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 R^{n}(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 S^{n−1} in R^{n}. Let T be a positive real number or T = +∞. If A is a
subset ofR^{n}, we setAT =A×(0, T). For each multi-indexα= (α1, . . . , αn)∈N^{n},
set|α|=α1+· · ·+αn, and∂^{α}=∂_{x}^{α}=∂^{α}_{x}_{1}^{1}. . . ∂_{x}^{α}_{n}^{n}.

Let us introduce some functional space used in this paper. Letlbe a nonnegative
integer. We denote by W_{2}^{l}(G) the usual Sobolev space of functions defined in G
with the norm

kuk_{W}l

2(G)=Z

G

X

|α|≤m

|∂_{x}^{α}u|^{2}dx^{1/2}
,

and byW_{2}^{l−}^{1}^{2}(S) the space of traces of functions fromW_{2}^{l}(G) onS with the norm
kuk

W^{l−}

1 2

2 (S)= inf
kvk_{W}l

2(G):v∈W_{2}^{l}(G), v|S=u .

We define the weighted Sobolev spaceV_{2,γ}^{l} (K) (γ∈R) as the closure ofC_{0}^{∞}(K\{0})
with respect to the norm

kuk_{V}l

2,γ(K)= X

|α|≤l

Z

K

r^{2(γ+|α|−l)}|∂_{x}^{α}u|^{2}dx^{1/2}

, (2.1)

wherer=|x|= Pn

k=1x^{2}_{k}1/2

. 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_{V}l

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−}^{1}^{2}(∂K). The weighted spaces V_{2,γ}^{l} (G), Vγ^{l−}^{1}^{2}(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. ByW_{2}^{h}((0, T);X) we
denote the Sobolev space ofX-valued functions defined on (0, T) with

kfk_{W}h

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

k=0

Z T O

d^{k}f(t)
dt^{k}

2 Xdt1/2

<+∞.

For short, we set

W_{2}^{h}((0, T)) =W_{2}^{h}((0, T);C), W_{2}^{l,h}(ΩT) =W_{2}^{h}((0, T);W_{2}^{l}(Ω)),
V_{2,γ}^{l,h}(GT) =W_{2}^{h}((0, T);V_{2,γ}^{l} (G)), V^{l−}

1 2,h

2,γ (ST) =W_{2}^{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 ofR^{n} 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 constantc_{0}such that

|L^{◦}(x, t, ξ)| ≥c0|ξ|^{2m} (2.3)
for all ξ ∈ R^{n} 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 onS^{n−1}. Thus we
can writeL(t, ∂x),Bj(t, ∂x) in the form

L(t, ∂x) =r^{−2m}L(ω, t, ∂ω, r∂r),
Bj(t, ∂x) =r^{−µ}^{j}Bj(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 ≡W_{2}^{l}(Ω) intoY ≡W_{2}^{l−2m}(Ω)×

m

Y

j=1

W^{l−µ}^{j}^{−}

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}+· · ·+A_{0}(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, t_{0}) andϕ_{0} ∈ X is called an eigenvector corresponding to λ_{0}. Λ =
dim kerU(λ_{0}, t_{0}) is called the geometric multiplicity of the eigenvalueλ_{0}.

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

σ

X

q=0

1 q!

d^{q}

dλ^{q}U(λ, t0)|λ=λ_{0}ϕ_{σ−q} = 0 forσ= 1, . . . , s,

then the ordered collectionϕ_{0}, ϕ_{1}, . . . , ϕ_{s}is said to be a Jordan chain corresponding
to the eigenvalueλ_{0} of the lengths+ 1. The rank of the eigenvectorϕ_{0} (rankϕ_{0})
is the maximal length of the Jordan chains corresponding to the eigenvectorϕ_{0}.

