Asymptotic expansions of solutions of the first initial boundary-value problem for Schrödinger systems in domains with conical points. II
We consider asymptotic expansions of solutions of the first initial boundary-value problem for strong Schrödinger systems near a conical point of the boundary of a domain.
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| author | Cung, The Anh Nguen, Van Hung Кунг, Тхе Анх Нгуєн, Ван Хунг |
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UDC 517.9
Nguyen Manh Hung, Cung The Anh (Hanoi Univ. Education, Vietnam)
ASYMPTOTIC EXPANSIONS OF SOLUTIONS
OF THE FIRST INITIAL BOUNDARY-VALUE PROBLEM
FOR SCHRÖDINGER SYSTEMS IN DOMAINS
WITH CONICAL POINTS. II*
АСИМПТОТИЧН РОЗКЛАДИ РОЗВ’ЯЗКIВ
ПЕРШОЇ ПОЧАТКОВОЇ КРАЙОВОЇ ЗАДАЧI
ДЛЯ СИСТЕМ ШРЕДIНГЕРА В ОБЛАСТЯХ
З КОНIЧНИМИ ТОЧКАМИ. II
This paper is concerned with asymptotic expansions of solutions of the first initial boundary-value problem
for strongly Schrödinger systems near a conical point of the domain boundary.
Розглянуто асимптотичнi розклади розв’язкiв першої початкової крайової задачi для сильно шредiнге-
рових систем бiля конiчної точки межi областi.
1. Introduction and notations. At the present there exists a comprehensive theory of
boundary-value problems for elliptic, parabolic, and hyperbolic equations and systems
with a smooth boundary. One of the central results of this theory consists in the fact
that if the coefficients of the equation and of the boundary operators, its right-hand side,
and the boundary of the domain are sufficiently smooth, then the solution itself of the
problem is correspondingly smooth. (In the parabolic and hyperbolic cases, the initial
and boundary conditions must also satisfy the so-called compatibility condition; see,
e.g., [1, 2]).
However, many important applied problems reduce to the study of boundary-value
problems for partial differential equations in non-smooth domains. Such questions have
been discussed extensively in the literature since the appearance of the fundamental work
[3] of Kondartiev in 1967. By now the theory of boundary-value problems for elliptic
equations in non-smooth domains has been worked out in much detail, with a large
literature on it. We refer the survey paper [4] and the monographs [5, 6] for the results.
Parallel with this theory, the boundary-value problems for non-stationnary equations and
systems have been studied by many athors, such as Melinikov [7], Ngok [8], Eskin [9],
Kokotov and Plamenevskii [10, 11], Matyukevich and Plamenevskii [12], . . . . In these
works, they used results and methods of elliptic boundary-value problems in nonsmooth
domains to prove the assertions on the unique solvability, on the smoothness and the
asymptotic expansions of solutions near the singularities on the boundary.
Boundary-value problems for Schrödinger equations and Schrödinger systems in a
finite cylinder ΩT = Ω × (0, T ) have been studied by many authors (see, e.g., [1, 2,
13, 14]). In this paper, we continue the investigation presented in [15 – 18], in which we
considered the first initial boundary-value problem for strongly Schrödinger systems in
an infinite cylinder Ω∞ = Ω×(0,∞), where Ω is a bounded domain with conical points.
The existence, uniqueness and smoothness of generalized solutions to the problem were
*This work was supported by the Vietnam National Foundation for Science and Technology Development
(NAFO STED).
c© NGUYEN MANH HUNG, CUNG THE ANH, 2009
1640 ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 12
ASYMPTOTIC EXPANSIONS OF SOLUTIONS OF THE FIRST INITIAL BOUNDARY-VALUE ... 1641
given in [15, 16, 18]. The aim of [17] and this paper is to derive the asymptotic expansion
of the generalized solution of this problem in a neighbourhood of the singular point.
