We give here as an example the initial-boundary-value problem for the parabolic equation ut+L(x, t, ∂x)u=f in GT, (1.1) Bj(x, t, ∂x)u= 0, onST, j= 1

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

ASYMPTOTIC FORMULAS FOR SOLUTIONS OF PARAMETER-DEPENDING ELLIPTIC BOUNDARY-VALUE

PROBLEMS IN DOMAINS WITH CONICAL POINTS

NGUYEN THANH ANH, NGUYEN MANH HUNG

Abstract. In this article, we study elliptic boundary-value problems, depend- ing on a real parameter, in domains with conical points. We present asymptotic formulas for solutions near singular points, as linear combinations of special singular functions and regular functions. These functions and the coefficients of the linear combination are regular with respect to the parameter.

1. Introduction

Elliptic boundary-value problems in domains with point singularities were thor- oughly investigated (see, e.g, [3] and the extensive bibliography in this book). We are concerned with elliptic boundary-value problems depending on a real param- eter in domains with conical points. These problems arise in considering initial- boundary-value problems for non-stationary equations with coefficients depending on time (see, e.g, [5], where the initial-boundary-value problem for strongly hyper- bolic systems with Dirichlet boundary conditions was considered). We give here as an example the initial-boundary-value problem for the parabolic equation

ut+L(x, t, ∂x)u=f in GT, (1.1) B_j(x, t, ∂_x)u= 0, onS_T, j= 1, . . . , m, (1.2)

u|t=0=ϕ onG, (1.3)

where the setsG, G_T, S_T, and the operatorsL, B_j are introduced in Section 2. For this problem we have first dealt with the unique solvability and the regularity of the generalized solution with respect to the time variable t (see [6]). After that, to investigate the regularity and the asymptotic of the solution, (1.1) and (1.2) are rewritten in the form

L(x, t, ∂x)u=f−ut in GT, (1.4) Bj(x, t, ∂x)u= 0, onST, j= 1, . . . , m. (1.5) Then (1.4), (1.5) can be regarded as a elliptic boundary-value problem depending on the parametert. This approach was suggested in [2].

2000Mathematics Subject Classification. 35J40, 35B40, 35P99, 47A55, 47A56.

Key words and phrases. Elliptic boundary problem; nonsmooth domains; conical point;

asymptotic behaviour.

c

2009 Texas State University - San Marcos.

Submitted March 5, 2009. Published September 4, 2009.

1

In the present paper we are concerned with asymptotic behaviour of the solutions near the singular points. Firstly, applying the results of the analytic perturbation theory of linear operators ([1]) and the method of linearization of polynomial op- erator pencils ([10]), we establish the smoothness with respect to the parameter of the eigenvalues, the eigenvectors of the operator pencils generated by the problems.

After that, applying the well-known results for elliptic boundary-value problems (without parameter) in the considered domains, we receive the asymptotic formu- las of the solutions as a sum of a linear combination of special singular functions and a regular function in which this functions and the coefficients of the linear combinations are regular with respect to the parameter. The present results will be applied to deal with the asymptotic behaviour of the solutions of initial-boundary- value problems for parabolic equations in cylinders with bases containing conical points in a forthcoming work.

Our paper is organized as follows. In Section 2, we introduce some needed notation and definitions. We study the spectral properties of the operator pencil generated by the problem in Section 3. Section 4 is devoted to establishing the asymptotic behaviour of the solutions in a neighborhood of the conical point.

2. Preliminaries

LetGbe a bounded domain in Rⁿ(n≥2) with the boundary ∂G. We suppose that S =∂G\ {0} is a smooth manifold andG in a neighborhood of the origin 0 coincides with the coneK ={x:x/|x| ∈Ω}, where Ω is a smooth domain on the unit sphere Sⁿ⁻¹ in Rⁿ. Let T be a positive real number or T = +∞. If A is a subset ofRⁿ, we setAT =A×(0, T). For each multi-indexα= (α1, . . . , αn)∈Nⁿ, set|α|=α1+· · ·+αn, and∂^α=∂_x^α=∂^α_x1¹. . . ∂_x^α_nⁿ.

Let us introduce some functional space used in this paper. Letlbe a nonnegative integer. We denote by W₂^l(G) the usual Sobolev space of functions defined in G with the norm

kuk_Wl

2(G)=Z

G

X

|α|≤m

|∂_x^αu|²dx^1/2 ,

and byW₂^l−12(S) the space of traces of functions fromW₂^l(G) onS with the norm kuk

W^l−

1 2

2 (S)= inf kvk_Wl

2(G):v∈W₂^l(G), v|S=u .

We define the weighted Sobolev spaceV_2,γ^l (K) (γ∈R) as the closure ofC₀^∞(K\{0}) with respect to the norm

kuk_Vl

2,γ(K)= X

|α|≤l

Z

K

r^{2(γ+|α|−l)}|∂_x^αu|²dx^1/2

, (2.1)

wherer=|x|= Pn

k=1x²_k1/2

. Ifl≥1, thenV^l−

1

γ 2(∂K) denote the space consisting of traces of functions fromV_2,γ^l (K) on the boundary∂K with the norm

kuk

V^l−

1

γ 2(∂K)= inf kvk_Vl

2,γ(K):v∈V_2,γ^l (K), v|∂K =u . (2.2) It is obvious from the definition that the spaceV_2,γ+k^l+k (K) is continuously imbed- ded into the space V_2,γ^l (K) for an arbitrary nonnegative integer k. An analogous

assertion holds for the space V_2,γ^l−12(∂K). The weighted spaces V_2,γ^l (G), Vγ^l−12(S) are defined similarly as in (2.1), (2.2) withK, ∂K replaced byG, S, respectively.