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 operatorA_{0}(t) is invertible. Indeed, ifλ_{0}is 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^{−1}_{0} (t)U(λ, t), D_{k}(t) =A^{−1}_{0} (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_{k}λ^{k−1} P2m

k=2D_{k}λ^{k−2} . . . D_{2m−1}+D_{2m}λ
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_{k}λ^{k−1} −P2m

k=2D_{k}λ^{k−2} . . . −(D2m−1+D_{2m}λ)
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

D_{1}:={λ∈C:γ_{1}<Reλ < γ_{2}} (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 D_{2} = (C\ D)∩ D_{1}. Moreover, the boundary ∂D_{2} of D_{2}
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 I_{0} be the maximal interval ofσ(t) considered and suppose that
t_{1} is the right end ofI_{0} and 0< t_{1}< T. Since σ(t) does not go out of the domain
D_{0}and the spectrum ofA(t_{1}) consists only a finite set of its eigenvalues,σ(t) must
converge to an eigenvalue bσ_{0} ∈ D_{2} ofA(t_{1}) as t↑t_{1} (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 =W_{2}^{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_{j}uB^{0}_{j+m}vds=
Z

K

uL^{+}vdx+

m

X

j=1

Z

∂K

B_{j+m}uB^{0}_{j}vds (3.8)
holds foru, v ∈C_{0}^{∞}(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}^{α}u,

and B^{0}_{j} = B^{0}_{j}(t, ∂x) are boundary operators of order µ^{0}_{j} = 2m−1−µj+m if
j ≤ m, and of order µ^{0}_{j} = 2m−1−µ_{j−m} if j ≥ m+ 1. The coefficients of
B^{0}_{j} =B^{0}_{j}(t, ∂x), j= 1, . . . ,2m, are independent of the variablexand dependent on
tanalytically on [0, T].

The operatorsL^{+}(t, ∂x),B^{0}_{j}(t, ∂x) can be written in the form

L^{+}(t, ∂x) =r^{−2m}L^{+}(ω, t, ∂ω, r∂r), (3.9)
B^{0}_{j}(t, ∂x) =r^{−µ}^{0}^{j}Bj^{0}(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)euB^{0}j+m(t,−λ+ 2m−n)evds

= Z

ΩueL^{+}(t,−λ+ 2m−n)evdx+

m

X

j=1

Z

∂Ω

Bj+m(t, λ)ueB^{0}j(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)}(λ_{0}(t), t)ϕ(t), ψ(t))_{Ω}
+

m

X

j=1

B^{(1)}_{j} (λ_{0}(t), t)ϕ(t),Bj+m^{0} (−λ_{0}(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, t_{0})u=U^{(1)}(λ, t_{0})ϕ(t_{0})

≡(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),B^{0}j+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−λ_{0}(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+m}^{0} (−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}^{l}^{1}^{,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}^{l}^{2}^{−2m,h}

2 (G_{T}), g_{j} ∈ V_{2,β}^{l}^{2}^{−µ}^{j}^{−}^{1}^{2}^{,h}

2 (S_{T}), l_{1}, l_{2} ≥ 2m, β_{1}−l_{1} > β_{2}−l_{2}.
Suppose that the linesReλ=−βi+li−^{n}_{2}(i= 1,2) do not contain eigenvalues of
the pencil U(λ, t), and all eigenvalues of this pencil in the strip −β1+l1−^{n}_{2} <

Reλ <−β2+l_{2}− ^{n}_{2} are simple for all t ∈[0, T] which are chosen to be analytic
functionsλ_{1}(t), λ_{2}(t), . . . , λ_{N}(t)defined on[0, T]as the result of Lemma 3.1. Then

there exists a neighborhoodV of the origin ofR^{n} such that inVT the solutionuhas
representation

u(x, t) =

N

X

k=1

ck(t)r^{λ}^{k}^{(t)}ϕk(ω, t) +w(x, t), (4.3)
where w∈V_{2,β}^{l}^{2}^{,h}

2(K_{T}),c_{k}(t)∈W_{2}^{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^{−λ}^{k}^{(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 ∈ R^{n} 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=−β_{2}+l_{2}−^{n}_{2}

r^{λ}U^{−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+m}^{0} vk(., t)

∂K

= f(., t), v_{k}(., t)

G+

m

X

j=1

g_{j}(., t), B^{0}_{j+m}v_{k}(., t)

S

fork= 1, . . . , N, whereFe= (r]^{2m}f ,r]^{µ}^{1}g1,. . .,r^^{µ}^{m}gm).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−1}g(ω, r, t)dr.

