ( x ) , ν (2)3 ( ξ ) (cid:1)p ε | h ′ ( x ) | , i = 2 , (cid:0) ν (3)1 ( ξ ) , ν (3)2 ( ξ ) , − εh ′ ( x ) (cid:1)p ε | h ′ ( x ) | , i = 3 , where ν i ( x i ε ) is the outward normal for the disk Υ ( i ) ε ( x i ) := { ξ i ∈ R : | ξ i | < h i ( x i ) } . Taking the view of the outer unit normal into account and putting the sum (4.1) into the third relationof the problem (2.1), we get with the help of Taylor’s formula for the function κ i the following relation: ε∂ ν i ( ξ i ) u ( i )2 ( ξ i , x i , t ) = ε h ′ i ( x i ) ∂ω ( i )0 ∂x i ( x i , t ) − ε α i κ i (cid:16) ω ( i )0 ( x i , t ) , x i , t (cid:17) + ε β i ϕ ( i ) ( ξ i , x i , t ) . (4.4)Relations (4.3) and (4.4) form the linear inhomogeneous Neumann boundary-value problem − ∆ ξ i u ( i )2 ( ξ i , x i , t ) = − ∂ω ( i )0 ∂t ( x i , t ) + ∂ ω ( i )0 ∂x i ( x i , t ) − k (cid:16) ω ( i )0 ( x i , t ) (cid:17) + f ( i )0 ( x i , t ) , ξ i ∈ Υ i ( x i ) ,∂ ν ξi u ( i )2 ( ξ i , x i , t ) = h ′ i ( x i ) ∂ω ( i )0 ∂x i ( x i , t ) − δ α i , κ i (cid:16) ω ( i )0 ( x i , t ) , x i , t (cid:17) + δ β i , ϕ ( i ) ( ξ i , x i , t ) ,ξ i ∈ ∂ Υ i ( x i ) , h u ( i )2 ( · , x i , t ) i Υ i ( x i ) = 0 , (4.5)to define u ( i )2 . Here h u ( · , x i , t ) i Υ i ( x i ) := R Υ i ( x i ) u ( ξ i , x i , t ) dξ i , the variables ( x i , t ) are regarded as param-eters from I ( i ) ε × (0 , T ) , where I ( i ) ε := { x : x i ∈ ( εℓ , ℓ i ) , x i = (0 , } . We add the third relation in (4.5)for the uniqueness of a solution.
Writing down the necessary and sufficient conditions for the solvability of the problem (4.5), we derivethe differential equation πh i ( x i ) ∂ω ( i )0 ∂t ( x i , t ) − π ∂∂x i h i ( x i ) ∂ω ( i )0 ∂x i ( x i , t ) ! + πh i ( x i ) k (cid:16) ω ( i )0 ( x i , t ) (cid:17) +2 π δ α i , h i ( x i ) κ i (cid:16) ω ( i )0 ( x i , t ) , x i , t (cid:17) = πh i ( x i ) f ( i )0 ( x i , t ) + δ β i , Z ∂ Υ i ( x i ) ϕ ( i ) ( ξ i , x i , t ) dl ξ i , ( x i , t ) ∈ I ( i ) ε × (0 , T ) , (4.6)to define ω ( i )0 ( i ∈ { , , } ) . Let ω ( i )0 be a solution of the differential equation (4.6) (its existence will be proved in the subsection 4.2.1).Thus, there exists a unique solution to the problem (4.5) for each i ∈ { , , } . For determination of the coefficients u ( i )3 , i = 1 , , , we similarly obtain the following problems: − ∆ ξ i u ( i )3 ( ξ i , x i , t ) = − ∂ω ( i )1 ∂t ( x i , t ) + ∂ ω ( i )1 ∂x i ( x i , t ) − k ′ (cid:16) ω ( i )0 ( x i , t ) (cid:17) ω ( i )1 ( x i , t ) + f ( i )1 ( ξ i , x i , t ) ,ξ i ∈ Υ i ( x i ) ,∂ ν ξi u ( i )3 ( ξ i , x i , t ) = h ′ i ( x i ) dω ( i )1 dx i ( x i , t ) − δ α i , ∂ s κ i (cid:16) ω ( i )0 ( x i , t ) , x i , t (cid:17) ω ( i )1 ( x i , t ) − δ α i , κ i (cid:16) ω ( i )0 ( x i , t ) , x i , t (cid:17) + δ β i , ϕ ( i ) ( ξ i , x i , t ) , ξ i ∈ ∂ Υ i ( x i ) , h u ( i )3 ( · , x i , t ) i Υ i ( x i ) = 0 , (4.7)for each i ∈ { , , } . Here f ( i )1 ( ξ i , x i , t ) = X j =1 (1 − δ ij ) ξ j ∂∂x j f ( x, t ) | x i =(0 , . Repeating the previous reasoning, we find that the coefficients { ω ( i )1 } i =1 have to be solutions to therespective linear ordinary differential equation πh i ( x i ) ∂ω ( i )1 ∂t ( x i , t ) − π ∂∂x i h i ( x i ) ∂ω ( i )1 ∂x i ( x i , t ) ! + πh i ( x i ) k ′ (cid:16) ω ( i )0 ( x i , t ) (cid:17) ω ( i )1 ( x i , t ) + 2 π δ α i , h i ( x i ) ∂ s κ i (cid:16) ω ( i )0 ( x i , t ) , x i , t (cid:17) ω ( i )1 ( x i , t )= Z Υ i ( x i ) f ( i )1 ( ξ i , x i , t ) dξ i − π δ α i , h i ( x i ) κ i (cid:16) ω ( i )0 ( x i , t ) , x i , t (cid:17) + δ β i , Z ∂ Υ i ( x i ) ϕ ( i ) ( ξ i , x i , t ) dl ξ i , ( x i , t ) ∈ I ( i ) ε × (0 , T ) (cid:0) i ∈ { , , } (cid:1) . (4.8)4.2. Inner part.
To obtain conditions for the functions { ω ( i ) n } i =1 , n ∈ { , } at the point (0 , , , weintroduce the inner part of the asymptotic approximation (4.2) in a neighborhood of the node Ω (0) ε . If wepass to the “fast variables” ξ = xε and tend ε to 0 , the domain Ω ε is transformed into the unboundeddomain Ξ that is the union of the domain Ξ (0) and three semibounded cylindersΞ ( i ) = { ξ = ( ξ , ξ , ξ ) ∈ R : ℓ < ξ i < + ∞ , | ξ i | < h i (0) } , i = 1 , , , i.e., Ξ is the interior of S i =0 Ξ ( i ) (see Fig. 4). PPROXIMATION FOR THE SOLUTION TO A PARABOLIC PROBLEM IN A THIN STAR-SHAPED JUNCTION 13 ξ ξ ξ Figure 4.
The domain ΞLet us introduce the following notation for parts of the boundary of the domain Ξ :Γ i = { ξ ∈ R : ℓ < ξ i < + ∞ , | ξ i | = h i (0) } , i = 1 , , , and Γ = ∂ Ξ \ (cid:16) [ i =1 Γ i (cid:17) . Substituting (4.2) into the problem (2.1) and equating coefficients at the same powers of ε , we derive thefollowing relations for N n , ( n ∈ { , , } ) : − ∆ ξ N n ( ξ, t ) = F n ( ξ, t ) , ξ ∈ Ξ ,∂ ν ξ N n ( ξ, t ) = B (0) n ( ξ, t ) , ξ ∈ Γ ,∂ ν ξi N n ( ξ, t ) = B ( i ) n ( ξ, t ) , ξ ∈ Γ i , i = 1 , , ,N n ( ξ, t ) ∼ ω ( i ) n (0 , t ) + Ψ ( i ) n ( ξ, t ) , ξ i → + ∞ , ξ i ∈ Υ i (0) , i = 1 , , . (4.9)Here F ≡ F ≡ , F ( ξ, t ) = − ∂ t N − k ( N ) + f (0 , t ) , ξ ∈ Ξ ,B (0)0 ≡ , B (0)1 = − δ α , κ ( N ) + δ β , ϕ (0) ( ξ, t ) ,B (0)2 ( ξ, t ) = − δ α , κ ′ ( N ) N − δ α , κ ( N ) + δ β , ϕ (0) ( ξ, t ) , ξ ∈ Γ ,B ( i )0 ≡ B ( i )1 ≡ , B ( i )2 ( ξ, t ) = − δ α i , κ i ( N , , t ) + δ β i , ϕ ( i ) ( ξ i , , t ) , ξ ∈ Γ i , i = 1 , , . The variable t is regarded as parameter from (0 , T ) . The right hand sides in the differential equation andboundary conditions on { Γ i } of the problem (4.9) are obtained with the help of the Taylor’s formula forthe functions f, k and ϕ ( i ) , κ , κ i at the points x = 0 , s = N and x i = 0 , i = 1 , , , respectively.The fourth condition in (4.9) appears by matching the regular and inner asymptotics in a neighborhood ofthe node, namely the asymptotics of the terms { N n } as ξ i → + ∞ have to coincide with the correspondingasymptotics of the terms { ω ( i ) n } as x i = εξ i → +0 , i = 1 , , , respectively. Expanding formally each termof the regular asymptotics in the Taylor series at the points x i = 0 and collecting the coefficients of the same powers of ε, we get Ψ ( i )0 ≡ , Ψ ( i )1 ( ξ, t ) = ξ i ∂ω ( i )0 ∂x i (0 , t ) , i = 1 , , , Ψ ( i )2 ( ξ, t ) = ξ i ∂ ω ( i )0 ∂x i (0 , t ) + ξ i ∂ω ( i )1 ∂x i (0 , t ) + u ( i )2 ( ξ i , , t ) , i = 1 , , . (4.10)A solution of the problem (4.9) at n = 1 , N n ( ξ, t ) = X i =1 Ψ ( i ) n ( ξ, t ) χ i ( ξ i ) + e N n ( ξ, t ) , (4.11)where χ i ∈ C ∞ ( R + ) , ≤ χ i ≤ χ i ( ξ i ) = ( , if ξ i ≤ ℓ , , if ξ i ≥ ℓ , i = 1 , , . Then e N n has to be a solution of the problem − ∆ ξ e N n ( ξ, t ) = e F n ( ξ, t ) , ξ ∈ Ξ ,∂ ν ξ e N n ( ξ, t ) = e B (0) n ( ξ, t ) , ξ ∈ Γ ,∂ ν ξi e N n ( ξ, t ) = e B ( i ) n ( ξ, t ) , ξ ∈ Γ i , i = 1 , , , (4.12)where e F ( ξ, t ) = X i =1 (cid:16) ξ i ∂ω ( i )0 ∂x i (0 , t ) χ ′′ i ( ξ i ) + 2 ∂ω ( i )0 ∂x i (0 , t ) χ ′ i ( ξ i ) (cid:17) , e F ( ξ, t ) = X i =1 "(cid:18) ξ i d ω ( i )0 dx i (0 , t ) + ξ i ∂ω ( i )1 ∂x i (0 , t ) + u ( i )2 ( ξ i , , t ) (cid:19) χ ′′ i ( ξ i )+ 2 (cid:18) ξ i ∂ ω ( i )0 ∂x i (0 , t ) + ∂ω ( i )1 ∂x i (0 , t ) (cid:19) χ ′ i ( ξ i ) − ∂ t N − k (cid:0) N (cid:1) + X i =1 (cid:16) ∂ t ω ( i )0 (0 , t ) + k (cid:0) ω ( i )0 (0 , t ) (cid:1)(cid:17) χ i ( ξ i ) + (cid:18) − X i =1 χ i ( ξ i ) (cid:19) f (0 , t ) , and e B (0)1 = − δ α , κ ( N ) + δ β , ϕ (0) ( ξ, t ) , e B (0)2 ( ξ, t ) = − δ α , κ ′ ( N ) N − δ α , κ ( N ) + δ β , ϕ (0) ( ξ, t ) , e B ( i )1 ≡ , e B ( i )2 ( ξ, t ) = − δ α i , (cid:16) κ i (cid:0) N , , t (cid:1) − κ i (cid:0) ω ( i )0 (0 , t ) , , t (cid:1) χ i ( ξ i ) (cid:17) + δ β i , ϕ ( i ) ( ξ i , , t ) (cid:0) − χ i ( ξ i ) (cid:1) , for i ∈ { , , } . In addition, we demand that e N n satisfies the following stabilization conditions: e N n ( ξ, t ) → ω ( i ) n (0 , t ) as ξ i → + ∞ , ξ i ∈ Υ i (0) , i = 1 , , . (4.13)The existence of a solution to the problem (4.12) in the corresponding energetic space can be obtainedfrom general results about the asymptotic behavior of solutions to elliptic problems in domains with differentexits to infinity (see e.g. [48, 49]). We will use approach proposed in [49, 50].Let C ∞ ,ξ (Ξ) be a space of functions infinitely differentiable in Ξ and finite with respect to ξ , i.e., ∀ v ∈ C ∞ ,ξ (Ξ) ∃ R > ∀ ξ ∈ Ξ ξ i ≥ R, i = 1 , , v ( ξ ) = 0 . PPROXIMATION FOR THE SOLUTION TO A PARABOLIC PROBLEM IN A THIN STAR-SHAPED JUNCTION 15
We now define a space H := (cid:16) C ∞ ,ξ (Ξ) , k · k H (cid:17) , where k v k H = sZ Ξ |∇ v ( ξ ) | dξ + Z Ξ | v ( ξ ) | | ρ ( ξ ) | dξ , and the weight function ρ ∈ C ∞ ( R ) , ≤ ρ ≤ ρ ( ξ ) = (cid:26) , if ξ ∈ Ξ (0) , | ξ i | − , if ξ i ≥ ℓ + 1 , ξ ∈ Ξ ( i ) , i = 1 , , . Definition 4.1.