Let h be a nonnegative integer and X be a Banach space. Denote by B(X) the set of all continuous linear operators fromX into itself. ByW₂^h((0, T);X) we denote the Sobolev space ofX-valued functions defined on (0, T) with

kfk_Wh

2((0,T);X)=X^h

k=0

Z T O

d^kf(t) dt^k

2 Xdt1/2

<+∞.

For short, we set

W₂^h((0, T)) =W₂^h((0, T);C), W₂^l,h(ΩT) =W₂^h((0, T);W₂^l(Ω)), V_2,γ^l,h(GT) =W₂^h((0, T);V_2,γ^l (G)), V^l−

1 2,h

2,γ (ST) =W₂^h((0, T);V^l−

1 2

2,γ (S)).

Recall that aX-valued functionf(t) defined on [0,+∞) is said to be continuous or analytic att= +∞if the functiong(t) =f(¹_t) is continuous or analytic, respec- tively, att= 0 with a suitable definition ofg(0)∈X. In these cases we can regard f(t) as a function defined on [0,+∞] withf(+∞) =g(0). Denote byC^a([0, T];X) the set of allX-valued functions defined and analytic on [0, T] (recall that T is a positive real number or T = +∞). It is clear that if f ∈ C^a([0, T];X), then f together with all its derivatives are bounded on [0, T].

Let A be a subset ofRⁿ and f(x, t) be a complex-valued function defined on A_T =A×[0, T]. We will say that f belongs to the classC^∞,a(A_T) if and only if f ∈C^a([0, T];C^l(A)) for all nonnegative integerl.

Let

L=L(x, t, ∂_x) = X

|α|≤2m

a_α(x, t)∂_x^α

be a differential operator of order 2m defined in Q with coefficients belonging to C^∞,a(G_T) (G_T =G×[0, T]). Suppose that L(x, t, ∂_x) is elliptic on G uniformly with respect tot on [0, T], i.e, there is a positive constantc₀such that

|L^◦(x, t, ξ)| ≥c0|ξ|^2m (2.3) for all ξ ∈ Rⁿ and for all (x, t) ∈ GT. Here L^◦(x, t, ∂x) is principal part of the operatorL(x, t, ∂x); i.e,

L^◦(x, t, ∂x) = X

|α|=2m

aα(x, t)∂_x^α. Let

Bj =Bj(x, t, ∂x) = X

|α|≤µj

bj,α(x, t)∂_x^α, j= 1, . . . , m,

be a system of boundary operators onS with coefficients belonging toC^∞,a(∂G× [0, T]), ordBj =µj ≤2m−1, j = 1, . . . , m. Suppose that {Bj(x, t, ∂x)}^m_j=1 is a normal system on S uniformly with respect to t on [0, T]; i.e, the two following conditions are satisfied:

(i) µj6=µk forj 6=k,

(ii) there are positive constantscj such that

|B_j^◦(x, t, ν(x))| ≥cj, j= 1, . . . , m, (2.4) for all (x, t)∈ST. HereB^◦_j(x, t, ∂x) is the principal part ofBj(x, t, ∂x) and ν(x) is the unit outer normal toS at pointx.

In this paper, we consider asymptotic behaviour near the conical point of solu- tions of the elliptic boundary-value problem depending on the parametert:

L(x, t, ∂_x)u=f inG_T, (2.5)

Bj(x, t, ∂x)u=gj onST, j= 1, . . . , m. (2.6) 3. Spectral properties of the pencil operator generated LetL=L(t, ∂x),Bj=Bj(t, ∂x) be the principal homogenous parts ofL(x, t, ∂x), Bj(x, t, ∂x) atx= 0; i.e,

L=L(t, ∂_x) = X

|α|=2m

a_α(0, t)∂_x^α, Bj =Bj(t, ∂x) = X

|α|=µj

bjα(0, t)∂_x^α, j= 1, . . . , m.

It can be directly verified that the derivative∂_x^α has the form

∂_x^α=r^−|α|

|α|

X

p=0

P_α,p(ω, ∂_ω)(r∂r)^p, (3.1) where Pα,p(ω, ∂ω) are differential operators of order ≤ |α| −pwith smooth coeffi- cients on Ω,r=|x|, ω is an arbitrary local coordinate system onSⁿ⁻¹. Thus we can writeL(t, ∂x),Bj(t, ∂x) in the form

L(t, ∂x) =r^−2mL(ω, t, ∂ω, r∂r), Bj(t, ∂x) =r^−µjBj(ω, t, ∂ω, r∂r).

We introduce the operator

U(λ, t) = (L(ω, t, ∂ω, λ),Bj(ω, t, ∂ω, λ)), λ∈C, t∈[0, T] of the parameter-depending elliptic boundary-value problem

L(ω, t, ∂_ω, λ)u=f in Ω,

Bj(ω, t, ∂ω, λ)u=gj on∂Ω, j= 1, . . . , m

(Here the parameters areλ andt). For every fixedλ∈C, t∈[0, T] this operator continuously maps

X ≡W₂^l(Ω) intoY ≡W₂^l−2m(Ω)×

m

Y

j=1

W^l−µj−

1 2

2 (∂Ω) (l≥2m).