Let Ω be a bounded domain in Rn. Its boundary ∂Ω is assumed to be an infinitely
differentiable surface everywhere except the coordinate origin, in a neighbourhood of
which Ω coincides with the cone K =
{
x : x/|x| ∈ G
}
, where G is a smooth domain
on the unit sphere Sn−1. We begin by recalling some notations and functional spaces
which will be frequenly used in this paper:
ΩT = Ω × (0, T ), ST = ∂Ω × (0, T ), Ω∞ = Ω × (0,∞), S∞ = ∂Ω × (0,∞),
x = (x1, . . . , xn) ∈ Ω, u(x, t) = (u1(x, t), . . . , us(x, t)) is a vector complex function,
|Dαu|2 =
s∑
i=1
|Dαui|2, utj =
(
∂ju1
∂tj
, . . . ,
∂jus
∂tj
)
,
|utj |2 =
s∑
i=1
∣∣∣∣∂jui
∂tj
∣∣∣∣2 , dx = dx1 . . . dxn, r = |x| =
√
x2
1 + . . .+ x2
n;
H l
β(Ω) — the space of all functions u(x) = (u1(x), . . . , us(x)) which have generali-
zed derivatives Dαui, |α| 6 l, 1 6 i 6 s, satisfying
‖u‖2Hl
β(Ω) =
l∑
|α|=0
∫
Ω
r2(β+|α|−l)|Dαu|2dx < +∞;
H l,k(e−γt,Ω∞) — the space of all functions u(x, t) which have generalized deri-
vatives Dαui,
∂jui
∂tj
, |α| 6 l, 1 6 j 6 k, 1 6 i 6 s, satisfying
‖u‖2Hl,k(e−γt,Ω∞) =
∫
Ω∞
l∑
|α|=0
|Dαu|2 +
k∑
j=1
|utj |2
e−2γtdxdt < +∞;
in particular
‖u‖2Hl,0(e−γt,Ω∞) =
l∑
|α|=0
∫
Ω∞
|Dαu|2e−2γtdxdt;
◦
H l,k(e−γt,Ω∞) — the closure inH l,k(e−γt,Ω∞) of the set of all infinitely differenti-
able in Ω∞ functions which belong to H l,k(e−γt,Ω∞) and vanish near S∞;
H l,k
β (e−γt,Ω∞) — the space of all functions u(x, t) which have generalized deri-
vatives Dαui,
∂jui
∂tj
, |α| 6 l, 1 6 j 6 k, 1 6 i 6 s, satisfying
‖u‖2
Hl,k
β (e−γt,Ω∞)
=
∫
Ω∞
l∑
|α|=0
r2(β+|α|−l)|Dαu|2 +
k∑
j=1
|utj |2
e−2γtdxdt < +∞;
H l
β(e−γt,Ω∞) — the space of all functions u(x, t) which have generalized derivatives
Dα(ui)tj , |α|+ j 6 l, 1 6 i 6 s, satisfying
‖u‖2Hl
β(e−γt,Ω∞) =
l∑
|α|+j=0
∫
Ω∞
r2(β+|α|+j−l)|Dαutj |2e−2γtdxdt < +∞;
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 12
1642 NGUYEN MANH HUNG, CUNG THE ANH
Let X be a Banach space. Denote by L∞(0,∞;X) the space consisting of all
measurable functions u : (0,∞) −→ X, t 7−→ u(x, t) satisfying
‖u‖L∞(0,∞;X) = ess sup
t>0
∥∥u(x, t)∥∥
X
< +∞.
Consider the differential operator of order 2m
L(x, t,D) =
m∑
|p|,|q|=0
Dp
(
apq(x, t)Dq
)
,
where apq are s × s-matrices of measurable bounded in Ω∞ complex functions, apq =
= (−1)|p|+|q|a∗qp. Suppose that apq are continuous in x ∈ Ω uniformly with respect to
t ∈ [0,∞) if |p| = |q| = m, and for each t ∈ [0,∞) the operator L(x, t,D) is uniformly
elliptic in Ω with ellipticity constant a0 independent of time t, i.e., we have∑
|p|=|q|=m
apq(x, t)ξpξqηη ≥ a0|ξ|2m|η|2,
for all ξ ∈ Rn \ {0}, η ∈ Cs \ {0} and (x, t) ∈ Ω∞.
In this paper we study the following problem: Find a function u(x, t) such that
(−1)m−1iL(x, t,D)u− ut = f(x, t) in Ω∞, (1.1)
u|t=0 = 0, (1.2)
∂ju
∂νj
∣∣∣
S∞
= 0, j = 0, . . . ,m− 1, (1.3)
where ν is the outer unit normal to S∞.
A function u(x, t) is called a generalized solution of the problem (1.1) – (1.3) in the
space
◦
Hm,0(e−γt,Ω∞) if and only if u(x, t) belongs to
◦
Hm,0(e−γt,Ω∞) and for each
T > 0 the following equality holds
(−1)m−1i
m∑
|p|,|q|=0
(−1)|p|
∫
ΩT
apqD
quDpηdxdt+
∫
ΩT
uηtdxdt =
∫
ΩT
fηdxdt, (1.4)
for all test function η ∈
◦
Hm,1(ΩT ), η(x, T ) = 0.
Putting
B(u, u)(t) =
m∑
|p|,|q|=0
(−1)|p|
∫
Ω
apqD
quDpudx, u(x, t) ∈
◦
Hm,0(e−γt,Ω∞).
For a.e. t ∈ [0,∞), the function x 7→ u(x, t) belongs to
◦
Hm(Ω). On the other hand,
since the principal coefficients apq are continuous in x ∈ Ω uniformly with respect
to t ∈ [0,∞) and the ellipticity constant a0 independent of t, repeating the proof of
Garding’s inequality [19, p. 44] we have the following lemma.