A function e N n from the space H is called a weak solution of the problem (4.12) if theidentity Z Ξ ∇ e N n · ∇ v dξ = Z Ξ e F n v dξ + X i =0 Z Γ i e B ( i ) n v dσ ξ holds for all v ∈ H . Similarly as in [50], we prove the following proposition.
Proposition 4.1.
Let ρ − e F n ( · , t ) ∈ L (Ξ) , e B (0) n ( · , t ) ∈ L (Γ ) , ρ − e B ( i ) n ( · , t ) ∈ L (Γ i ) , i = 1 , , , fora.e. t ∈ (0 , T ) . Then there exist a weak solution of problem (4.12) if and only if Z Ξ e F n dξ + X i =0 Z Γ i e B ( i ) n dσ ξ = 0 . (4.14) This solution is defined up to an additive constant. The additive constant can be chosen to guarantee theexistence and uniqueness of a weak solution of problem (4.12) with the following differentiable asymptotics: b N n ( ξ, t ) = O (exp( − γ ξ )) as ξ → + ∞ , δ (2) n ( t ) + O (exp( − γ ξ )) as ξ → + ∞ , δ (3) n ( t ) + O (exp( − γ ξ )) as ξ → + ∞ , (4.15) where γ i , i = 1 , , are positive constants. The values δ (2) n and δ (3) n in (4.15) are defined as follows: δ ( i ) n ( t ) = Z Ξ N i ( ξ ) e F n ( ξ, t ) dξ + X j =0 Z Γ j N i ( ξ ) e B ( j ) n ( ξ, t ) dσ ξ , i = 2 , , n ∈ { , , } , (4.16)where N and N are special solutions to the corresponding homogeneous problem − ∆ ξ N = 0 in Ξ , ∂ ν N = 0 on ∂ Ξ , (4.17)for the problem (4.12). Proposition 4.2.
The problem (4.17) has two linearly independent solutions N and N that do not belongto the space H and they have the following differentiable asymptotics: N ( ξ ) = − ξ πh (0) + O (exp( − γ ξ )) as ξ → + ∞ ,ξ πh (0) + C (2)2 + O (exp( − γ ξ )) as ξ → + ∞ ,C (3)2 + O (exp( − γ ξ )) as ξ → + ∞ , (4.18) N ( ξ ) = − ξ πh (0) + O (exp( − γ ξ )) as ξ → + ∞ ,C (2)3 + O (exp( − γ ξ )) as ξ → + ∞ ,ξ πh (0) + C (3)3 + O (exp( − γ ξ )) as ξ → + ∞ , (4.19) Any other solution to the homogeneous problem, which has polynomial growth at infinity, can be presentedas a linear combination c + c N + c N . Proof.
The solution N is sought in the form of a sum N ( ξ ) = − ξ πh (0) χ ( ξ ) + ξ πh (0) χ ( ξ ) + e N ( ξ ) , where e N ∈ H and e N is the solution to the problem (4.12) with right-hand sides e F ∗ ( ξ ) = πh (0) (cid:16)(cid:0) ξ χ ′ ( ξ ) (cid:1) ′ + χ ′ ( ξ ) (cid:17) , ξ ∈ Ξ (1) , − πh (0) (cid:16)(cid:0) ξ χ ′ ( ξ ) (cid:1) ′ + χ ′ ( ξ ) (cid:17) , ξ ∈ Ξ (2) , , ξ ∈ Ξ (0) ∪ Ξ (3) . It is easy to verify that the solvability condition (4.14) is satisfied. Thus, by virtue of Proposition 4.1 thereexist a unique solution e N ∈ H that has the asymptotics e N ( ξ ) = (1 − δ j ) C ( j )2 + O (exp( − γ j ξ j )) as ξ j → + ∞ , j = 1 , , . Similar we can prove the existence of the solution N with the asymptotics (4.19).Obviously, that N and N are linearly independent and any other solution to the homogeneous problem,which has polynomial growth at infinity, can be presented as c + c N + c N . (cid:3) Remark 4.1.
To obtain formulas (4.16) it is necessary to substitute the functions b N n , N and b N n , N inthe second Green-Ostrogradsky formula Z Ξ R (cid:0) b N ∆ ξ N − N ∆ ξ b N (cid:1) dξ = Z ∂ Ξ R (cid:0) b N ∂ ν ξ N − N ∂ ν ξ b N (cid:1) dσ ξ respectively, and then pass to the limit as R → + ∞ . Here Ξ R = Ξ ∩ { ξ : | ξ i | < R, i = 1 , , } . Limit problem.
The problem (4.9) at n = 0 is as follows: − ∆ ξ N ( ξ, t ) = 0 , ξ ∈ Ξ ,∂ ν ξ N ( ξ, t ) = 0 , ξ ∈ Γ ,∂ ν ξi N ( ξ, t ) = 0 , ξ ∈ Γ i , i = 1 , , ,N ( ξ, t ) −→ ω ( i )0 (0 , t ) , ξ i → + ∞ , ξ i ∈ Υ i (0) , i = 1 , , . It is ease to verify that δ (2)0 = δ (3)0 = 0 and b N ≡ . Thus, this problem has a solution in H if and only if ω (1)0 (0 , t ) = ω (2)0 (0 , t ) = ω (3)0 (0 , t ); (4.20)in this case N = e N = ω (1)0 (0 , t ) . In the problem (4.12) at n = 1 the solvability condition (4.14) reads as follows: πh (0) ∂ω (1)0 ∂x (0 , t ) + πh (0) ∂ω (2)0 ∂x (0 , t ) + πh (0) ∂ω (3)0 ∂x (0 , t ) − δ α , (cid:12)(cid:12) Γ (cid:12)(cid:12) κ (cid:0) ω (1)0 (0 , t ) (cid:1) = − d ∗ ( t ) , (4.21) PPROXIMATION FOR THE SOLUTION TO A PARABOLIC PROBLEM IN A THIN STAR-SHAPED JUNCTION 17 where d ∗ ( t ) = δ β , Z Γ ϕ (0) ( ξ, t ) dσ ξ . Substituting (4.1) into the forth condition in (2.1) and neglecting terms of order of O ( ε ) , we arrive tothe following boundary conditions: ω ( i )0 ( ℓ i , t ) = 0 , i = 1 , , . (4.22)Thus, taking into account (4.6), (4.20), (4.21) and (4.22), we obtain for { ω ( i )0 } i =1 the following semi-linearproblem: πh i ( x i ) ∂ω ( i )0 ∂t ( x i , t ) − π ∂∂x i h i ( x i ) ∂ω ( i )0 ∂x i ( x i , t ) ! + πh i ( x i ) k (cid:16) ω ( i )0 ( x i , t ) (cid:17) + 2 πδ α i , h i ( x i ) κ i (cid:16) ω ( i )0 ( x i , t ) , x i , t (cid:17) = b F ( i )0 ( x i , t ) , ( x i , t ) ∈ I i × (0 , T ) , i = 1 , , ,ω ( i )0 ( ℓ i , t ) = 0 , t ∈ (0 , T ) , i = 1 , , ,ω (1)0 (0 , t ) = ω (2)0 (0 , t ) = ω (3)0 (0 , t ) , t ∈ (0 , T ) , P i =1 πh i (0) ∂ω ( i )0 ∂x i (0 , t ) − δ α , (cid:12)(cid:12) Γ (cid:12)(cid:12) κ (cid:0) ω (1)0 (0 , t ) (cid:1) = − d ∗ ( t ) , t ∈ (0 , T ) ,ω ( i )0 ( x i ,
0) = 0 , x i ∈ I i , i = 1 , , , (4.23)where I i := { x : x i ∈ (0 , ℓ i ) , x i = (0 , } and b F ( i )0 ( x i , t ) := πh i ( x i ) f ( x, t ) (cid:12)(cid:12) x i =(0 , + δ β i , Z ∂ Υ i ( x i ) ϕ ( i ) ( ξ i , x i , t ) dl ξ i , x ∈ I i . (4.24)The problem (4.23) is called the limit problem for problem (2.1).For functions e φ ( x ) = φ (1) ( x ) , if x ∈ I ,φ (2) ( x ) , if x ∈ I ,φ (3) ( x ) , if x ∈ I , defined on the graph I := I ∪ I ∪ I , we introduce the Sobolev space H := n e φ : φ ( i ) ∈ H ( I i ) , φ ( i ) ( ℓ i ) = 0 , i = 1 , , , and φ (1) (0) = φ (3) (0) = φ (3) (0) o with the scalar product ( e φ, e ψ ) := X i =1 π Z ℓ i h i ( x i ) dφ ( i ) dx i dψ ( i ) dx i dx i , e φ, e ψ ∈ H . Definition 4.2.
A function e ω ∈ L (0 , T ; H ) , with e ω ′ ∈ L (0 , T ; H ∗ ) , is called a weak solution to theproblem (4.23) if it satisfies the integral identity π X i =1 Z ℓ i h i ( x i ) ∂ t ω ( i ) ( x i , t ) ψ ( i ) ( x i ) dx i + ( e ω, e ψ ) + δ α , (cid:12)(cid:12) Γ (cid:12)(cid:12) κ (cid:0) ω (1)0 (0 , t ) (cid:1) ψ (1) (0)+ X i =1 (cid:16) π Z ℓ i h i ( x i ) k ( ω ( i ) ( x i , t )) ψ ( i ) ( x i ) dx i + 2 πδ α i , Z ℓ i h i ( x i ) κ i ( ω ( i ) ( x i , t ) , x i , t ) ψ ( i ) ( x i ) dx i (cid:17) = d ∗ ( t ) ψ (1) (0) + X i =1 Z ℓ i b F ( i )0 ( x i , t ) ψ ( i ) ( x i ) dx i (4.25) for any function e ψ ∈ H and a.e. t ∈ (0 , T ) , and e ω | t =0 = 0 . Similarly as was done in Section 3, the integral identity (4.25) can be rewritten as follows (cid:10) ∂ t e ω, e ψ (cid:11) + (cid:10) A ( t ) e ω, e ψ (cid:11) = (cid:10) F ( t ) , e ψ (cid:11) , for all e ψ ∈ H and a.e. t ∈ (0 , T ) , and e ω | t =0 = 0 . Here the nonlinear operator A ( t ) : H
7→ H ∗ is definedthrough the relation (cid:10) A ( t ) φ ( i ) , ψ ( i ) (cid:11) = ( e φ, e ψ ) + δ α , (cid:12)(cid:12) Γ (cid:12)(cid:12) κ (cid:0) ω (1)0 (0 , t ) (cid:1) ψ (1) (0)+ X i =1 π Z ℓ i h i k ( φ ( i ) ) ψ ( i ) dx i + 2 π δ α i , Z ℓ i h i κ i ( φ ( i ) , x i , t ) ψ ( i ) dx i ! for all e φ, e ψ ∈ H , and the linear functional F ( t ) ∈ H ∗ is defined by (cid:10) F ( t ) , e ψ (cid:11) = d ∗ ( t ) ψ (1) (0) + X i =1 Z ℓ i b F ( i )0 ψ ( i ) dx i ∀ e ψ ∈ H , where h· , ·i is the duality pairing of the dual space H ∗ and H .Using (2.2) and (2.4), we can prove that the operator A is bounded, strongly monotone, hemicontinuousand coercive. As a result, the existence and uniqueness of the weak solution to the problem (4.23) followdirectly from Corollary 4.1 (see [47, Chapter 3]).4.2.2. Problem for { e ω } . Let us verify the solvability condition (4.14) for the problem (4.12) at n = 2 .Knowing that N ≡ ω (1)0 (0 , t ) and taking into account the third relation in problem (4.5), the equality (4.14) PPROXIMATION FOR THE SOLUTION TO A PARABOLIC PROBLEM IN A THIN STAR-SHAPED JUNCTION 19 can be re-written as follows: X i =1 " πh i (0) ℓ +2 Z ℓ +1 (cid:18) ξ i ∂ ω ( i )0 ∂x i (0 , t ) + ∂ω ( i )1 ∂x i (0 , t ) (cid:19) χ ′ i ( ξ i ) dξ i − ℓ +2 Z ℓ (1 − χ i ( ξ i )) Z Υ i (0) (cid:16) ∂ t ω (1)0 (0 , t ) + k (cid:0) ω (1)0 (0 , t ) (cid:1) − f (0 , t ) (cid:17) dξ i dξ i − ℓ +2 Z ℓ (1 − χ i ( ξ i )) Z ∂ Υ i (0) (cid:16) δ α i , κ i (cid:0) ω (1)0 (0 , t ) , , t (cid:1) − δ β i , ϕ ( i ) ( ξ i , , t ) (cid:17) dl ξ i dξ i − δ α , Z Γ κ ′ (cid:0) ω (1)0 (0 , t ) (cid:1) N ( ξ, t ) dσ ξ − δ α , Z Γ κ (cid:0) ω (1)0 (0 , t ) (cid:1) dσ ξ + δ β , Z Γ ϕ (0) ( ξ, t ) dσ ξ − Z Ξ (0) (cid:16) ∂ t ω (1)0 (0 , t ) + k (cid:0) ω (1)0 (0 , t ) (cid:1) − f (0 , t ) (cid:17) dξ = 0 . Whence, integrating by parts in the first three integrals with regard to (4.6), we obtain the following relationsfor { ω ( i )1 } i =1 : X i =1 πh i (0) ∂ω ( i )1 ∂x i (0 , t ) = d ∗ ( t ) , (4.26)where d ∗ ( t ) = − ℓ X i =1 πh i (0) (cid:16) ∂ t ω ( i )0 (0 , t ) + k (cid:0) ω (1)0 (0 , t ) (cid:1) − f (0 , t ) (cid:17) + 2 π δ α i , h i (0) κ i (cid:0) ω (1)0 (0 , t ) , , t (cid:1) − δ β i , Z ∂ Υ i (0) ϕ ( i ) ( ξ i , , t ) dl ξ i ! + δ α , κ ′ (cid:0) ω (1)0 (0 , t ) (cid:1) Z Γ N ( ξ, t ) dσ ξ + δ α , (cid:12)(cid:12) Γ (cid:12)(cid:12) κ (cid:0) ω (1)0 (0 , t ) (cid:1) − δ β , Z Γ ϕ (0) ( ξ, t ) dσ ξ + (cid:12)(cid:12) Ξ (0) (cid:12)(cid:12) (cid:16) ∂ t ω (1)0 (0 , t ) + k (cid:0) ω (1)0 (0 , t ) (cid:1) − f (0 , t ) (cid:17) . (4.27)Hence, if the functions { ω ( i )1 } i =1 satisfy (4.26), then there exist a weak solution e N of the problem (4.12).According to Proposition 4.1, it can be chosen in a unique way to guarantee the asymptotics (4.15).It remains to satisfy the stabilization conditions (4.13) at n = 1 . For this, we represent a weak solutionof the problem (4.12) in the following form: e N ( ξ, t ) = ω (1)1 (0 , t ) + b N ( ξ, t ) , ξ ∈ Ξ (1) . Taking into account the asymptotics (4.15), we have to put ω (1)1 (0 , t ) = ω (2)1 (0 , t ) − δ (2)1 ( t ) = ω (3)1 (0 , t ) − δ (3)1 ( t ) . (4.28)As a result, we get the solution of the problem (4.9) with the following asymptotics: N ( ξ, t ) = ω ( i )1 (0 , t ) + Ψ ( i )1 ( ξ, t ) + O (exp( − γ i ξ i )) as ξ i → + ∞ , i = 1 , , . (4.29)Let us denote by G ( ξ, t ) := ω ( i )1 (0 , t ) + Ψ ( i )1 ( ξ, t ) , ( ξ, t ) ∈ Ξ ( i ) × (0 , T ) , i = 1 , , . Remark 4.2.