We can writeU(λ, t) in the form

U(λ, t) =A_2m(t)λ^2m+A_2m−1(t)λ^2m−1+· · ·+A₀(t),

whereAk(t), k= 2m,2m−1, . . . ,0 are differential operators in Ω of order 2m−k with coefficients belonging toC^∞,a(ΩT), especially

A2m(t) = ( X

|α|=2m

aα(0, t)ω^α,0, . . . ,0). (3.2) We mention now some well-known definitions ([3]). Let t0 ∈ [0, T] fixed. If λ0 ∈ C, ϕ0 ∈ X such that ϕ0 6= 0,U(λ0, t0)ϕ0 = 0, then λ0 is called an eigen- value of U(λ, t₀) andϕ₀ ∈ X is called an eigenvector corresponding to λ₀. Λ = dim kerU(λ₀, t₀) is called the geometric multiplicity of the eigenvalueλ₀.

If the elementsϕ1, . . . , ϕs ofX satisfy the equations

σ

X

q=0

1 q!

d^q

dλ^qU(λ, t0)|λ=λ₀ϕ_σ−q = 0 forσ= 1, . . . , s,

then the ordered collectionϕ₀, ϕ₁, . . . , ϕ_sis said to be a Jordan chain corresponding to the eigenvalueλ₀ of the lengths+ 1. The rank of the eigenvectorϕ₀ (rankϕ₀) is the maximal length of the Jordan chains corresponding to the eigenvectorϕ₀.

A canonical system of eigenvectors ofU(λ0, t0) corresponding to the eigenvalue λ0is a system of eigenvectors ϕ1,0, . . . , ϕΛ,0 such that rankϕ1,0 is maximal among the rank of all eigenvectors corresponding to λ0 and rankϕj,0 is maximal among the rank of all eigenvectors in any direct complement in kerU(λ0, t0) to the linear span of the vectorsϕ1,0, . . . , ϕ_j−1,0(j = 2, . . . ,Λ). The numberκj = rankϕj,0(j= 1, . . . ,Λ) are called the partial multiplicities and the sumκ=κ1+· · ·+κΛis called the algebraic multiplicity of the eigenvalueλ0.

The eigenvalue ofλ0 is called simple if both its geometric multiplicity and the rank of the corresponding eigenvector equal to one.

For each t ∈ [0, T] fixed the set of all complex number λsuch that U(λ, t) is not invertible is called the spectrum ofU(λ, t). It is known that the spectrum of U(λ, t) is an enumerable set of its eigenvalues (see [3, Th. 5.2.1]). Moreover, there are constantsδ, R such thatU(λ, t) is invertible for allt∈[0, T] and all λin the set

D:={λ∈C:|Reλ| ≤δ|Imλ|,|λ| ≥R} (3.3) (see [3, Thm. 3.6.1]).

Now we use method of linearization to investigate the smoothness of the eigen- values and the eigenvectors ofU(λ, t) with respect tot. Without loss of generality we can assume that the operatorA₀(t) is invertible. Indeed, ifλ₀is an eigenvalue ofU(λ, t) for all t∈[0, T], then

U(λ0+λ, t) =

2m

X

k=0

Aek(t)λ^k, whereAe0(t) =U(λ0, t) is invertible for allt∈[0, T]. Setting

V(λ, t) =A⁻¹₀ (t)U(λ, t), D_k(t) =A⁻¹₀ (t)A_k(t), k= 1, . . . ,2m,

we have the pencils of continuous operatorsV(λ, t),Dk(t),k= 1, . . . ,2m, fromX into itself, and

V(λ, t) =D2m(t)λ^2m+D_2m−1(t)λ^2m−1+· · ·+D1(t)λ+I, (3.4) where I is the identical operator in X. The eigenvalues and the eigenvectors of V(λ, t) are defined analogously as ofU(λ, t).

We can verify directly (or see [10, Le. 12.1]) that

I −λA(t) =C(t)E(λ, t)









 V(λ, t)

I . ..

I











F(λ, t), (3.5)

where

A =











−D1 . . . −D_2m−1 −D2m

I . ..

I











, C =











I −D1 . . . −D_2m−1 I

. .. I









 ,

E =











I P2m

k=1D_kλ^k−1 P2m

k=2D_kλ^k−2 . . . D_2m−1+D_2mλ I

. ..

I









 ,

F =









 I

−λI I . .. . ..

−λI I









 ,

(in operator matrices all the elements not indicated are assumed to be zero, and the argumentthas been omitted for the sake of brevity) andIis the identical operator inX^2m. Verifying directly we see thatC(t),E(λ, t),F(λ, t) are invertible elements ofB(X^2m) with

C⁻¹=











I D1 . . . D_2m−1 I

. .. I











, F⁻¹=









 I

−λI I . .. . ..

−λI I









 ,

E⁻¹=











I −P2m

k=1D_kλ^k−1 −P2m

k=2D_kλ^k−2 . . . −(D2m−1+D_2mλ) I

. ..

I









 .

It follows from the assumption on the analyticity of coefficients of differential operators L(x, t, ∂x) and Bj(x, t, ∂x), j = 1, . . . , m, that V(λ, t) is of the class C^a([0, T];B(X)) andA(t) is of the classC^a([0, T];B(X^2m)).