We will prove below thatw∈V_{2,β}^{l}^{2}^{,h}

2(KT),ck(t)∈W_{2}^{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−^{n}_{2}, from the proof of [3, Th. 3.6.1] we
have the estimate

kU^{−1}(λ, t)eΨk^{2}_{W}l

2(Ω,λ)≤C
kηke ^{2}

W_{2}^{l−2m}(Ω,λ)+

m

X

j=1

kηe_{j}k^{2}

W^{l−}^{µj}^{−}

1 2 2 (∂Ω,λ)

(4.5)
for all λon the line Reλ =−β2+l_{2}−^{n}_{2}, t ∈[0, T], and all Ψ = (e η,e ηe_{1}, . . . ,ηe_{m})

∈W_{2}^{l−2m}(Ω)×Qm

k=1W^{l−µ}^{k}^{−}

1 2

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

kuk_{W}l

2(Ω,λ)=kuk_{W}l

2(Ω)+|λ|^{l}kuk_{L}_{2}_{(Ω)},
kuk_{W}l

2(∂Ω,λ)=kuk

W^{l−}

1 2

2 (∂Ω)+|λ|^{l−}^{1}^{2}kuk_{L}_{2}_{(∂Ω)},

which are equivalent to the norms inW_{2}^{l}(Ω),W_{2}^{l−}^{1}^{2}(∂Ω), respectively, for arbitrary
fixed complex numberλ.

We will prove by induction onhthat
k(U^{−1})_{t}h(λ, t)Ψke ^{2}_{W}l

2(Ω,λ)≤C(h)

kηke ^{2}_{W}l−2m
2 (Ω,λ)+

m

X

j=1

kηejk^{2}

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})_{t}k(λ, t) +U(λ, t)(U^{−1})_{t}h(λ, t) = 0.

Rewrite this equality in the form
(U^{−1})_{t}h(λ, t) =−U^{−1}(λ, t)

h−1

X

k=0 h−1

k

Ut^{h−k}(λ, t)(U^{−1})_{t}k(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|||_{V}l

2,β(K)= 1 2πi

Z

Reλ=−β+l−^{n}_{2}

ku(., λ)ke ^{2}_{W}l

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−^{n}_{2}

keu(., λ)k^{2}

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)k^{2}

V_{2,β}^{l}^{2}

2(K)

≤ C 2πi

Z

Reλ=−β2+l_{2}−^{n}_{2}

kw(., λ, t)ke ^{2}

W_{2}^{l}^{2}(Ω,λ)dλ

= C

2πi Z

Reλ=−β_{2}+l_{2}−^{n}_{2}

kU^{−1}(λ, t)F(., λ, t)ke _{W}l2
2 (Ω,λ)dλ

≤ C 2πi

Z

Reλ=−β2+l_{2}−^{n}_{2}

kr^{2m}^f(., t)k^{2}_{W}l−2m
2 (Ω,λ)

+

m

X

j=1

kr^{µ}^^{j}gj(., t)k^{2}

W^{l−}^{µj}^{−}

1 2 2 (∂Ω,λ)

dλ (4.7)

≤C

kr^{2m}f(., t)k^{2}

V_{2,β}^{l}^{2}^{−2m}

2−2m(K)+

m

X

j=1

kr^{µ}^{j}g_{j}(., t)k^{2}

V^{l}^{2}^{−}^{µj}^{−}

1 2 2,β2−µj (∂K)

≤C

kf(., t)k^{2}

V_{2,β}^{l}^{2}^{−2m}

2 (K)+

m

X

j=1

kg_{j}(., t)k^{2}

V^{l}^{2}^{−}^{µj}^{−}

1 2 2,β2 (∂K)

=C

kf(., t)k^{2}

V_{2,β}^{l}^{2}^{−2m}

2 (G)+

m

X

j=1

kgj(., t)k^{2}

V^{l}^{2}^{−}^{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}^{l}^{2}

2(KT) and
kwk^{2}

V_{2,β}^{l}^{2}^{,0}

2(K_{T})≤C
kfk^{2}

V_{2,β}^{l}^{2}^{−2m,0}

2 (G_{T})+

m

X

j=1

kgjk^{2}

V^{l}^{2}^{−}^{µj}^{−}

1
2,0
2,β2 (S_{T})

.