Due to (4.29), the function N − G are exponentially decrease as ξ i → + ∞ , i = 1 , , . Relations (4.28) and (4.26) are the first and second transmission conditions for { ω ( i )1 } i =1 at x = 0 . Thus,the second term of the regular asymptotics e ω is determined from the linear problem πh i ( x i ) ∂ω ( i )1 ∂t ( x i , t ) − π ∂∂x i h i ( x i ) ∂ω ( i )1 ∂x i ( x i , t ) ! + πh i ( x i ) k ′ (cid:16) ω ( i )0 ( x i , t ) (cid:17) ω ( i )1 ( x i , t )+ 2 π δ α i , h i ( x i ) ∂ s κ i (cid:16) ω ( i )0 ( x i , t ) , x i , t (cid:17) ω ( i )1 ( x i , t ) = b F ( i )1 ( x i , t ) , ( x i , t ) ∈ I i × (0 , T ) , i = 1 , , ,ω ( i )1 ( ℓ i , t ) = 0 , t ∈ (0 , T ) , i = 1 , , ,ω (1)1 (0 , t ) = ω (2)1 (0 , t ) − δ (2)1 ( t ) = ω (3)1 (0 , t ) − δ (3)1 ( t ) , t ∈ (0 , T ) , P i =1 πh i (0) ∂ω ( i )1 ∂x i (0 , t ) = d ∗ ( t ) , t ∈ (0 , T ) ,ω ( i )1 ( x i ,
0) = 0 , x i ∈ I i , i = 1 , , , (4.30)where b F ( i )1 ( x i , t ) = Z Υ i ( x i ) f ( i )1 ( ξ i , x i , t ) dξ i − π δ α i , h i ( x i ) κ i (cid:16) ω ( i )0 ( x i , t ) , x i , t (cid:17) + δ β i , Z ∂ Υ i ( x i ) ϕ ( i ) ( ξ i , x i , t ) dl ξ i , ( x i , t ) ∈ I i × (0 , T ) , i = 1 , , . (4.31)The values δ (2)1 and δ (3)1 are uniquely determined (see Remark 4.1) by formula δ ( i )1 ( t ) = Z Ξ N i ( ξ ) X j =1 (cid:18) ξ j ∂ω ( j )0 ∂x j (0 , t ) χ ′′ j ( ξ j ) + 2 ∂ω ( j )0 ∂x j (0 , t ) χ ′ j ( ξ j ) (cid:19) dξ, + Z Γ N i ( ξ ) (cid:16) − δ α , κ (cid:0) ω (1)0 (0 , t ) (cid:1) + δ β , ϕ (0) ( ξ, t ) (cid:17) dσ ξ , i = 2 , . (4.32)With the help of the substitutions φ (1)1 ( x , t ) = ω (1)1 ( x , t ) , φ (2)1 ( x , t ) = ω (2)1 ( x , t ) − δ (2)1 ( t )( ℓ − x ) ,φ (3)1 ( x , t ) = ω (3)1 ( x , t ) − δ (3)1 ( t )( ℓ − x ) , we reduce the problem (4.30) to the respective linear parabolicproblem in the space L (cid:0) , T ; H (cid:1) . Thus the existence and uniqueness of the solution to the problem (4.30)follow from the classical theory of linear parabolic problems.5.
Justification
With the help of e ω , e ω , N and smooth cut-off functions defined by formulas χ ( i ) ℓ ( x i ) = (cid:26) , if x i ≥ ℓ , , if x i ≤ ℓ , i = 1 , , , (5.1) PPROXIMATION FOR THE SOLUTION TO A PARABOLIC PROBLEM IN A THIN STAR-SHAPED JUNCTION 21 we construct the following asymptotic approximation: U (1) ε ( x, t ) = X i =1 χ ( i ) ℓ (cid:16) x i ε a (cid:17) (cid:16) ω ( i )0 ( x i , t ) + ε ω ( i )1 ( x i , t ) (cid:17) + − X i =1 χ ( i ) ℓ (cid:16) x i ε a (cid:17)! (cid:18) ω (1)0 (0 , t ) + εN (cid:16) xε , t (cid:17) (cid:19) , x ∈ Ω ε × (0 , T ) , (5.2)where a is a fixed number from the interval (cid:0) , (cid:1) . Theorem 5.1.
Let assumptions made in the statement of the problem ( 2 . are satisfied. Then the sum ( 5 . is the asymptotic approximation for the solution u ε to the boundary-value problem ( 2 . , i.e., ∃ C > ∃ ε > ∀ ε ∈ (0 , ε ) :max t ∈ [0 ,T ] (cid:13)(cid:13)(cid:13) U (1) ε ( · , t ) − u ε ( · , t ) (cid:13)(cid:13)(cid:13) L (Ω ε ) + (cid:13)(cid:13)(cid:13) U (1) ε − u ε (cid:13)(cid:13)(cid:13) L (0 ,T ; H (Ω ε )) ≤ C µ ( ε ) , (5.3) where µ ( ε ) = o ( ε ) as ε → and µ ( ε ) = (cid:18) ε a + X i =1 (cid:16) (1 − δ α i , ) ε α i + (1 − δ β i , ) ε β i (cid:17) + (1 − δ α , ) ε α +1 + (1 − δ β , ) ε β +1 (cid:19) . (5.4) Proof.
Substituting U (0) ε in the equations and the boundary conditions of problem (2.1), we find ∂ t U (1) ε − ∆ x U (1) ε + k (cid:16) U (1) ε (cid:17) − f = b R ε in Ω ε × (0 , T ) ,∂ ν U (1) ε + ε α κ (cid:16) U (1) ε (cid:17) − ε β ϕ (0) ε = ˘ R (0) ε on Γ (0) ε × (0 , T ) ,∂ ν U (1) ε + ε α i κ i (cid:16) U (1) ε , x i , t (cid:17) − ε β i ϕ ( i ) ε = ˘ R ( i ) ε on Γ ( i ) ε × (0 , T ) , i = 1 , , ,U (1) ε = 0 on Υ ( i ) ε ( ℓ i ) × (0 , T ) , i = 1 , , , (5.5)where b R ε ( x, t ) = − X i =1 ε − a dχ ( i ) ℓ dζ i ( ζ i ) (cid:12)(cid:12)(cid:12)(cid:12) ζ i = xiε a (cid:18) ∂ω ( i )0 ∂x i ( x i , t ) − ∂ω ( i )0 ∂x i (0 , t ) + ε ∂ω ( i )1 ∂x i ( x i , t ) − (cid:16) ∂N ∂ξ i ( ξ, t ) − ∂G ∂ξ i ( ξ, t ) (cid:17)(cid:12)(cid:12)(cid:12) ξ = xε (cid:19) + ε − a d χ ( i ) ℓ dζ i ( ζ i ) (cid:12)(cid:12)(cid:12)(cid:12) ζ i = xiε a (cid:18) ω ( i )0 ( x i , t ) − ω ( i )0 (0 , t ) − x i ∂ω ( i )0 ∂x i (0 , t ) + εω ( i )1 ( x i , t ) − εω ( i )1 (0 , t ) − εN (cid:16) xε , t (cid:17) + εG (cid:16) xε , t (cid:17)(cid:19) + χ ( i ) ℓ (cid:16) x i ε a (cid:17) (cid:18) − ∂ω ( i )0 ∂t ( x i , t ) + ∂ ω ( i )0 ∂x i ( x i , t ) − ε ∂ω ( i )1 ∂t ( x i , t ) + ε ∂ ω ( i )1 ∂x i ( x i , t ) (cid:19)! + − X i =1 χ ( i ) ℓ (cid:16) x i ε a (cid:17)! ∂ω ( i )0 ∂t (0 , t ) + ε ∂N ∂t (cid:16) xε , t (cid:17)! + k (cid:16) U (1) ε ( x, t ) (cid:17) − f ( x, t ) , and˘ R (0) ε ( x, t ) = ε α κ (cid:16) U (1) ε ( x, t ) (cid:17) − δ α , κ (cid:0) ω (1)0 (0 , t ) (cid:1) − ε β ϕ (0) ε ( x, t ) + δ β , ϕ (0) ε ( x, t ) , ˘ R ( i ) ε ( x, t ) = − εh ′ i ( x i ) p ε | h ′ i ( x i ) | χ ( i ) ℓ (cid:16) x i ε a (cid:17) ∂ω ( i )0 ∂x i ( x i , t ) + ε ∂ω ( i )1 ∂x i ( x i , t ) ! + ε α i κ i (cid:16) U (1) ε ( x, t ) , x i , t (cid:17) − ε β i ϕ ( i ) ε ( x, t ) , i = 1 , , . Since ω ( i )0 ( x i ,
0) = ω ( i )1 ( x i ,
0) = 0 , x i ∈ I i , i = 1 , , , it follows from (4.9) at n = 1 that N (cid:12)(cid:12) t =0 = 0 . As result, asymptotic approximation (5.2) leaves no residualsin the initial condition, i.e., U (1) ε (cid:12)(cid:12) t =0 = 0 in Ω ε . From (5.5) we derive the following integral relation: Z Ω ε ∂ t U (1) ε v dx + Z Ω ε ∇ U (1) ε · ∇ v dx + Z Ω ε k ( U (1) ε ) v dx + ε α Z Γ (0) ε κ ( U (1) ε ) v dσ x + X i =1 ε α i Z Γ ( i ) ε κ i ( U (1) ε , x i , t ) v dσ x − Z Ω ε f v dx − X i =0 ε β i Z Γ ( i ) ε ϕ ( i ) ε v dσ x = R ε ( v ) , (5.6)for all v ∈ L (0 , T ; H ε ) and a.e. t ∈ (0 , T ) . Here R ε ( v ) = Z Ω ε b R ε v dx + X i =0 Z Γ ( i ) ε ˘ R ( i ) ε v dσ x . From (4.5) and (4.7) we deduce that integral identities Z Υ i ( x i ) (cid:18) − ∂ω ( i )0 ∂t + ∂ ω ( i )0 ∂x i (cid:19) η dξ i = Z Υ i ( x i ) ∇ ξ i u ( i )2 · ∇ ξ i η dξ i − Z ∂ Υ i ( x i ) h ′ i ∂ω ( i )0 ∂x i η dl ξ i + Z Υ i ( x i ) k (cid:0) ω ( i )0 (cid:1) η dξ i + δ α i , Z ∂ Υ i ( x i ) κ i (cid:0) ω ( i )0 , x i , t (cid:1) η dl ξ i − Z Υ i ( x i ) f ( i )0 η dξ i − δ β i , Z ∂ Υ i ( x i ) ϕ ( i ) η dl ξ i (5.7)and Z Υ i ( x i ) (cid:18) − ∂ω ( i )1 ∂t + ∂ ω ( i )1 ∂x i (cid:19) η dξ i = Z Υ i ( x i ) ∇ ξ i u ( i )3 · ∇ ξ i η dξ i − Z ∂ Υ i ( x i ) h ′ i ∂ω ( i )1 ∂x i η