It is obvious thatU(λ, t) andV(λ, t) have the same eigenvalues with the same multiplicities and the same corresponding eigenvectors. It follows from (3.5) that the spectra except the zero of the pencilV(λ, t) and the operatorA(t) coincide for allt ∈[0, T]. We now show that for each t ∈[0, T] all eigenvalues of V(λ, t) and A(t) are nonzero. It is obvious for these ofV(λ, t). Supposeϕ= (ϕ⁽¹⁾, . . . , ϕ^(2m))∈ X^2m, ϕ6= 0 such thatA(t)ϕ= 0 for somet∈[0, T]. Thenϕ⁽¹⁾=· · ·=ϕ^(2m−1)= 0 and D2m(t)ϕ^(2m) = 0. This impliesA2m(t)ϕ^(2m)= 0, but this do not occur since kerA2m(t) ={0}which follows from (3.2).

Now we can apply [10, Le. 12.5, 12.8] to conclude that the complex number λ0

is an eigenvalue of the pencilV(λ, t) (for somet∈[0, T]) if and only ifσ0= (λ0)⁻¹ is an eigenvalue of the operatorA(t) with the same multiplicities. Hence for each t ∈ [0, T] the spectrum of the operator A(t) is a bounded set consisting nonzero eigenvalues with finite multiplicities.

Lemma 3.1. Letγ1, γ2 be real numbers,γ1< γ2 such that the linesReλ=γj, j= 1,2, do not contain any eigenvalue ofU(λ, t) and all eigenvalues of this pencil in the strip

D₁:={λ∈C:γ₁<Reλ < γ₂} (3.6) are simple for all t ∈ [0, T]. Then there are complex-valued functions λk(t) and X-valued functions ϕk(t),k= 1, . . . , N, which are analytic on[0, T]such that, for each t ∈ [0, T], {λ1(t), . . . , λN(t)} is the set of all eigenvalues of U(λ, t) in D1

and ϕk(t) are eigenvectors corresponding to the eigenvalues λk(t), k = 1, . . . , N, respectively.

Proof. Since the setDdefined in (3.3) does not contains eigenvalues ofU(λ, t) for all t ∈ [0, T], the eigenvalues of this pencil in the strip D_∞ actually are located in the bounded domain D₂ = (C\ D)∩ D₁. Moreover, the boundary ∂D₂ of D₂ contains no eigenvalues of U(λ, t) for all t ∈[0, T]. Let M be a positive number such thatkA(t)k< M for allt∈[0, T]. PutD0={σ∈C: (σ)⁻¹∈ D2,|σ|< M}.

ThenD0 is a connected bounded domain inCand for eacht∈[0, T] the spectrum of the operator A(t) consists a finite set of its simple eigenvalues and does not intersect with the boundary∂D0.

Now let t0 ∈ [0, T] andσ0 ∈ D0 be a simple eigenvalue of the operator A(t0).

Then according to the results on analytic perturbation of linear operators (see [11, Th. XII.8]), there exists a complex-valuedσ(t) defined and analytic on a subinterval containingt0of [0, T] such thatσ(t) is a simple eigenvalue ofA(t) for alltin such subinterval. We show now thatσ(t) may be continued to be defined on [0, T].

To see this, let I₀ be the maximal interval ofσ(t) considered and suppose that t₁ is the right end ofI₀ and 0< t₁< T. Since σ(t) does not go out of the domain D₀and the spectrum ofA(t₁) consists only a finite set of its eigenvalues,σ(t) must converge to an eigenvalue bσ₀ ∈ D₂ ofA(t₁) as t↑t₁ (see [1, VII.3.5]). Thus,σ(t) must coincide with the analytic functionσ(t) representing eigenvalues ofb A(t) in a subinterval containing t1,σ(tb 1) =σb0. This implies thatσ(t) admits an analytic continuation beyondt1, contradicting the supposition thatt1is the right end of the maximal intervalI0ofσ(t).

Treating the other eigenvalues ofA(t0) inD0in the same way, we receive func- tionsσ1(t), . . . , σN(t) analytic on [0, T] such that σk(t), k = 1, . . . , N, are simple eigenvalues of A(t) for all t ∈ [0, T]. One can also chooseX^2m-valued functions ηk(t),k= 1, . . . , N, analytic on [0, T] such thatηk(t) are eigenvectors corresponding to the eigenvaluesσk(t) (see [11, Th. XII.8]). Setλk(t) = (δk(t))⁻¹,k= 1, . . . , N. Then these functions are analytic functions on [0, T] and{λ1(t), . . . , λN(t)} is the set of all eigenvalues ofU(λ, t) in the stripD1 for eacht∈[0, T].

Rewrite the function η_k(t) in the form of column vector (η⁽¹⁾_k (t), . . . , η_k^(2m)(t)) (k = 1, . . . , N). Then X-valued function ϕk(t) = η_k⁽¹⁾(t) is analytic on [0, T] and ϕk(t) is an eigenvector of U(λ, t) corresponding to eigenvalues λk(t) for eacht ∈ [0, T]. Remember thatX =W₂^l(Ω),l is an arbitrary nonnegative integer. Thus, by Sobolev imbedding theorem, we haveη⁽¹⁾_k (t)∈C^∞,a(ΩT). The proof is complete.

From the assumption on the coefficients of the operatorsB_j and the assumption (2.4), we have

|B_j^◦(0, t, ν(x))| ≥ |B^◦_j(x, t, ν(x))| − |B^◦_j(0, t, ν(x))−B_j^◦(x, t, ν(x))|>0 (3.7)

(j = 1, . . . , m), for all x ∈ S sufficiently near the origin and for all t ∈ [0, T].