Differentiating (4.4)htimes with respect tot we have
w_{t}h(x, t) = 1

2πi Z

Reλ=−β2+l_{2}−^{n}_{2}

r^{λ}

h

X

k=0 h k

(U^{−1})_{t}k(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_{t}hk^{2}

V_{2,β}^{l}^{2}^{,0}

2(K_{T})≤C

h

X

k=0

kf_{t}kk^{2}

V_{2,β}^{l}^{2}^{−2m,0}

2 (G_{T})+

m

X

k=1

k(gj)_{t}kk^{2}

V^{l}^{2}^{−}^{µj}^{−}

1
2,0
2,β2 (S_{T})

. (4.8)
Therefore,w∈V_{2,β}^{l}^{2}^{,h}

2(K_{T}) and
kwk^{2}

V_{2,β}^{l}^{2}^{,h}

2(KT)≤C
kfk^{2}

V_{2,β}^{l}^{2}^{−2m,h}

2 (GT)+

m

X

j=1

kg_{j}k^{2}

V^{l}^{2}^{−}^{µj}^{−}

12,h
2,β2 (S_{T})

. (4.9)
Now we verify thatck(t)∈W_{2}^{h}((0, T)) fork= 1, . . . , N. For some suchk put

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

∂^{α}v=r^{−|α|}

|α|

X

p=0

(r∂r)^{p}r^{−λ}^{k}^{(t)+2m−n}Pα,pψk

=r^{−|α|−λ}^{k}^{(t)+2m−n}

|α|

X

p=0

(−λk(t) + 2m−n)^{p}P_{α,p}ψ_{k}.

(4.11)

Since Reλ_{k}(t)<−β_{2}+l_{2}−^{n}_{2} for allt∈[0, T] andλ_{k}(t) is analytic on [0, T], then
there is a real number >0 such that Reλ_{k}(t)≤ −β_{2}+l_{2}−^{n}_{2}−2for allt∈[0, T].

Thus, it follows from (4.11) that

|r^{−γ}^{2}^{+l}^{2}^{−2m+|α|}∂^{α}v(x, t)| ≤Cr^{−}^{n}^{2}^{+}

|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_{V}l

2,−γ2 +l2−2m+l(G)≤Ckψ_{k}(., t)k_{W}l
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_{t}p=

p

X

q=0

p q

r^{−λ}^{k}^{(t)+2m−n}

t^{p−q}(ψ_{k})_{t}q

=r^{−λ}^{k}^{(t)+2m−n}

p

X

q=0

p q

X n!

m_{1}!. . . m_{n}!(lnr)^{m}^{1}^{+···+m}^{n}

×

n

Y

s=1

−λ^{(s)}_{k} (t)
s!

ms

(ψk)t^{q},

where the second sum is over alln−tuples (m_{1}, . . . , 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_{V}l

2,−γ2 +l2−2m+l(G)≤C

p

X

q=0

k(ψk)t^{q}k_{W}l
2(Ω).
Thus, we have

sup

t∈[0,T]

kvk_{V}l,p

2,−γ2 +l2−2m+l(G)≤C

p

X

q=0

sup

t∈[0,T]

k(ψk)t^{q}k_{W}l

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^{β}^{2}^{−l}^{2}^{+2m}ft^{q}k^{2}_{L}

2(G)

X^{p}

q=0

kr^{−β}^{2}^{+l}^{2}^{−2m}vt^{q}k^{2}_{L}

2(G)