dl ξ i + Z Υ i ( x i ) k ′ (cid:0) ω ( i )0 (cid:1) ω ( i )1 η dξ i + δ α i , Z ∂ Υ i ( x i ) ∂ s κ i (cid:0) ω ( i )0 , x i , t (cid:1) ω ( i )1 η dl ξ i − Z Υ i ( x i ) f ( i )1 η dξ i + δ α i , Z ∂ Υ i ( x i ) κ i (cid:0) ω ( i )0 , x i , t (cid:1) η dl ξ i − δ β i , Z ∂ Υ i ( x i ) ϕ ( i ) η dl ξ i (5.8)hold for all η ∈ H (Υ i ( x i )) and for all ( x i , t ) ∈ I ( i ) ε × (0 , T ) , i = 1 , , . Using (5.7) and (5.8), we rewrite R ε in the form R ε ( v ) = X j =1 R ε,j ( v ) , PPROXIMATION FOR THE SOLUTION TO A PARABOLIC PROBLEM IN A THIN STAR-SHAPED JUNCTION 23 where R ε, ( v ) = Z Ω ε (cid:18) k (cid:0) U (1) ε ( x, t ) (cid:1) − X i =1 χ ( i ) ℓ (cid:16) x i ε a (cid:17) (cid:16) k (cid:0) ω ( i )0 ( x i , t ) (cid:1) + ε k ′ (cid:0) ω ( i )0 ( x i , t ) (cid:1) ω ( i )1 ( x i , t ) (cid:17) (cid:19) v ( x ) dx,R ε, ( v ) = − Z Ω ε f ( x, t ) − X i =1 χ ( i ) ℓ (cid:16) x i ε a (cid:17) (cid:18) f ( i )0 ( x i , t ) + εf ( i )1 (cid:16) x i ε , x i , t (cid:17)(cid:19)! v ( x ) dx,R ε, ( v ) = ε α Z Γ (0) ε (cid:16) κ (cid:0) U (1) ε ( x, t ) (cid:1) − δ α , κ (cid:0) ω (1)0 (0 , t ) (cid:1)(cid:17) v ( x ) dσ x − ε β Z Γ (0) ε (1 − δ β , ) ϕ (0) ε ( x, t ) v ( x ) dσ x ,R ε, ( v ) = X i =1 ε α i Z Γ ( i ) ε (cid:18) κ i (cid:0) U (1) ε ( x, t ) , x i , t (cid:1) − χ ( i ) ℓ (cid:16) x i ε a (cid:17) (cid:16) δ α i , κ i (cid:0) ω ( i )0 ( x i , t ) , x i , t (cid:1) + ε δ α i , ∂ s κ i (cid:0) ω ( i )0 ( x i , t ) , x i , t (cid:1) ω ( i )1 ( x i , t ) + δ α i , κ i (cid:0) ω ( i )0 ( x i , t ) , x i , t (cid:1)(cid:17)(cid:19) v ( x ) dσ x ,R ε, ( v ) = − X i =1 ε β i Z Γ ( i ) ε (cid:18) − χ ( i ) ℓ (cid:16) x i ε a (cid:17) ( δ β i , + δ β i , ) (cid:19) ϕ ( i ) ε ( x, t ) v ( x ) dσ x ,R ε, ( v ) = Z Ω ε (cid:18) − X i =1 χ ( i ) ℓ (cid:16) x i ε a (cid:17) (cid:19) ∂ω ( i )0 ∂t (0 , t ) + ε ∂N ∂t (cid:16) xε , t (cid:17)! v ( x ) dx,R ε, ( v ) = ε X i =1 Z Γ ( i ) ε h ′ i ( x i ) ∂ω ( i )0 ∂x i ( x i , t ) + ε ∂ω ( i )1 ∂x i ( x i , t ) ! − p ε | h ′ i ( x i ) | ! χ ( i ) ℓ (cid:16) x i ε a (cid:17) v ( x ) dσ x ,R ε, ( v ) = − ε − a X i =1 Z Ω ε dχ ( i ) ℓ dζ i ( ζ i ) (cid:12)(cid:12)(cid:12)(cid:12) ζ i = xiε a (cid:18) ∂ω ( i )0 ∂x i ( x i , t ) − ∂ω ( i )0 ∂x i (0 , t ) + ε ∂ω ( i )1 ∂x i ( x i , t ) (cid:19) v ( x ) dx,R ε, ( v ) = − ε − a X i =1 Z Ω ε d χ ( i ) ℓ dζ i ( ζ i ) (cid:12)(cid:12)(cid:12)(cid:12) ζ i = xiε a · (cid:18) ω ( i )0 ( x i , t ) − ω ( i )0 (0 , t ) − x i ∂ω ( i )0 ∂x i (0 , t ) + εω ( i )1 ( x i , t ) − εω ( i )1 (0 , t ) (cid:19) v ( x ) dx,R ε, ( v ) = − ε X i =1 Z I ( i ) ε Z Υ i ( x i ) χ ( i ) ℓ (cid:16) x i ε a (cid:17) ∇ ξ i u ( i )2 ( ξ i , x i , t ) · ∇ ξ i v ( x, t ) dξ i dx i ,R ε, ( v ) = − ε X i =1 Z I ( i ) ε Z Υ i ( x i ) χ ( i ) ℓ (cid:16) x i ε a (cid:17) ∇ ξ i u ( i )3 ( ξ i , x i , t ) · ∇ ξ i v ( x, t ) dξ i dx i ,R ε, ( v ) = − X i =1 Z Ω ε ε − a dχ ( i ) ℓ dζ i ( ζ i ) (cid:18) ∂N ∂ξ i ( ξ, t ) − ∂G ∂ξ i ( ξ, t ) (cid:19) + ε − a d χ ( i ) ℓ dζ i ( ζ i ) (cid:16) N ( ξ, t ) − G ( ξ, t ) (cid:17)!(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ζ i = xiε a , ξ = xε v ( x ) dx. Let us estimate the value R ε . Using (3.2), (3.6) and (2.4), we deduce the following estimates: | R ε,j ( v ) | ≤ ˇ C X i =1 r πℓ i max x i ∈ I i h i ( x i ) ε + vuut | Ξ (0) | + 3 πℓ X i =1 h i (0) ε a k v k L (Ω ε ) , j = 1 , , (5.9) | R ε, ( v ) | ≤ ˇ C p | Γ | (cid:0) ε α + δ α , + ε β (1 − δ β , ) (cid:1) ε k v k H (Ω ε ) , (5.10) | R ε, ( v ) | ≤ ˇ C X i =1 (cid:18)q πℓ i max x i ∈ I i h i ( x i ) ε α i +( δ αi, + δ αi, ) + p πℓ h i (0) ε α i + a (cid:19) k v k H (Ω ε ) , (5.11) | R ε, ( v ) | ≤ ˇ C X i =1 (cid:18) (1 − δ β i , − δ β i , ) q πℓ i max x i ∈ I i h i ( x i ) ε β i + ( δ β i , + δ β i , ) p πℓ h i (0) ε β i + a (cid:19) k v k H (Ω ε ) , (5.12) | R ε, ( v ) | ≤ ˇ C vuut | Ξ (0) | + 3 πℓ X i =1 h i (0) · ε a k v k L (Ω ε ) , (5.13) | R ε, ( v ) | ≤ ˇ C X i =1 q πℓ i max x i ∈ I i h i ( x i ) ε k v k H (Ω ε ) , (5.14) | R ε,j ( v ) | ≤ ˇ C X i =1 q πℓ h i (0) ε a k v k L (Ω ε ) , j = 8 , , (5.15) | R ε, ( v ) | ≤ ˇ Cε k∇ x v k L (Ω ε ) , | R ε, ( v ) | ≤ ˇ Cε k∇ x v k L (Ω ε ) . (5.16)Due to the exponential decreasing of functions N − G (see Remark 4.2) and the fact that the support ofthe derivative of χ ( i ) ℓ belongs to the set { x i : 2 ℓ ε a ≤ x i ≤ ℓ ε a } , we arrive that | R ε, ( v ) | ≤ ˇ Cε − − a exp (cid:18) − ℓ ε − a min i =1 , , γ i (cid:19) k v k L (Ω ε ) . (5.17)Subtracting the integral identity ( 2 . . t ∈ (0 , τ ) , where τ ∈ (0 , T ] , we obtain Z τ (cid:16)(cid:10) ∂ t U (1) ε − ∂ t u ε , v (cid:11) ε + (cid:10) A ε ( t ) U (1) ε − A ε ( t ) u ε , v (cid:11) ε (cid:17) dt = Z τ R ε ( v ) dt ∀ v ∈ L (0 , T ; H ε ) . (5.18)Now set v = U (1) ε − u ε in (5.18). Then, taking into account that A ε is strongly monotone and (5.9)–(5.17),we arrive to the inequality (cid:13)(cid:13)(cid:13) U (1) ε ( · , τ ) − u ε ( · , τ ) (cid:13)(cid:13)(cid:13) L (Ω ε ) + (cid:13)(cid:13)(cid:13) U (1) ε − u ε (cid:13)(cid:13)(cid:13) L (0 ,τ ; H ε ) ≤ Cµ ( ε ) (cid:13)(cid:13)(cid:13) U (1) ε − u ε (cid:13)(cid:13)(cid:13) L (0 ,τ ; H (Ω ε )) , whence thanks to (3.1) it follows (5.3). (cid:3) Corollary 5.1.
The differences between the solution u ε of problem ( 2 . and the sum U (0) ε ( x, t ) = X i =1 χ ( i ) ℓ (cid:16) x i ε a (cid:17) ω ( i )0 ( x i , t ) + − X i =1 χ ( i ) ℓ (cid:16) x i ε a (cid:17)! ω (1)0 (0 , t ) , x ∈ Ω ε × (0 , T ) admit the following asymptotic estimate: max t ∈ [0 ,T ] k u ε ( · , t ) − U (0) ε ( · , t ) k L (Ω ε ) + k u ε − U (0) ε k L (0 ,T ; H (Ω ε )) ≤ e C µ ( ε ) , (5.19) PPROXIMATION FOR THE SOLUTION TO A PARABOLIC PROBLEM IN A THIN STAR-SHAPED JUNCTION 25 where µ ( ε ) is defined in (5.4), and a is a fixed number from the interval (cid:0) , (cid:1) . In each thin cylinder Ω ( i ) ε, a := Ω ( i ) ε ∩ (cid:8) x ∈ R : x i ∈ I ( i ) ε, a := (3 ℓ ε a , ℓ i ) (cid:9) , ( i = 1 , , the following estimateholds: max t ∈ [0 ,T ] k u ε ( · , t ) − ω ( i )0 ( · , t ) k L (Ω ( i ) ε, a ) + k u ε − ω ( i )0 k L (0 ,T ; H (Ω ( i ) ε, a )) ≤ e C µ ( ε ) , (5.20) where { ω ( i )0 } i =1 is the solution of the limit problem ( 4 .
23 ) . In the neighbourhood Ω (0) ε,ℓ := Ω ε ∩ (cid:8) x : x i < ℓ ε, i = 1 , , (cid:9) of the node Ω (0) ε , we get estimates k ∇ x u ε − ∇ ξ N k L (cid:0) Ω (0) ε,ℓ × (0 ,T ) (cid:1) ≤ e C µ ( ε ) . (5.21) Proof.