Bj(t, ν(x)) can be regarded as defined onKT, andBj(t, ν(x))6= 0 for allx∈∂KT

and for all t ∈ [0, T] since ν(x) are the same on each axis of the cone K. Thus, the system {Bj(t, ∂_x)}^m_j=1 is normal on ∂K for each t ∈ [0, T]. Therefore, there are boundary operatorsBj(t, ∂x),ordBj(t, ∂x) =µj<2m, j=m+ 1, . . .2m, such that the system {Bj(x, t, ∂x)}^2m_j=1 is a Dirichlet system of order 2m(for definition see [3], p. 63) on∂K for eacht∈[0, T], and the following classical Green formula

Z

K

Luvdx+

m

X

j=1

Z

∂K

B_juB⁰_j+mvds= Z

K

uL⁺vdx+

m

X

j=1

Z

∂K

B_j+muB⁰_jvds (3.8) holds foru, v ∈C₀^∞(K\ {0}) and for each t ∈[0, T]. Here L⁺ =L⁺(t, ∂x) is the formal adjoint operator ofL, i.e,

L⁺u= (−1)^2m X

|α|=2m

aα(0, t)∂_x^αu,

and B⁰_j = B⁰_j(t, ∂x) are boundary operators of order µ⁰_j = 2m−1−µj+m if j ≤ m, and of order µ⁰_j = 2m−1−µ_j−m if j ≥ m+ 1. The coefficients of B⁰_j =B⁰_j(t, ∂x), j= 1, . . . ,2m, are independent of the variablexand dependent on tanalytically on [0, T].

The operatorsL⁺(t, ∂x),B⁰_j(t, ∂x) can be written in the form

L⁺(t, ∂x) =r^−2mL⁺(ω, t, ∂ω, r∂r), (3.9) B⁰_j(t, ∂x) =r^−µ0jBj⁰(ω, t, ∂ω, r∂r). (3.10) From the Green formula (3.8) we get the following Green formula

Z

Ω

L(t, λ)ueevdx+

m

X

j=1

Z

∂Ω

Bj(t, λ)euB⁰j+m(t,−λ+ 2m−n)evds

= Z

ΩueL⁺(t,−λ+ 2m−n)evdx+

m

X

j=1

Z

∂Ω

Bj+m(t, λ)ueB⁰j(t,−λ+ 2m−n)evds foru,e ve∈C^∞(Ω) and for allt∈[0, T] (see [3, p. 206]). Here for the sake of brevity, we have omitted the argumentsωand∂_ω in the operators of this formula.

We denote byU⁺(λ, t) the operator of the boundary-value problem

L⁺(ω, t, ∂ω,−λ+ 2m−n)v=f in Ω, (3.11) Bj⁺(ω, t, ∂ω,−λ+ 2m−n)v=gj on∂Ω, j = 1, . . . , m. (3.12) Letλ0(t) be an analytic function on [0, T] such thatλ0(t) be a simple eigenvalue ofU(λ, t) for eacht∈[0, T] and letϕ∈C^∞,a(Ω_T) such thatϕ(t) be an eigenvector ofU(λ, t) corresponding to the eigenvalueλ0(t) for eacht∈[0, T]. Thenλ0(t) are simple eigenvalues of the pencilU⁺(λ, t) for allt∈[0, T] (see [3, 6.1.6]). Moreover, there exists a functionψ∈C^∞,a(ΩT) such thatψ(t) is an eigenvector ofU⁺(λ, t) corresponding to the eigenvaluesλ0(t) for eacht∈[0, T] which is analogous to the case of the pencilU(λ, t). We claim that

(L⁽¹⁾(λ₀(t), t)ϕ(t), ψ(t))_Ω +

m

X

j=1

B⁽¹⁾_j (λ₀(t), t)ϕ(t),Bj+m⁰ (−λ₀(t) + 2m−n, t)ψ(t)

∂Ω6= 0, (3.13)

for allt∈[0, T], where L⁽¹⁾(λ, t) = d

dλL(λ, t),B_j⁽¹⁾(λ, t) = d

dλBj(λ, t), j= 1, . . . , m.

We prove this by contradiction. If (3.13) is not true for some t0∈[0, T], then one can solve with respect touthe following elliptic boundary-value problem

U(λ, t₀)u=U⁽¹⁾(λ, t₀)ϕ(t₀)

≡(L⁽¹⁾(λ, t0)ϕ(t0),B1⁽¹⁾(λ, t0)ϕ(t0), . . . ,Bm⁽¹⁾(λ, t0)ϕ(t0)).

This implies that the eigenvalueλ0(t0) is not simple which is not possible. It follows from (3.13) that we can chooseψ(t)∈C^∞,a(ΩT) such that

(L⁽¹⁾(λ0(t), t)ϕ(t), ψ(t))Ω

+

m

X

j=1

B⁽¹⁾j (λ0(t), t)ϕ(t),B⁰j+m(−λ0(t) + 2m−n, t)ψ(t)

∂Ω= 1 (3.14) for all t ∈ [0, T]. Moreover, applying [3, Th. 5.1.1], we assert that there exists a neighborhoodU of the originO such that inU_T =U×[0, T] the inverseU⁻¹(λ, t) has the following representation

U⁻¹(λ, t) = P₋₁(t)

λ−λ₀(t)+P(λ, t), (3.15) whereP−1(t) is a 1-dimensional operator from fromYintoXdepending analytically ont∈[0, T] defined by

P₋₁(t)v=hhv, ψ(t)iiϕ(t), v∈ Y, (3.16) andP(λ, t) is a pencil of continuous operators fromYintoXdepending analytically on bothλ∈Candt∈[0, T]. Here

hhv, ψ(t)ii:= (v0, ψ(t))Ω+

m

X

j=1

vj, B_j+m⁰ (−λ0(t) + 2m−n, t)ψ(t)

∂Ω

forv= (v0, v1, . . . , vm)∈ Y.