≤C

p

X

q=0

kf_{t}qk^{2}

V_{2,β}^{l}^{2}^{−2m}

2 (G).
This impliesc(t)∈W_{2}^{h}((0, T)) and

kck_{W}h

2((0,T))≤Ckfk_{V}l2−2m,h

2,β2 (GT). (4.13)

Now setcj(t) = (gj, B_{j+m}^{0} 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})_{t}p−q(., t), v_{t}q(., t))_{S}

2

≤C

p

X

q=0

kr^{β}^{2}^{−l}^{2}^{+µ}^{j}^{+}^{1}^{2}(gj)t^{q}k^{2}_{L}

2(G)

p

X

q=0

kr^{−β}^{2}^{+l}^{2}^{−µ}^{j}^{−}^{1}^{2}(B_{j+m}^{0} v)t^{q}k^{2}_{L}

2(G)

≤CX^{p}

q=0

k(gj)t^{q}k^{2}

V^{l}^{2}^{−}^{µj}^{−}

1 2 2,β2 (S)

.X^{p}

q=0

kvt^{q}k^{2}

V_{2,−β}^{2m−}^{µj}

2 +l2−µj(G)

≤C

p

X

q=0

k(gj)_{t}qk^{2}

V^{l}^{2}^{−}^{µj}^{−}

1 2 2,β2 (S)

(p≤h).

This impliescj ∈W_{2}^{h}((0, T)) and
kcjk_{W}h

2((0,T)) ≤Ckgjk

V^{l}^{2}^{−}^{µj}^{−}

1 2,h

2,β2 (ST). (4.14)

From (4.13) and (4.14), we can conclude thatck(t)∈W_{2}^{h}((0, T)) and
kckk_{W}h

2((0,T))≤C

kfk_{V}l2−2m,h
2,β2 (GT)+

m

X

j=1

kgjk

V^{l}^{2}^{−}^{µj}^{−}

1 2,h 2,β2 (ST)

. (4.15)

(ii) Now we consider the caseu∈V_{2,β}^{l}^{1}^{,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 subdomainG^{0} ofG with the smooth boundary
such thatG∩U \V ⊂G^{0}. We will show thatu∈W_{2}^{l}^{2}^{,h}(G^{0}_{T}) for allG^{0} ∈G. To
this end, we will prove by induction onhthat

u_{t}k∈W_{2}^{l}^{2}^{,0}(G^{0}_{T}) fork= 0, . . . , handG^{0}∈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)∈W_{2}^{l}^{2}(G^{0}) for each
t∈[0, T] and

ku(., t)k_{W}l2

2 (G^{0})≤C

ku(., t)k_{W}l1

2 (G^{00})+kf(., t)k_{W}l2−2m
2 (G^{00})

+

m

X

j=1

kgj(., t)k

W^{l}^{2}^{−}^{µj}^{−}

1 2

2 (S∩∂G^{00})

,

whereG^{00}∈Gsuch thatG^{0}⊂S∪G^{00}andCis a constant independent ofu,f,g_{j}and
t. Integrating this inequality with respect to tfrom 0 to T we getu∈W_{2}^{l}^{2}^{,0}(G^{0}_{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_{t}h =f_{t}h−

h−1

X

k=0

h k

L_{t}h−ku_{t}k ∈W_{2}^{l}^{2}^{−2m,0}(G^{00}_{T}),

Bju_{t}h= (gj)_{t}h−

h−1

X

k=0

h k

(Bj)_{t}h−ku_{t}k ∈W^{l}^{2}^{−µ}^{j}^{−}

1 2,0

2,β2 (ST ∩∂G^{00}_{T}),

where G^{0}, G^{00} ∈ G, G^{0} ⊂S∪G^{00}. Applying the arguments above for u_{t}h, we get
u_{t}h ∈W_{2}^{l}^{2}^{,0}(G^{0}_{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∈W_{2}^{l}^{2}^{,h}(G^{0}_{T})
for allG^{0}∈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}^{l}^{2}^{−2m,h}

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

1 2,h

2,β_{2} (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.