Denote by χ ( i ) ℓ , a ,ε ( · ) := χ ( i ) ℓ ( · ε a ) (the function χ ( i ) ℓ is determined in (5.1)) and k v k ∗ Ω := max t ∈ [0 ,T ] k v ( · , t ) k L (Ω) + k v k L (0 ,T ; H (Ω)) . Using the smoothness of the functions { ω ( i )1 } i =1 and the exponential decay of the functions { N − G } , i = 1 , , , at infinity, we deduce the inequality (5.19) from estimate (5.3), namely (cid:13)(cid:13)(cid:13) u ε − U (0) ε (cid:13)(cid:13)(cid:13) ∗ Ω ε ≤ (cid:13)(cid:13)(cid:13) u ε − U (1) ε (cid:13)(cid:13)(cid:13) ∗ Ω ε + ε (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X i =1 χ ( i ) ℓ , a ,ε ω ( i )1 + (cid:18) − X i =1 χ ( i ) ℓ , a ,ε (cid:19) N (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∗ Ω ε ≤ C µ ( ε ) + ε X i =1 (cid:13)(cid:13)(cid:13)(cid:13) χ ( i ) ℓ , a ,ε ω + (cid:0) − χ ( i ) ℓ , a ,ε (cid:1) N (cid:13)(cid:13)(cid:13)(cid:13) ∗ Ω ( i ) ε + ε k N k ∗ Ω (0) ε ≤ C µ ( ε ) + X i =1 (cid:13)(cid:13)(cid:13)(cid:13)(cid:0) − χ ( i ) ℓ , a ,ε (cid:1) x i dω ( i )0 dx i (0 , · ) (cid:13)(cid:13)(cid:13)(cid:13) ∗ Ω ( i ) ε + ε X i =1 (cid:13)(cid:13)(cid:13)(cid:13)(cid:0) − χ ( i ) ℓ , a ,ε (cid:1) (cid:16) ω ( i )1 (0 , · ) − ω ( i )1 (cid:17)(cid:13)(cid:13)(cid:13)(cid:13) ∗ Ω ( i ) ε + ε X i =1 (cid:13)(cid:13)(cid:13) ω ( i )1 (cid:13)(cid:13)(cid:13) ∗ Ω ( i ) ε + ε X i =1 (cid:13)(cid:13)(cid:13) (cid:0) − χ ( i ) ℓ , a ,ε (cid:1) ( N − G ) (cid:13)(cid:13)(cid:13) ∗ Ω ( i ) ε + ε max t ∈ [0 ,T ] k N ( · , t ) k L (Ξ (0) ) + ε k N k L (0 ,T ; H (Ξ (0) )) ≤ e C µ ( ε ) . Also with the help of estimate (5.3), we derive (cid:13)(cid:13)(cid:13) u ε − ω ( i )0 (cid:13)(cid:13)(cid:13) ∗ Ω ( i ) ε, a ≤ (cid:13)(cid:13)(cid:13) u ε − U (1) ε (cid:13)(cid:13)(cid:13) ∗ Ω ε + ε (cid:13)(cid:13)(cid:13) ω ( i )1 (cid:13)(cid:13)(cid:13) ∗ Ω ( i ) ε, a ≤ e C µ ( ε ) , whence we get (5.20).The energetic estimate (5.21) in a neighbourhood of the node Ω (0) ε follows directly from (5.3). (cid:3) Using the Cauchy-Buniakovskii-Schwarz inequality and the continuously embedding of the space H ( I ( i ) ε, a )in C (cid:0) [3 ℓ ε a , ℓ i ] (cid:1) , it follows from (5.20) the following corollary. Corollary 5.2. If h i ( x i ) ≡ h i ≡ const, i = 1 , , , then max t ∈ [0 ,T ] k ( E ( i ) ε u ε )( · , t ) − ω ( i )0 ( · , t ) k L ( I ( i ) ε, a ) + k E ( i ) ε u ε − ω ( i )0 k L (cid:0) ,T ; C ([3 ℓ ε a ,ℓ i ]) (cid:1) ≤ C µ ( ε ) ε , (5.22) where µ ( ε ) is defined in (5.4) and (cid:0) E ( i ) ε u ε (cid:1) ( x i , t ) = 1 πε h i Z Υ ( i ) ε (0) u ε ( x, t ) dx i . Asymptotic approximation in the case α < , α i ≥ , i ∈ { , , } Due to (3.12) we conclude that ω ( i )0 (0 , t ) = 0 , i ∈ { , , } , and consequently also N ≡ . Thus thelimit problem ( 4 .
23 ) splits into the following three independent problems: πh i ( x i ) ∂ω ( i )0 ∂t ( x i , t ) − π ∂∂x i h i ( x i ) ∂ω ( i )0 ∂x i ( x i , t ) ! + πh i ( x i ) k (cid:16) ω ( i )0 ( x i , t ) (cid:17) + 2 πδ α i , h i ( x i ) κ i (cid:16) ω ( i )0 ( x i , t ) , x i , t (cid:17) = b F ( i )0 ( x i , t ) , ( x i , t ) ∈ I i × (0 , T ) ,ω ( i )0 (0 , t ) = ω ( i )0 ( ℓ i , t ) = 0 , t ∈ (0 , T ) ,ω ( i )0 ( x i ,
0) = 0 , x i ∈ I i , (6.1)where { b F ( i )0 } i =1 is defined in (4.24), i ∈ { , , } . However, to construct an asymptotic approximation and to obtain asymptotic estimates in this case, weneed extra assumptions. Namely, if α ∈ [ − q, − q + 1) , q ∈ N we assume the following more strongercondition of zero-absorption: κ ∈ C q +1 ( R ) , d q +1 κ ds q +1 ∈ L ∞ ( R ) , κ (0) = dκ ds (0) = . . . = d q − κ ds q − (0) = 0 , ∃ k − > ∀ s ∈ R : d q κ ds q ( s ) ≥ k − ; (6.2)in addition, if α = − q then f ∈ C q − x (cid:0) Ω a × [0 , T ] (cid:1) ∩ C qx i (cid:16) Ω ( i ) a × [0 , T ] (cid:17) , k ∈ C q +1 ( R ) , d q kds q ∈ L ∞ ( R ) , q ∈ { , . . . , q + 1 } ,κ i ∈ C q +1 ,q − , (cid:0) R × [0 , ℓ i ] × [0 , T ] (cid:1) , d q κ i ds q ( · , x i , t ) ∈ L ∞ ( R ) , q ∈ { , . . . , q + 1 } , (6.3)uniformly with respect to x i ∈ [0 , ℓ i ] and t ∈ [0 , T ] (cid:0) i ∈ { , , } (cid:1) . Proposition 6.1.
Under conditions (6.2) ε Z Ω (0) ε × (0 ,T ) u ε dx dt ≤ C ε min { , − α q +1 } −→ as ε → . (6.4) Proof.
With the help of Taylor’s formula (with Lagrange form of the remainder) and (6.2), we obtain | κ ( s ) s | ≥ k − q ! | s | q +1 , s ∈ R . Knowing that κ ( s ) s ≥ s ∈ R (see (2.4)), we get κ ( s ) s ≥ k − q ! | s | q +1 , s ∈ R . (6.5)Similarly as in subsection 3.1, from the integral identity (2.3) and inequalities (2.4), (3.2), (3.1), (3.6) and(3.10) it follows ε α Z Γ (0) ε × (0 ,T ) κ ( u ε ) u ε dσ x dt ≤ C ε . Thanks to (6.5) and H¨older’s inequality, we get Z Γ (0) ε × (0 ,T ) u ε dσ x dt ≤ (cid:16) ε | Γ | T (cid:17) q − q +1 (cid:18) Z Γ (0) ε × (0 ,T ) | u ε | q +1 dσ x dt (cid:19) q +1 PPROXIMATION FOR THE SOLUTION TO A PARABOLIC PROBLEM IN A THIN STAR-SHAPED JUNCTION 27 ≤ (cid:16) ε | Γ | T (cid:17) q − q +1 (cid:16) q ! k − (cid:17) q +1 (cid:18) Z Γ (0) ε × (0 ,T ) κ ( u ε ) u ε dσ x dt (cid:19) q +1 ≤ C ε − α q +1 . Now with the help of (3.4) we get Z Ω (0) ε × (0 ,T ) u ε dx dt ≤ C ε Z Ω (0) ε × (0 ,T ) |∇ x u ε | dxdt + ε Z Γ (0) ε × (0 ,T ) u ε dσ x dt ! ≤ C ε ϑ , where ϑ := min { , − α q +1 } . (cid:3) Thus, in consequence of (6.4) we have the same three independent problems (6.1) to determine ω (1)0 , ω (2)0 and ω (3)0 if conditions (6.2) take place instead of C3 (a) . To avoid cumbersome formulas and calculations, we consider the case q = 1 , i.e. α ∈ [ − , , that ismore typical and realistic.6.1. The case α ∈ ( − , . For the regular parts of the approximation in each thin cylinder Ω ( i ) ε ( i ∈{ , , } ) , we propose the following ansatz: X n ∈ A ε n ω ( i ) n ( x i , t ) + ε n +2 u ( i ) n +2 (cid:18) x i ε , x i , t (cid:19) ! (6.6)where the index set A = { , − α , − α , α , , − α } ; and for the inner part of the approximationin a neighborhood of the node Ω (0) ε the ansatz looks as follows ε − α V − α ( t ) + ε − α V − α ( t ) + X n ∈ I (cid:18) ε n V n ( t ) + ε n N n (cid:16) xε , t (cid:17) (cid:19) (6.7)where the index set I = { , − α , α , } . Similarly as was done in the subsection 4.1, we obtain the linear inhomogeneous Neumann boundary-valueproblems to define coefficients { u ( i ) n +2 } : − ∆ ξ i u ( i ) n +2 = − ∂ω ( i ) n ∂t + ∂ ω ( i ) n ∂x i − δ , n k (cid:0) ω ( i )0 (cid:1) − (1 − δ n , ) (cid:18) k ′ (cid:0) ω ( i )0 (cid:1) ω ( i ) n + k ′′ (cid:0) ω ( i )0 (cid:1) K n (cid:16) { ω ( i ) j } j < n (cid:17)(cid:19) + δ , n f ( i )0 + δ , n f ( i )1 in Υ i ( x i ) ,∂ ν ξi u ( i ) n +2 = h ′ i dω ( i ) n dx i − P m ∈ A δ α i , m +1 δ m , n κ i (cid:0) ω ( i )0 , x i , t (cid:1) + (1 − δ m , n ) (cid:18) ∂ s κ i (cid:0) ω ( i )0 , x i , t (cid:1) ω ( i ) n − m + ∂ ss κ i (cid:0) ω ( i )0 , x i , t (cid:1) K n − m (cid:16) { ω ( i ) j } j < n − m (cid:17)(cid:19)! + δ β i , n +1 ϕ ( i ) on ∂ Υ i ( x i ) , (6.8)where n , j ∈ A , i ∈ { , , } ; ∂ ss κ i = ∂ κ i /∂s ; and functions K n := K n (cid:16) { z j } j < n (cid:17) , n ∈ A , are defined bythe formulas K ≡ K − α ≡ K α ≡ , K = z α z − α , K − α = z − α , K − α = z z − α + z α z − α . If n − m / ∈ A , then ω ( i ) n − m ≡ , K n − m ≡ . Also if α i = 2 + α , then ω ( i )1+ α ≡ . In (6.8) the right-hand sides f ( i )0 , f ( i )1 are defined in the subsection 4.1 and the variables ( x i , t ) areregarded as parameters from I ( i ) ε × (0 , T ) . Also, we should add conditions h u ( i ) n +2 ( · , x i , t ) i Υ i ( x i ) = 0 to theseproblems to guarantee the uniqueness of the solution. By the same way as in subsection 4.2, the coefficients N n , n ∈ I , of the inner part of the asymptotics(6.7) are determined from the following relations: − ∆ ξ N n ( ξ, t ) = F n ( ξ, t ) , ξ ∈ Ξ ,∂ ν ξ N n ( ξ, t ) = B (0) n ( ξ, t ) , ξ ∈ Γ ,∂ ν ξi N n ( ξ, t ) = B ( i ) n ( ξ, t ) , ξ ∈ Γ i , i = 1 , , ,V n ( t ) + N n ( ξ, t ) ∼ ω ( i ) n (0 , t ) + Ψ ( i ) n ( ξ, t ) , ξ i → + ∞ , ξ i ∈ Υ i (0) , i = 1 , , . (6.9)Whence, using the representation (4.11) (at n = n ∈ I ) , we get the problem − ∆ ξ e N n ( ξ, t ) = e F n ( ξ, t ) , ξ ∈ Ξ ,∂ ν ξ e N n ( ξ, t ) = e B (0) n ( ξ, t ) , ξ ∈ Γ ,∂ ν ξi e N n ( ξ, t ) = e B ( i ) n ( ξ, t ) , ξ ∈ Γ i , i = 1 , , , (6.10)to determine e N n . As before, we demand that e N n satisfies the following stabilization conditions: V n ( t ) + e N n ( ξ, t ) → ω ( i ) n (0 , t ) as ξ i → + ∞ , ξ i ∈ Υ i (0) , i = 1 , , . (6.11)The variable t in (6.9) and (6.10) is regarded as parameter from (0 , T ) . The right hand sides in the differ-ential equations and boundary conditions on { Γ i } of the problems (6.9), (6.10) and the fourth conditionsin (6.9) are similarly obtained as in subsection 4.2. As a result, we getΨ ( i ) n ( ξ, t ) = ξ i ∂ω ( i ) n − ∂x i (0 , t ) , n ∈ I \ { } , Ψ ( i )2 ( ξ, t ) = ξ i ∂ ω ( i )0 ∂x i (0 , t ) + ξ i ∂ω ( i )1 ∂x i (0 , t ) + u ( i )2 ( ξ i , , t ) , i = 1 , , .F n ≡ , n ∈ I \ { } , F ( ξ, t ) = − k (0) + f (0 , t ) , e F n ( ξ, t ) = X i =1 (cid:16) ξ i ∂ω ( i ) n − ∂x i (0 , t ) χ ′′ i ( ξ i ) + 2 ∂ω ( i ) n − ∂x i (0 , t ) χ ′ i ( ξ i ) (cid:17) , n ∈ I \ { } , e F ( ξ, t ) = X i =1 "(cid:18) ξ i d ω ( i )0 dx i (0 , t ) + ξ i ∂ω ( i )1 ∂x i (0 , t ) + u ( i )2 ( ξ i , , t ) (cid:19) χ ′′ i ( ξ i )+ 2 (cid:18) ξ i ∂ ω ( i )0 ∂x i (0 , t ) + ∂ω ( i )1 ∂x i (0 , t ) (cid:19) χ ′ i ( ξ i ) + (cid:18) − X i =1 χ i ( ξ i ) (cid:19)(cid:16) f (0 , t ) − k (0) (cid:17) ,B (0)1 ( ξ, t ) = e B (0)1 ( ξ, t ) = − κ ′ (0) V − α ( t ) + δ β , ϕ (0) ( ξ, t ) ,B (0)1 − α ( ξ, t ) = e B (0)1 − α ( ξ, t ) = − κ ′ (0) V − α ( t ) − κ ′′ (0) V − α ( t ) + δ β , − α ϕ (0) ( ξ, t ) ,B (0)2+ α ( ξ, t ) = e B (0)2+ α ( ξ, t ) = − κ ′ (0) (cid:0) V ( t ) + N ( ξ, t ) (cid:1) + δ β , α ϕ (0) ( ξ, t ) ,B (0)2 ( ξ, t ) = e B (0)2 ( ξ, t ) = − κ ′ (0) (cid:0) V − α ( t ) + N − α ( ξ, t ) (cid:1) − κ ′′ (0) (cid:0) V ( t ) + N ( ξ, t ) (cid:1) V − α ( t ) + δ β , ϕ (0) ( ξ, t ) ,B ( i ) n ≡ e B ( i ) n ≡ , n ∈ I \ { } , B ( i )2 ( ξ, t ) = − δ α i , κ i (0 , , t ) + δ β i , ϕ ( i ) ( ξ i , , t ) , e B ( i )2 ( ξ, t ) = (cid:16) − δ α i , κ i (cid:0) , , t (cid:1) + δ β i , ϕ ( i ) ( ξ i , , t ) (cid:17) (cid:0) − χ i ( ξ i ) (cid:1) , i = 1 , , . PPROXIMATION FOR THE SOLUTION TO A PARABOLIC PROBLEM IN A THIN STAR-SHAPED JUNCTION 29
The existence of a solution of the problem (6.10) in H follows from Proposition 4.1. In order to satisfysolvability conditions (4.14) of the problem (6.10) we choose the values V n − − α , n ∈ I as follows: V α ≡ ,V − α ( t ) = 1 κ ′ (0) | Γ | (cid:18) P i =1 ∂ω ( i )0 ∂x i (0 , t ) + δ β , R Γ ϕ (0) ( ξ, t ) dσ ξ (cid:19) ,V − α ( t ) = 1 κ ′ (0) | Γ | (cid:18) P i =1 ∂ω ( i ) − α ∂x i (0 , t ) − κ ′′ (0) | Γ | V − α ( t ) + δ β , − α R Γ ϕ (0) ( ξ, t ) dσ ξ (cid:19) ,V ( t ) = 1 κ ′ (0) | Γ | (cid:18) P i =1 ∂ω ( i )1+ α ∂x i (0 , t ) − κ ′′ (0) R Γ N ( ξ, t ) dσ ξ + δ β , α R Γ ϕ (0) ( ξ, t ) dσ ξ (cid:19) ,V − α ( t ) = 1 κ ′ (0) | Γ | (cid:18) P i =1 ∂ω ( i )1 ∂x i (0 , t ) − κ ′ (0) R Γ N − α ( ξ, t ) dσ ξ − κ ′′ (0) V − α ( t ) R Γ (cid:0) V ( t ) + N ( ξ, t ) (cid:1) dσ ξ + ℓ P i =1 (cid:16) πh i (0) (cid:0) k (0) − f (0 , t ) (cid:1) + 2 π δ α i , h i (0) κ i (0 , , t ) − δ β i , R Υ i (0) ϕ ( i ) ( ξ i , , t ) dl ξ i (cid:17) − | Ξ (0) | (cid:0) k (0) − f (0 , t ) (cid:1) + δ β , R Γ ϕ (0) ( ξ, t ) dσ ξ (cid:19) . (6.12)Again, according to Proposition 4.1, the solution can be chosen in a unique way to guarantee the asymptotics(4.15) with values δ (2) n and δ (3) n (at n = n ∈ I ) . It remains to satisfy the stabilization conditions (6.11) at n ∈ { , − α } . Taking into account theasymptotics (4.15), we have to put ω (1) n (0 , t ) = V n ( t ) , ω (2) n (0 , t ) = V n ( t ) + δ (2) n ( t ) , ω (3) n (0 , t ) = V n ( t ) + δ (3) n ( t ) , n ∈ { , − α } . (6.13)As a result, we get the solution of the problem (6.9) with the following asymptotics: N n ( ξ, t ) = − V n ( t ) + ω ( i ) n (0 , t ) + Ψ ( i ) n ( ξ, t ) + O (exp( − γ i ξ i )) as ξ i → + ∞ , i = 1 , , . (6.14)To complete matching the regular and inner asymptotics, we put ω ( i )1+ α (0 , t ) = 0 , ω ( i ) − α (0 , t ) = V − α ( t ) , ω ( i ) − α (0 , t ) = V − α ( t ) , i = 1 , , . (6.15)With the help of the necessary and sufficient condition for the solvability of the problem (6.8) and conditions(6.13), (6.15), we get the following problems for ω (1) n , ω (2) n and ω (3) n ( n ∈ A \ { } ) : πh i ( x i ) ∂ω ( i ) n ∂t ( x i , t ) − π ∂∂x i h i ( x i ) ∂ω ( i ) n ∂x i ( x i , t ) ! + πh i ( x i ) k ′ (cid:16) ω ( i )0 ( x i , t ) (cid:17) ω ( i ) n ( x i , t )+ 2 π δ α i , h i ( x i ) ∂ s κ i (cid:16) ω ( i )0 ( x i , t ) , x i , t (cid:17) ω ( i ) n ( x i , t ) = b F ( i ) n ( x i , t ) , ( x i , t ) ∈ I i × (0 , T ) ,ω ( i ) n (0 , t ) = V n ( t ) + δ ( i ) n ( t ) , ω ( i ) n ( ℓ i , t ) = 0 , t ∈ (0 , T ) ,ω ( i ) n ( x i ,
0) = 0 , x i ∈ I i , (6.16) for each i ∈ { , , } . Here the values V n are defined in (6.12), b F ( i ) n ( x i , t ) = − πh i ( x i ) k ′′ (cid:16) ω ( i )0 ( x i , t ) (cid:17) K n (cid:16) { ω ( i ) j ( x i , t ) } j < n (cid:17) − πh i ( x i ) X m ∈ R δ α i , m +1 δ m , n κ i (cid:16) ω ( i )0 ( x i , t ) , x i , t (cid:17) + (1 − δ m , n ) (cid:18) ∂ s κ i (cid:16) ω ( i )0 ( x i , t ) , x i , t (cid:17) ω ( i ) n − m + ∂ ss κ i (cid:16) ω ( i )0 ( x i , t ) , x i , t (cid:17) K n − m (cid:16) { ω ( i ) j ( x i , t ) } j < n − m (cid:17)(cid:19)! + δ , n Z Υ i ( x i ) f ( i )1 ( ξ i , x i , t ) dξ i + δ β i , n +1 Z ∂ Υ i ( x i ) ϕ ( i ) ( ξ i , x i , t ) dl ξ i , ( x i , t ) ∈ I i × (0 , T ) , i = 1 , , δ ( i )1+ α = δ ( i ) − α = δ ( i ) − α = 0 , i ∈ { , , } , δ (1)1 = δ (1)1 − α = 0 , and δ (2)1 , δ (3)1 and δ (2)1 − α , δ (3)1 − α are uniquely determined (see Remark 4.1) by formulas δ ( i )1 ( t ) = Z Ξ N i ( ξ ) X j =1 (cid:18) ξ j ∂ω ( j )0 ∂x j (0 , t ) χ ′′ j ( ξ j ) + 2 ∂ω ( j )0 ∂x j (0 , t ) χ ′ j ( ξ j ) (cid:19) dξ − κ ′ (0) V − α ( t ) Z Γ N i ( ξ ) dσ ξ + δ β , Z Γ N i ( ξ ) ϕ (0) ( ξ, t ) dσ ξ , i = 2 , , (6.17) δ ( i )1 − α ( t ) = Z Ξ N i ( ξ ) X j =1 (cid:18) ξ j ∂ω ( j ) − α ∂x j (0 , t ) χ ′′ j ( ξ j ) + 2 ∂ω ( j ) − α ∂x j (0 , t ) χ ′ j ( ξ j ) (cid:19) dξ − (cid:16) κ ′ (0) V − α ( t ) + κ ′′ (0) V − α ( t ) (cid:17) Z Γ N i ( ξ ) dσ ξ + δ β , − α Z Γ N i ( ξ ) ϕ (0) ( ξ, t ) dσ ξ , i = 2 , , (6.18)where N and N are defined in Proposition 4.2.The determination of the terms of the asymptotics is carried out according to the following scheme: { ω ( i )1+ α } i =1 N α { ω ( i )0 } i =1 N V { ω ( i )1 } i =1 N V − α { ω ( i ) − α } i =1 N − α V − α { ω ( i )1 − α } i =1 V − α { ω ( i ) − α } i =1PPROXIMATION FOR THE SOLUTION TO A PARABOLIC PROBLEM IN A THIN STAR-SHAPED JUNCTION 31 Comments to the scheme.
The arrows indicate the order for determining the terms of theasymptotics. We start with elements { ω ( i )0 } i =1 (see (6.1)) and move across the arrows. Herethe terms { ω ( i ) n } i =1 , n ∈ A \ { } and N n , n ∈ I are determined from the problems (6.16)and (6.9), respectively; the values V n − − α , n ∈ I are defined in (6.12). If α j = 2 + α forsome j ∈ { , , } , then ω ( j )1+ α ≡ ω ( j )1 − α doesnot depend on ω ( j ) − α . If α j = 2 + α for all j ∈ { , , } , then the dashed arrows disappearand we don’t need to find the elements { ω ( i ) − α } i =1 . The approximation does not contain theterms N α and N , they are only needed to find the values V and V − α . Thus, the asymptotic approximation in the case α ∈ ( − ,
0) has the following form: U (1 − α ) ε ( x, t ) = X i =1 χ ( i ) ℓ (cid:16) x i ε a (cid:17) (cid:16) ω ( i )0 ( x i , t ) + ε − α ω ( i ) − α ( x i , t ) + ε ω ( i )1 ( x i , t ) + ε − α ω ( i )1 − α ( x i , t ) (cid:17) + − X i =1 χ ( i ) ℓ (cid:16) x i ε a (cid:17)! ε − α V − α ( t ) + ε (cid:18) V ( t ) + N (cid:16) xε , t (cid:17) (cid:19) + ε − α (cid:18) V − α ( t ) + N − α (cid:16) xε , t (cid:17) (cid:19)! , ( x, t ) ∈ Ω ε × (0 , T ) , (6.19)where a is a fixed number from the interval (cid:0) , (cid:1) , and { χ ( i ) ℓ } i =1 are defined in (5.1). Theorem 6.1.
Let assumptions made in the statement of the problem (2.1) and (6.2), (6.3) at q = 1 aresatisfied. Then the sum (6.19) is the asymptotic approximation for the solution u ε to the boundary-valueproblem ( 2 . , i.e., ∃ C > ∃ ε > ∀ ε ∈ (0 , ε ) :max t ∈ [0 ,T ] (cid:13)(cid:13)(cid:13) U (1 − α ) ε ( · , t ) − u ε ( · , t ) (cid:13)(cid:13)(cid:13) L (Ω ε ) + (cid:13)(cid:13)(cid:13) U (1 − α ) ε − u ε (cid:13)(cid:13)(cid:13) L (0 ,T ; H (Ω ε )) ≤ C µ ( ε ) , (6.20) where µ ( ε ) = o ( ε ) as ε → and µ ( ε ) = (cid:18) ε a + X i =1 (cid:16) (1 − δ α i , − δ α i , − α ) ε α i + (1 − δ β i , − δ β i , − α ) ε β i (cid:17) + ε α + ε − α + (1 − δ β , − δ β , − α ) ε β +1 (cid:19) . (6.21) Proof.