4. Asymptotic behaviour of the solutions

In this section we investigate the behaviour of the solutions of the problem (2.5), (2.6) in a neighborhood of the conical point. First, let us introduce some needed lemmas.

Lemma 4.1. Let u∈V_2,β^l1,h

1(GT)be a solution of the problem

L(t, ∂x)u=f inGT, (4.1) B_j(t, ∂_x)u=g_j on S_T, j= 1, . . . , m, (4.2) where f ∈ V_2,β^l2−2m,h

2 (G_T), g_j ∈ V_2,β^{l2−µj−12,h}

2 (S_T), l₁, l₂ ≥ 2m, β₁−l₁ > β₂−l₂. Suppose that the linesReλ=−βi+li−ⁿ₂(i= 1,2) do not contain eigenvalues of the pencil U(λ, t), and all eigenvalues of this pencil in the strip −β1+l1−ⁿ₂ <

Reλ <−β2+l₂− ⁿ₂ are simple for all t ∈[0, T] which are chosen to be analytic functionsλ₁(t), λ₂(t), . . . , λ_N(t)defined on[0, T]as the result of Lemma 3.1. Then

there exists a neighborhoodV of the origin ofRⁿ such that inVT the solutionuhas representation

u(x, t) =

N

X

k=1

ck(t)r^λk(t)ϕk(ω, t) +w(x, t), (4.3) where w∈V_2,β^l2,h

2(K_T),c_k(t)∈W₂^h((0, T))and ϕ_k ∈C^∞,a(Ω_T) are eigenvectors of U(λ, t) corresponding to the eigenvaluesλk(t),k= 1, . . . , N.

Proof. For eachk= 1, . . . , N, letψk(t) be eigenvectors the pencil U⁺(λ, t) corre- sponding to the eigenvaluesλk(t) (k= 1, . . . , N) having the properties as in (3.14).

Setvk=r^{−λk(t)+2m−n}ψk fork= 1, . . . , N.

(i) First, we assume that the functionuhas the support contained inUT, where U is a certain neighborhood of 0 ∈ Rⁿ in which the domain G coincides with the cone K. By extension by zero toKT (respectively, ∂KT) we can regard u, f (respectively,gj) as functions defined inKT (respectively,∂KT).

For each t ∈ [0, T] fixed, according to results for elliptic boundary problem in a cone (see, e.g, [3, Th. 6.1.4, Th. 6.1.7]), the solution u(x, t) admits the representation (4.3) inK with

w(x, t) = 1 2πi

Z

Reλ=−β₂+l₂−ⁿ₂

r^λU⁻¹(λ, t)Fe(ω, λ, t)dλ (4.4) and

ck(t) = f(., t), vk(., t)

K+

m

X

j=1

gj(., t), B_j+m⁰ vk(., t)

∂K

= f(., t), v_k(., t)

G+

m

X

j=1

g_j(., t), B⁰_j+mv_k(., t)

S

fork= 1, . . . , N, whereFe= (r]^2mf ,r]^µ1g1,. . .,r^^µmgm).Hereeg(ω, λ, t) denotes the Mellin transformation with respect to the variablerofg(ω, r, t); i.e,

eg(ω, λ, t) = Z +∞

0

r^−λ−1g(ω, r, t)dr.

We will prove below thatw∈V_2,β^l2,h

2(KT),ck(t)∈W₂^h((0, T)).

Now we make clear the first one. Since there are no eigenvalues of the operator pencilU(λ, t) on the line Reλ=−β2+l2−ⁿ₂, from the proof of [3, Th. 3.6.1] we have the estimate

kU⁻¹(λ, t)eΨk²_Wl

2(Ω,λ)≤C kηke ²

W₂^l−2m(Ω,λ)+

m

X

j=1

kηe_jk²

W^l−µj−

1 2 2 (∂Ω,λ)

(4.5) for all λon the line Reλ =−β2+l₂−ⁿ₂, t ∈[0, T], and all Ψ = (e η,e ηe₁, . . . ,ηe_m)

∈W₂^l−2m(Ω)×Qm

k=1W^l−µk−

1 2

2 (∂Ω), where the constant C is independent of λ, t andΨ. Heree

kuk_Wl

2(Ω,λ)=kuk_Wl

2(Ω)+|λ|^lkuk_L2(Ω), kuk_Wl

2(∂Ω,λ)=kuk

W^l−

1 2

2 (∂Ω)+|λ|^l−12kuk_L2(∂Ω),

which are equivalent to the norms inW₂^l(Ω),W₂^l−12(∂Ω), respectively, for arbitrary fixed complex numberλ.

We will prove by induction onhthat k(U⁻¹)_th(λ, t)Ψke ²_Wl

2(Ω,λ)≤C(h)

kηke ²_Wl−2m 2 (Ω,λ)+

m

X

j=1

kηejk²

W^l−µj−

1 2 2 (∂Ω,λ)

. (4.6) It holds forh= 0 by (4.5). Assume that it holds forh−1. From the equality

U(λ, t)U⁻¹(λ, t) =I,

differentiating both sides of ith(h≥1) times with respect totwe obtain

h−1

X

k=0 h−1

k

Ut^h−k(λ, t)(U⁻¹)_tk(λ, t) +U(λ, t)(U⁻¹)_th(λ, t) = 0.

Rewrite this equality in the form (U⁻¹)_th(λ, t) =−U⁻¹(λ, t)

h−1

X

k=0 h−1

k

Ut^h−k(λ, t)(U⁻¹)_tk(λ, t).

Then (4.6) follows from this equality and the inductive assumption. It is well-known (see [3, Le. 6.1.4]) that the norm (2.1) is equivalent to

|||u|||_Vl

2,β(K)= 1 2πi

Z

Reλ=−β+l−ⁿ₂

ku(., λ)ke ²_Wl

2(Ω,λ)dλ1/2

,

and the norm (2.2) is equivalent to

|||u|||

V^l−

12

2,β (∂K)= 1 2πi

Z

Reλ=−β+l−ⁿ₂

keu(., λ)k²

W^l−

1 2 2 (∂Ω,λ)

dλ^1/2 . Using these with noting

w(ω, λ, t) =e U⁻¹(λ, t)Fe(., λ, t) (see [3, Le. 6.1.3]) and (4.5), we get from (4.4) that

kw(., t)k²

V_2,β^l2

2(K)

≤ C 2πi

Z

Reλ=−β2+l₂−ⁿ₂

kw(., λ, t)ke ²

W₂^l2(Ω,λ)dλ

= C

2πi Z

Reλ=−β₂+l₂−ⁿ₂

kU⁻¹(λ, t)F(., λ, t)ke _Wl2 2 (Ω,λ)dλ

≤ C 2πi

Z

Reλ=−β2+l₂−ⁿ₂

kr^2m^f(., t)k²_Wl−2m 2 (Ω,λ)

+

m

X

j=1

kr^µ^^jgj(., t)k²

W^l−µj−

1 2 2 (∂Ω,λ)

dλ (4.7)

≤C

kr^2mf(., t)k²

V_2,β^l2−2m

2−2m(K)+

m

X

j=1

kr^µjg_j(., t)k²

V^l2−µj−

1 2 2,β2−µj (∂K)

≤C

kf(., t)k²

V_2,β^l2−2m

2 (K)+

m

X

j=1

kg_j(., t)k²

V^l2−µj−

1 2 2,β2 (∂K)

=C

kf(., t)k²

V_2,β^l2−2m

2 (G)+

m

X

j=1

kgj(., t)k²

V^l2−µj−

12 2,β2 (S)

for all t ∈ [0, T]. Here, and sometimes later, for convenience, we denote different constants by the same symbol C. Integrating the last inequality with respect to t from 0 to +∞, we obtainw∈V_2,β^l2

2(KT) and kwk²

V_2,β^l2,0

2(K_T)≤C kfk²

V_2,β^l2−2m,0

2 (G_T)+

m

X

j=1

kgjk²

V^l2−µj−

1 2,0 2,β2 (S_T)

.

Differentiating (4.4)htimes with respect tot we have w_th(x, t) = 1

2πi Z

Reλ=−β2+l₂−ⁿ₂

r^λ

h

X

k=0 h k

(U⁻¹)_tk(λ, t)Fe_t^h−k(ω, λ, t)dλ.

Now using (4.6) and arguments the same as in (4.7) we arrive at kw_thk²

V_2,β^l2,0

2(K_T)≤C

h

X

k=0

kf_tkk²

V_2,β^l2−2m,0

2 (G_T)+

m

X

k=1

k(gj)_tkk²

V^l2−µj−

1 2,0 2,β2 (S_T)

. (4.8) Therefore,w∈V_2,β^l2,h

2(K_T) and kwk²

V_2,β^l2,h

2(KT)≤C kfk²

V_2,β^l2−2m,h

2 (GT)+

m

X

j=1

kg_jk²

V^l2−µj−

12,h 2,β2 (S_T)

. (4.9) Now we verify thatck(t)∈W₂^h((0, T)) fork= 1, . . . , N. For some suchk put

v(x, t) =r^{−λk(t)+2m−n}ψ_k(ω, t). (4.10) Using formula (3.1), we have

∂^αv=r^−|α|

|α|

X

p=0

(r∂r)^pr^{−λk(t)+2m−n}Pα,pψk

=r^{−|α|−λk(t)+2m−n}

|α|

X

p=0

(−λk(t) + 2m−n)^pP_α,pψ_k.

(4.11)

Since Reλ_k(t)<−β₂+l₂−ⁿ₂ for allt∈[0, T] andλ_k(t) is analytic on [0, T], then there is a real number >0 such that Reλ_k(t)≤ −β₂+l₂−ⁿ₂−2for allt∈[0, T].

Thus, it follows from (4.11) that

|r^{−γ2+l2−2m+|α|}∂^αv(x, t)| ≤Cr⁻ⁿ²⁺

|α|

X

p=0

|Pα,pψk(ω, t)|

for all (x, t)∈G_T and all multi-index α. This implies v(., t)∈V_2,−β^l

2+l2−2m+l(G) and

kv(., t)k_Vl

2,−γ2 +l2−2m+l(G)≤Ckψ_k(., t)k_Wl 2(Ω)

for an arbitrary integer l. Using Fa`a Di Bruno’s Formula for the higher order derivatives of composite functions (see, e.g, [7]), we have

v_tp=

p

X

q=0

p q

r^{−λk(t)+2m−n}

t^p−q(ψ_k)_tq

=r^{−λk(t)+2m−n}

p

X

q=0

p q

X n!

m₁!. . . m_n!(lnr)^m1+···+mn

×

n

Y

s=1

−λ^(s)_k (t) s!

ms

(ψk)t^q,

where the second sum is over alln−tuples (m₁, . . . , m_n) satisfying the condition m1+ 2m2+· · ·+nmn=n.