The proof of Theorem 6.1 repeats the proof of Theorem 5.1. To avoid huge amount of calculationswe note the main differences.The residual b R ε in the differential equation in the whole domain Ω ε and the residuals ˘ R ( i ) ε in theboundary conditions on the surfaces Γ i of the thin cylinders Ω ( i ) ε ( i ∈ { , , } ) can be similarly obtainedand estimated.Let us consider the residual that asymptotic approximation (6.19) leaves in the boundary condition onthe node. We get ∂ ν U (1 − α ) ε + ε α κ (cid:16) U (1 − α ) ε (cid:17) − ε β ϕ (0) ε = ˘ R (0) ε on Γ (0) ε × (0 , T ) , where˘ R (0) ε ( x, t ) = ε α κ (cid:16) U (1 − α ) ε ( x, t ) (cid:17) − κ ′ (0) V − α ( t ) − ε − α κ ′ (0) V − α ( t ) − ε − α κ ′′ (0) V − α ( t )+ ( δ β , + δ β , − α − ε β ϕ (0) ε ( x, t ) , ( x, t ) ∈ Γ (0) ε × (0 , T ) . Denote by N (1 − α ) ε ( x, t ) := ε − α V − α ( t ) + ε (cid:18) V ( t ) + N (cid:16) xε , t (cid:17)(cid:19) + ε − α (cid:18) V − α ( t ) + N − α (cid:16) xε , t (cid:17)(cid:19) . Taking into account that U (1 − α ) ε = N (1 − α ) ε on Γ (0) ε and using Taylor’s formula κ (cid:16) U (1 − α ) ε (cid:17) = κ ′ (0) N (1 − α ) ε + N (1 − α ε Z (cid:16) N (1 − α ) ε − s (cid:17) κ ′′ ( s ) ds, we rewrite ˘ R (0) ε in the following form:˘ R (0) ε ( x, t ) = ε α κ ′ (0) (cid:18) V ( t ) + N (cid:16) xε , t (cid:17)(cid:19) + εκ ′ (0) (cid:18) V − α ( t ) + N − α (cid:16) xε , t (cid:17)(cid:19) + ε α N (1 − α ε ( x,t ) Z (cid:16) N (1 − α ) ε ( x, t ) − s (cid:17) κ ′′ ( s ) ds − ε − α κ ′ (0) V − α ( t ) − ε − α κ ′′ (0) V − α ( t )+ ( δ β , + δ β , − α − ε β ϕ (0) ε ( x, t ) , ( x, t ) ∈ Γ (0) ε × (0 , T ) . With the help of (3.6), we obtain | R ε, ( v ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z Γ (0) ε ˘ R (0) ε v dσ x (cid:12)(cid:12)(cid:12)(cid:12) ≤ ˇ C p | Γ | (cid:0) ε α + ε − α + ε β (1 − δ β , − δ β , − α ) (cid:1) ε k v k H (Ω ε ) , for all v ∈ L (0 , T ; H ε ) and a.e. t ∈ (0 , T ) . (cid:3) The case α = − . In this case we take ansatzes (4.1) for the approximation in each thin cylinderΩ ( i ) ε ( i ∈ { , , } ) and entirely repeat all calculations from the subsection 4.1. In a neighborhood of thenode Ω (0) ε we consider only one term εN (cid:16) xε , t (cid:17) . Similarly as in subsection 4.2 we derive the following relations for N : − ∆ ξ N ( ξ, t ) = 0 , ξ ∈ Ξ ,∂ ν ξ N ( ξ, t ) + κ ′ (0) N ( ξ, t ) = B (0)1 ( ξ, t ) , ξ ∈ Γ ,∂ ν ξi N ( ξ, t ) = 0 , ξ ∈ Γ i , i = 1 , , ,N ( ξ, t ) ∼ ω ( i )1 (0 , t ) + Ψ ( i )1 ( ξ, t ) , ξ i → + ∞ , ξ i ∈ Υ i (0) , i = 1 , , . (6.22)With the help of the representation (4.11) (at n = 1) , we obtain the problem − ∆ ξ e N ( ξ, t ) = e F ( ξ, t ) , ξ ∈ Ξ ,∂ ν ξ e N ( ξ, t ) + κ ′ (0) e N ( ξ, t ) = e B (0)1 ( ξ, t ) , ξ ∈ Γ ,∂ ν ξi e N ( ξ, t ) = 0 , ξ ∈ Γ i , i = 1 , , , e N ( ξ, t ) → ω ( i )1 (0 , t ) as ξ i → + ∞ , ξ i ∈ Υ i (0) , i = 1 , , , (6.23)to determine e N . Here { Ψ ( i )1 } i =1 , e F are the same as in subsection 4.2, and B (0)1 = e B (0)1 = δ β , ϕ (0) . Similarly as in subsection 4.2, we introduce the space H and prove the existence of a unique weak solutionto the problem (6.23). But in contrast to the problem (4.12) we have the Robin condition on Γ . PPROXIMATION FOR THE SOLUTION TO A PARABOLIC PROBLEM IN A THIN STAR-SHAPED JUNCTION 33
Definition 6.1.
A function e N from the space H is called a weak solution of the problem (6.23) if theidentity Z Ξ ∇ e N · ∇ v dξ + κ ′ (0) Z Γ e N v dσ ξ = Z Ξ e F v dξ + Z Γ e B (0)1 v dσ ξ holds for all v ∈ H . Proposition 6.2.
Let ρ − e F ( · , t ) ∈ L (Ξ) , e B (0)1 ( · , t ) ∈ L (Γ ) for a.e. t ∈ (0 , T ) . Then there exist aunique weak solution of problem (6.23) with the following differentiable asymptotics: e N ( ξ, t ) = δ (1)1 ( t ) + O (cid:0) exp( − γ ξ ) (cid:1) as ξ → + ∞ , δ (2)1 ( t ) + O (cid:0) exp( − γ ξ ) (cid:1) as ξ → + ∞ , δ (3)1 ( t ) + O (cid:0) exp( − γ ξ ) (cid:1) as ξ → + ∞ , (6.24) where γ i , i = 1 , , are positive constants. The values { δ ( i )1 } i =1 in (6.24) are defined as follows: δ ( i )1 ( t ) = Z Ξ N i ( ξ ) e F ( ξ, t ) dξ + Z Γ N i ( ξ ) e B (0)1 ( ξ, t ) dσ ξ , i = 1 , , , (6.25)where { N i } i =1 are special solutions to the corresponding homogeneous problem − ∆ ξ N = 0 in Ξ , ∂ ν N + κ ′ (0) N = 0 on Γ , ∂ ν N = 0 on ∂ Ξ \ Γ (6.26)for the problem (6.23). Proposition 6.3.
The problem (6.26) has three linearly independent solutions { N i } i =1 that do not belongto the space H and they have the following differentiable asymptotics: N i ( ξ ) = C (1) i + δ i, ξ πh (0) + O (cid:0) exp( − γ ξ ) (cid:1) as ξ → + ∞ ,C (2) i + δ i, ξ πh (0) + O (cid:0) exp( − γ ξ ) (cid:1) as ξ → + ∞ ,C (3) i + δ i, ξ πh (0) + O (cid:0) exp( − γ ξ ) (cid:1) as ξ → + ∞ , i = 1 , , . (6.27) Any other solution to the homogeneous problem, which has polynomial growth at infinity, can be presentedas a linear combination c N + c N + c N . In order to satisfy the forth condition in (6.23), we have to put ω ( i )1 (0 , t ) = δ ( i )1 ( t ) , i = 1 , , . (6.28)As a result, we get the solution of the problem (6.22) with the following asymptotics: N ( ξ, t ) = ω ( i )1 (0 , t ) + Ψ ( i )1 ( ξ, t ) + O (exp( − γ i ξ i )) as ξ i → + ∞ , i = 1 , , . (6.29)Taking into account (6.28) we derive for each i ∈ { , , } the problem πh i ( x i ) ∂ω ( i )1 ∂t ( x i , t ) − π ∂∂x i h i ( x i ) ∂ω ( i )1 ∂x i ( x i , t ) ! + πh i ( x i ) k ′ (cid:16) ω ( i )0 ( x i , t ) (cid:17) ω ( i )1 ( x i , t )+ 2 π δ α i , h i ( x i ) ∂ s κ i (cid:16) ω ( i )0 ( x i , t ) , x i , t (cid:17) ω ( i )1 ( x i , t ) = b F ( i )1 ( x i , t ) , ( x i , t ) ∈ I i × (0 , T ) ,ω ( i )1 (0 , t ) = δ ( i )1 ( t ) , ω ( i )1 ( ℓ i , t ) = 0 , t ∈ (0 , T ) ,ω ( i )1 ( x i ,
0) = 0 , x i ∈ I i , (6.30) to determine uniquely ω ( i )1 . Here b F ( i )1 are defined in (4.31), and δ ( i )1 ( t ) = Z Ξ N i ( ξ ) X j =1 ∂ω ( j )0 ∂x j (0 , t ) (cid:16) ξ j χ ′′ j ( ξ j ) + 2 χ ′ j ( ξ j ) (cid:17) dξ + δ β , Z Γ N i ( ξ ) ϕ (0) ( ξ, t ) dσ ξ , i = 1 , , , (6.31)where { N i } i =1 are defined in Proposition 6.3.With the help of { ω , ω } i =1 , N (see (6.1), (6.30), (6.22), respectively) we construct the followingasymptotic approximation: U (1) ε ( x, t ) = X i =1 χ ( i ) ℓ (cid:16) x i ε a (cid:17) (cid:16) ω ( i )0 ( x i , t ) + ε ω ( i )1 ( x i , t ) (cid:17) + − X i =1 χ ( i ) ℓ (cid:16) x i ε a (cid:17)! εN (cid:16) xε , t (cid:17) , ( x, t ) ∈ Ω ε × (0 , T ) , (6.32)where a is a fixed number from the interval (cid:0) , (cid:1) , and { χ ( i ) ℓ } i =1 are defined in (5.1). Theorem 6.2.
Let assumptions made in the statement of the problem (2.1) and (6.2) at q = 1 are sat-isfied. Then the sum (6.32) is the asymptotic approximation for the solution u ε to the boundary-valueproblem ( 2 . , i.e., ∃ C > ∃ ε > ∀ ε ∈ (0 , ε ) :max t ∈ [0 ,T ] (cid:13)(cid:13)(cid:13) U (1) ε ( · , t ) − u ε ( · , t ) (cid:13)(cid:13)(cid:13) L (Ω ε ) + (cid:13)(cid:13)(cid:13) U (1) ε − u ε (cid:13)(cid:13)(cid:13) L (0 ,T ; H (Ω ε )) ≤ C µ ( ε ) , (6.33) where µ ( ε ) = o ( ε ) as ε → and µ ( ε ) = (cid:18) ε a + X i =1 (cid:16) (1 − δ α i , ) ε α i + (1 − δ β i , ) ε β i (cid:17) + (1 − δ β , ) ε β +1 (cid:19) . (6.34) Proof.
The proof of Theorem 6.2 repeats the proof of Theorem 5.1. The only difference is the residual onthe boundary of the node, namely˘ R (0) ε ( x, t ) = ε − κ (cid:16) U (1) ε ( x, t ) (cid:17) − κ ′ (0) N (cid:16) xε , t (cid:17) + ( δ β , − ε β ϕ (0) ε ( x, t ) , ( x, t ) ∈ Γ (0) ε × (0 , T ) . We estimate the value R ε, ( v ) = Z Γ (0) ε ˘ R (0) ε v dσ x with the help of (3.6) and Taylor’s formula. As a result, we get | R ε, ( v ) | ≤ ˇ C p | Γ | (cid:0) ε + ε β (1 − δ β , ) (cid:1) ε k v k H (Ω ε ) , for all v ∈ L (0 , T ; H ε ) and a.e. t ∈ (0 , T ) . (cid:3) Remark 6.1.