According to Lemma 3.1,λk(t) is analytic on [0, T]. Therefore, it together with its derivatives are bounded on [0, T]. Repeating the arguments as above, we get

kvt^p(., t)k_Vl

2,−γ2 +l2−2m+l(G)≤C

p

X

q=0

k(ψk)t^qk_Wl 2(Ω). Thus, we have

sup

t∈[0,T]

kvk_Vl,p

2,−γ2 +l2−2m+l(G)≤C

p

X

q=0

sup

t∈[0,T]

k(ψk)t^qk_Wl

2(Ω)<+∞ (4.12) for arbitrary nonnegative integersl, p.

Setc(t) = (f(., t), v(., t))_G. Forp≤h, using (4.12), we have

|ct^p(t)|²=

p

X

q=0 p q

(f_t^p−q(., t), vt^q(., t))G

2

≤CX^p

q=0

kr^β2−l2+2mft^qk²_L

2(G)

X^p

q=0

kr^{−β2+l2−2m}vt^qk²_L

2(G)

≤C

p

X

q=0

kf_tqk²

V_2,β^l2−2m

2 (G). This impliesc(t)∈W₂^h((0, T)) and

kck_Wh

2((0,T))≤Ckfk_Vl2−2m,h

2,β2 (GT). (4.13)

Now setcj(t) = (gj, B_j+m⁰ v)S, j= 1, . . . , m. Then also using (4.12), we have

|(cj)t^p(t)|²

=

p

X

q=0 p q

((g_j)_tp−q(., t), v_tq(., t))_S

2

≤C

p

X

q=0

kr^{β2−l2+µj+12}(gj)t^qk²_L

2(G)

p

X

q=0

kr^{−β2+l2−µj−12}(B_j+m⁰ v)t^qk²_L

2(G)

≤CX^p

q=0

k(gj)t^qk²

V^l2−µj−

1 2 2,β2 (S)

.X^p

q=0

kvt^qk²

V_2,−β^2m−µj

2 +l2−µj(G)

≤C

p

X

q=0

k(gj)_tqk²

V^l2−µj−

1 2 2,β2 (S)

(p≤h).

This impliescj ∈W₂^h((0, T)) and kcjk_Wh

2((0,T)) ≤Ckgjk

V^l2−µj−

1 2,h

2,β2 (ST). (4.14)

From (4.13) and (4.14), we can conclude thatck(t)∈W₂^h((0, T)) and kckk_Wh

2((0,T))≤C

kfk_Vl2−2m,h 2,β2 (GT)+

m

X

j=1

kgjk

V^l2−µj−

1 2,h 2,β2 (ST)

. (4.15)

(ii) Now we consider the caseu∈V_2,β^l1,h

1(GT) is arbitrary. Let η be an infinitely differential function with support in U, equal to one in a neighborhoodV of the origin. Denote by Gthe set of all subdomainG⁰ ofG with the smooth boundary such thatG∩U \V ⊂G⁰. We will show thatu∈W₂^l2,h(G⁰_T) for allG⁰ ∈G. To this end, we will prove by induction onhthat

u_tk∈W₂^l2,0(G⁰_T) fork= 0, . . . , handG⁰∈G. (4.16) According to the results on the regularity of solutions of elliptic boundary problems in smooth domains, we can conclude from (4.1), (4.2) thatu(., t)∈W₂^l2(G⁰) for each t∈[0, T] and

ku(., t)k_Wl2

2 (G⁰)≤C

ku(., t)k_Wl1

2 (G⁰⁰)+kf(., t)k_Wl2−2m 2 (G⁰⁰)

+

m

X

j=1

kgj(., t)k

W^l2−µj−

1 2

2 (S∩∂G⁰⁰)

,

whereG⁰⁰∈Gsuch thatG⁰⊂S∪G⁰⁰andCis a constant independent ofu,f,g_jand t. Integrating this inequality with respect to tfrom 0 to T we getu∈W₂^l2,0(G⁰_T).

Thus (4.16) holds for h = 0. Assume that it holds for h−1. Differentiating equalities (4.1), (4.2) with respect tot htimes and using the inductive assumption, we have

Lu_th =f_th−

h−1

X

k=0

h k

L_th−ku_tk ∈W₂^l2−2m,0(G⁰⁰_T),

Bju_th= (gj)_th−

h−1

X

k=0

h k

(Bj)_th−ku_tk ∈W^l2−µj−

1 2,0

2,β2 (ST ∩∂G⁰⁰_T),

where G⁰, G⁰⁰ ∈ G, G⁰ ⊂S∪G⁰⁰. Applying the arguments above for u_th, we get u_th ∈W₂^l2,0(G⁰_T).

From (4.1) we have

L(ηu) =ηf+ [L, η]uin GT, (4.17) where [L, η] =Lη−ηLis the commutator ofLandη. Noting that u∈W₂^l2,h(G⁰_T) for allG⁰∈Gand [L, η] is a differential expression (acting onu) of order≤2m−1 with coefficients having the supports contained in U \ V, we have [L, η]u is in W_2,β^l2−2m,h

2 (GT). So is the right-hand side of (4.17). Similarly, we have Bj(ηu) =ηgj+ [Bj, η]u∈W^l2−µj−

1 2,h

2,β₂ (ST)(j= 1, . . . , m). (4.18) Applying the the part (i) above for the function ηu, we conclude from (4.17) and (4.18) thatuadmits the decomposition (4.3) inVT. The theorem is proved.