As we have argued inequalities (5.20), (5.21) and (5.22), we can prove similar inequalities inthe case α ∈ [ − , using (6.20) and (6.33). Comments At first glance it may seem that there is no difference between the nonlinear Robin condition (1.1) inthe problem (2.1) and the corresponding linear Neumann condition, since the term κ i ( u ε , x i , t ) is multipliedby ε α i ( i ∈ { , , } ) . However, this is true only if α i > . If α i = 1 , then the new blow-up term2 π h i ( x i ) κ i (cid:16) ω ( i )0 ( x i , t ) , x i , t (cid:17) , which takes into account the curvilinearity of the thin cylinder Ω ( i ) ε through the function h i , appears inthe differential equation of the corresponding limit problem (see (4.23) and (6.1)). PPROXIMATION FOR THE SOLUTION TO A PARABOLIC PROBLEM IN A THIN STAR-SHAPED JUNCTION 35
What happens when α i < i ∈ { , , } ; to be specific we put α < . As in the case C3 (a)we additionally suppose that there is a constant k − such that 0 < k − ≤ κ ′ ( s, x , t ) for all s ∈ R uniformlywith respect to x ∈ [0 , ℓ ] and t ∈ [0 , T ] , and κ (0 , x , t ) = 0 . Then from the integral identity (2.3) andinequalities (2.4), (3.2), (3.1), (3.6) and (3.10) it follows ε α Z Γ (1) ε × (0 ,T ) u ε dσ x dt ≤ C (cid:16) | k (0) | ε + ε α +1 | κ (0) | + X i =2 ε α i max [0 ,ℓ i ] × [0 ,T ] | κ i (0 , x i , t ) | + k f k L (Ω ε × (0 ,T )) + ε β k ϕ (0) ε k L (Γ (0) ε × (0 ,T )) + X i =1 ε β i − k ϕ ( i ) ε k L (Γ ( i ) ε × (0 ,T )) (cid:17) k u ε k L (0 ,T ; H ε ) ≤ C ε . Now with the help of (3.3) we get Z Ω (1) ε × (0 ,T ) u ε dxdt ≤ C ε Z Ω (1) ε × (0 ,T ) |∇ x u ε | dxdt + ε − α ε α Z Γ (1) ε × (0 ,T ) u ε dσ x dt ! ≤ C ε ϑ , where ϑ := min { , − α } . This means that1 ε Z Ω (1) ε × (0 ,T ) u ε dxdt ≤ C ε min { , − α } −→ ε → . (7.1)Due to (7.1) we conclude that ω (1)0 ≡ . If α < , then we can state that there are two independentproblems (6.1) ( i = 2 and i = 3) to determine ω (2)0 and ω (3)0 . The view of the limit problem is stillunknown for α ∈ [0 , α in addition. From obtained results it follows that the asymptotic behaviour of the solution essentially depends onthe parameter α characterizing the intensity of processes at the boundary of the node. If α > β > , then the limit problem (4.23) does not feel both those processes and the node geometry. In this case,in order to take into account all these factors on the global level, we propose to consider a system consistingof the limit problem (4.23) and (4.30) on the graph. The coefficients d ∗ , δ (2)1 and δ (3)1 in the Kirchhofftransmission conditions of the problem (4.30) pay respect to all parameters { α i } i =0 , { β i } i =0 , and manyother features (see formulas (4.27) and (4.32)). This proposition is justified by Theorem 5.The same observation holds for the cases α ∈ ( − ,
0) and α = − x = 0 in the problems (6.16) indicate the dependence of these problemsboth on previous solutions ω (1) n , ω (2) n and ω (3) n ( n ∈ A ) and on other factors through the values V n and δ ( i ) n (see (6.12), (6.17) and (6.18)) for α ∈ ( − , α = − Thanks to estimates (5.20) and (5.21), we get the zero-order approximation of the gradient (flux) ofthe solution ∇ u ε ( x, t ) ∼ ∂ω ( i )0 ∂x i ( x i , t ) as ε → ( i ) ε, a ( i = 1 , ,
3) and ∇ u ε ( x, t ) ∼ ∇ ξ (cid:0) N ( ξ, t ) (cid:1)(cid:12)(cid:12)(cid:12) ξ = xε as ε → . in the neighbourhood Ω (0) ε,ℓ of the node.The estimate (5.21) is very important if we investigate processes occurring in a neighbourhood of thenode. In this case, in terms of practical application, we propose to apply numerical methods not to original problems in thin star-shaped junctions, as was done for instance in [14] without enough accuracy (see theIntroduction), and to the corresponding problem for N (see (4.9), (6.9) at n = 1 and (6.22)). An important problem of existing multi-scale methods is their stability and accuracy. The proof ofthe error estimate between the constructed approximation and the exact solution is a general principle thathas been applied to the analysis of the efficiency of a multi-scale method. In our paper, we have constructedand justified the asymptotic approximation for the solution to problem (2.1) and proved the correspondingestimates for different values of the parameters { α i } and { β i } . It should be noted here that we do notassume any orthogonality conditions for the right-hand sides in the equation and in the nonlinear Robinboundary conditions.The results obtained in Theorems 5.1, 6.1, 6.2 and Corollaries 5.1, 5.2 argue that in depending on { α i } and { β i } it is possible to replace the complex boundary-value problem (2.1) with the corresponding limitproblem (4.23) ((6.1)) on the graph I with sufficient accuracy measured by the parameter ε characterizingthe thickness and the local geometrical irregularity of the thin star-shaped junction Ω ε . References [1] J. von Below,
Classical solvability of linear parabolic equations in networks,
J. Differ. Equations, (1988) 316–337. 1[2] Y. Avishai and J. Luck, Quantum percolation and ballistic conductance on a lattice of wires , Phys. Rev. B, :3 (1992)1074–1095. 1[3] F. Ali-Mehmeti, Nonlinear waves in networks . Academie-Verlag, 1994. 1[4] M.K. Banda, M. Herty and A. Klar,
Gas Flow in Pipeline Networks.
Networks and Heter. Media, bf 1 (2006) 41–56. 1[5] Yu. Golovaty and V. Flyud,
Singularly perturbed hyperbolic problems on metric graphs: asymptotics of solutions , OpenMathematics, (2017) in print. 1[6] V. Kostrykin, J. Potthoff and R. Schrader,
Finite propagation speed for solutions of the wave equation on the metricgraphs,
J. Funct. Anal. (2012) 1198–1223. 1[7] D. Kowal, U. Sivan, O. Entin-Wohlman and Y. Imry,
Transmission through multiply-connected wire systems , Phys. Rev.B, :14 (1990) 9009–9018. 1[8] P. Kuchment, Quantum graphs I: Some basic structures , Waves Random Media, :1 (2004) 107–128. 1[9] S. Manko, Quantum-graph vertex couplings: some old and new approximations , Mathematica Bohemica, :2 (2014)259–267. 1[10] D. Pelinovsky, G. Schneider,
Bifurcations of standing localized waves on periodic graphs . Preprint, 2016,arXiv:1603.05463v2. 1[11] Yu.V. Pokornyi, O.M. Penkin, V.L. Pryadiev, A.V. Borovskikh, K.P. Lazarev and S.A. Shabrov,
Differential equations ongeometric graphs . Moskva: Fizmatlit. 2004. 1[12] D. Mugnolo,
Gaussian estimates for a heat equation on a network , Netw. Heterog.Media, (2007) 55–79. 1[13] D. Mugnolo and S. Romanelli, Dynamic and generalized Wentzell node conditions for network equations , MathematicalMethods in the Applied Sciences, (2007) 681–706. 1[14] Ø. Evju, K. Valen-Sendstad and K.A. Mardal, A study of wall shear stress in 12 aneurysms with respect to differentviscosity modelsand flow conditions , Journal of Biomechanics, (2013) 2802–2808. 1, 7[15] M.I. Freidlin and A.D. Wentzell, Diffusion processes on graphs and the averaging principle , The Annals of Probability, (1993) 2215–2245. 1[16] S. Albeverio and S. Kusuoka, Diffusion processes in thin tubes and their limits on graphs , The Annals of Probability , :5(2012) 2131–2167. 1[17] G.P. Panasenko, Method of asymptotic partial decomposition of domain , Mathematical Models and Methods in AppliedSciences, (1998) 139–156. 1[18] G.P. Panasenko, Multi-scale modelling for structures and composites . Springer, Dordrecht, 2005. 1[19] G. Cardone, A. Corbo-Esposito and G. Panasenko,
Asymptotic partial decomposition for diffusion with sorption in thinstructures , Nonlinear Analysis, (2006) 79–106. 1[20] G. Panasenko and K. Pileckas, Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure. I. Thecase without boundary-layer in time . Nonlinear Analysis, (2015) 125–168. 1[21] G. Panasenko and K. Pileckas,
Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure. II.General case.
Nonlinear Analysis, (2015) 582–607. 1[22] A. Gaudiello, G. Panasenko and A. Piatnitski,
Asymptotic analysis and domain decomposition for a biharmonic problemin a thin multi-structure , Communications in Contemporary Mathematics (2016) 1550057 (27 pages) 1 PPROXIMATION FOR THE SOLUTION TO A PARABOLIC PROBLEM IN A THIN STAR-SHAPED JUNCTION 37 [23] S.A. Nazarov and A.S. Slutskii,
Asymptotic analysis of an arbitrary spatial system of thin rods , Proceedings of the St.Petersburg Mathematical Society, (ed. N.N. Uraltseva), (2004) 59–109. 1[24] S.A. Nazarov and A.S. Slutskii, Arbitrary plane systems of anisotropic beams , Tr. Mat. Inst. Steklov., (2002) 234–261.1[25] P. Kuchment,
Graph models for waves in thin structures , Waves in Random Media, :4 (2002) 1–24. 1[26] P. Kuchment, H. Zeng, Convergence of spectra of mesoscopic systems collapsing onto a graph , J. Math. Anal. Appl. (2001) 671–700. 1[27] P. Duclos and P. Exner,
Curvature-induced bound states in quantum wavequides in two and three dimensions , Rev. Math.Phys. (1995) 73–102. 1[28] P. Exner and O. Post, Convergence of spectra of graph-like thin manifolds , J. Geom. Phys. (2005) 77–115. 1[29] D. Grieser, Spectra of graph neighborhoods and scattering , Proc. London Math. Soc. :3 (2008) 718–752. 1[30] S. Molchanov, B. Vainberg, Scattering solutions in networks of thin fibers: Small diameter asymptotics , Commun. Math.Phys. (2007) 533–559. 1[31] S. Molchanov, B. Vainberg,
Laplace operator in networks of thin fibers: Spectrum near the threshold , Preprint, 2007,arXiv:0704.2795. 1[32] O. Post,
Branched quantum wave guides with Dirichlet boundary conditions: the decoupling case , J. Phys. A, Math. Gen. (2005) 4917–4931. 1[33] J. Rubinstein and M. Schatzman, Variational problems on multiply connected thin strips. I: Basic estimates and conver-gence of the Laplacian spectrum , Arch. Ration. Mech. Anal. (2001) 271–308. 1[34] S.A. Nazarov, K. Ruotsalainen and P. Uusitalo,
Multifarious transmission conditions in the graph models of carbon nano-structures , Materials Physics and Mechanics, (2016) 107–115. 1[35] P. Kuchment and L. Kunyansky, Differential operators on graphs and photonic crystals , Adv. Comput. (2002) 263–290.1[36] H. Ueckera, D. Griesera, Z. Sobirovb, D. Babajanovc and D. Matrasulov, Soliton transport in tubular networks: transmis-sion at vertices in the shrinking limit , Preprint, 2015, arXiv:1406.0738v3 . 1[37] A.V. Klevtsovskiy and T.A. Mel’nyk,
Asymptotic expansion for the solution to a boundary-value problem in a thin cascadedomain with a local joint , Asymptotic Analysis, (2016) 265–290. 1.1, 4[38] A.V. Klevtsovskiy and T.A. Mel’nyk, Asymptotic approximations of the solution to a boundary-value problem in a thinaneurysm-type domain , Journal of Mathematical Sciences, 2017 (in print); Preprint (2017) arXiv:1702.02976v1 1.1[39] A. M. Il’in,
Matching of asymptotic expansions of solutions of boundary value problems . Translations of MathematicalMonographs, 102. American Mathematical Society, Providence, RI, 1992. 1.1[40] A.N. Kolmogorov, I. Petrovskii, and N. Piskunov,
A study of the diffusion equation with increase in the amount of substanceand its application to a biology problem , Moskow Univ. Bull. Math. A, :6 (1937) 1–25. 2.1[41] R.A. Fisher, The wave of advance of advantageous genes , Ann Eugenics, (1937) 355–369. 2.1[42] C. Conca, J.I. Diaz, A. Linan and C. Timofte, Homogenization in chemical reactive flows , Electron. J. Differential Equa-tions, 2004(40) (2004) 1–22. 2.1[43] C.V. Pao,
Nonlinear Parabolic and Elliptic Equations . Plenum Press, New York; 1992. 2.1[44] J.M. Arrieta, A.N. Carvalho, M.C. Pereira and R.P. Silva,
Semilinear parabolic problems in thin domains with a highlyoscillatory boundary , Nonlinear Analysis, (2011) 5111–5132. 2.1[45] M. Prizzi and K.P. Rybakowski, The effect of domain squeezing upon the dynamics of reaction-diffusion equations , Journalof Differential Equations, (2001) 271–320. 2.1, 2.1[46] T.A. Mel’nyk,
Homogenezation of a boundary-value problem with a nonlinear boundary condition in a thick junction oftype 3:2:1 , Math. Meth. Appl. Sci., (2008) 1005–1027. 3[47] R.E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations . Math. SurveysMonogr., vol. 49, American Mathematical Society, 1997. 2, 3, 4.2.1[48] V.A. Kondratiev and O.A. Oleinik,
Boundary-value problems for partial differential equations in non-smooth domains ,Russian Mathematical Surveys, :2 (1983) 1–86. 4.2[49] S.A. Nazarov, Junctions of singularly degenerating domains with different limit dimensions , J. Math. Sci., :6 (1996)1989–2034. 4.2[50] T.A. Mel’nyk, Homogenization of the Poisson equation in a thick periodic junction , Zeitschrift f¨ur Analysis und ihreAnwendungen, :4 (1999) 953–975. 4.2, 4.2 Faculty of Mathematics and Mechanics, Department of Mathematical Physics, Taras Shevchenko National Uni-versity of Kyiv, Volodymyrska str. 64, 01601 Kyiv, Ukraine
E-mail address ::