Lemma 1.1. There exist two constants µ0 and λ0 (µ0 > 0, λ0 ≥ 0) such that
(−1)mB(u, u)(t) ≥ µ0
∥∥u(x, t)∥∥2
Hm(Ω)
− λ0
∥∥u(x, t)∥∥2
L2(Ω)
(1.5)
for all u(x, t) ∈
◦
Hm,0(e−γt,Ω∞).
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 12
ASYMPTOTIC EXPANSIONS OF SOLUTIONS OF THE FIRST INITIAL BOUNDARY-VALUE ... 1643
Therefore, using the transformation u = eiλ0tv if necessary, we can assume that the
operator L(x, t,D) satisfies
(−1)mB(u, u)(t) ≥ µ0‖u‖2Hm(Ω) (1.6)
for all u(x, t) ∈
◦
Hm,0(e−γt,Ω∞). This inequality is a basic tool for proving the existence
and uniqueness of solutions of the problem.
2. Smoothness of generalized solutions. In this section we summarize the known
results on the smoothness of generalized solutions of the problem (1.1) – (1.3).
Denote by m∗ the number of multiindexes which have order not exceeding m, µ0 is
the constant in (1.6). The following theorem was proved in [18].
Theorem 2.1. Let
i) sup
{∣∣∣∣∂apq
∂t
∣∣∣∣ : (x, t) ∈ Ω∞ 0 6 |p|, |q| 6 m
}
= µ < +∞;
∣∣∣∣∂kapq
∂tk
∣∣∣∣ 6 µ1,
µ1 = const > 0, for 2 6 k 6 h+ 1;
ii) ftk ∈ L∞(0,∞;L2(Ω)), for k 6 h+ 1;
iii) ftk(x, 0) = 0, for k 6 h.
Then for every γ > γ0 =
m∗µ
2µ0
, the problem (1.1) – (1.3) has exactly one generalized
solution u(x, t) in the space
◦
Hm,0(e−γt,Ω∞). Moreover, u(x, t) has derivatives with
respect to t up to order h belonging to
◦
Hm,0(e−(2h+1)γt,Ω∞) and the following estimate
holds:
‖uth‖2Hm,0(e−(2h+1)γt,Ω∞) 6 C
h+1∑
k=0
‖ftk‖2L∞(0,∞;L2(Ω)),
where C is a positive constant independent of u and f.
From now forward, for the sake of brevity we will write γh instead of (2h+1)γ,
h = 1, 2, . . . .
In order to study the smoothness with respect to (x, t) and to establish asymptotic
formulas of solutions of the problem (1.1) – (1.3), we assume that coefficients apq(x, t)
of the operator L(x, t,D) are infinitely differentiable in Ω∞. Moreover, we also assume
that apq(x, t) and its all derivatives are bounded in Ω∞.
First, we recall two basic lemmas.
Lemma 2.1 [16]. Let f, ft, ftt ∈ L∞(0,∞;L2(K)) and f(x, 0) = ft(x, 0) = 0.
If u(x, t) ∈
◦
Hm,0(e−γt,Ω∞) is a generalized solution of the problem (1.1) – (1.3)
in the space
◦
Hm,0(e−γt,Ω∞) such that u ≡ 0 whenever |x| > R = const, then
u ∈ H2m,0
m (e−γ1t,K∞) and the following estimate holds:
‖u‖2
H2m,0
m (e−γ1t,K∞)
6
6 C
[
‖f‖2L∞(0,∞;L2(K)) + ‖ft‖2L∞(0,∞;L2(K)) + ‖ftt‖2L∞(0,∞;L2(K))
]
,
where C = const .
Denote by L0(0, t,D) the principal part of the operator L(x, t,D) at origin 0. We
consider the Dirichlet problem for the system
(−1)m−1L0(0, t,D)u = F (x, t), x ∈ K. (2.1)
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 12
1644 NGUYEN MANH HUNG, CUNG THE ANH
Lemma 2.2 [16]. Let u(x, t) be a generalized solution of the Dirichlet problem for
the system (2.1) for a.e. t ∈ [0,∞) such that u ≡ 0 whenever |x| > R = const,
and u(x, t) ∈ H2m+l−1,0
β−1 (e−γt,K∞). Let F ∈ H l,0
β (e−γt,K∞). Then u(x, t) ∈
∈ H2m+l,0
β (e−γt,K∞) and
‖u‖2
H2m+l,0
β (e−γt,K∞)
6 C
[
‖F‖2
Hl,0
β (e−γt,K∞)
+ ‖u‖2
H2m+l−1,0
β−1 (e−γt,K∞)
]
,
where C = const.
Let ω be a local coordinate system on Sn−1. The principal part of the operator
L(x, t,D) at origin 0 can be written in the form
L0(0, t,D) = r−2mQ(ω, t, rDr, Dω), Dr =
i∂
∂r
,
where Q is a linear operator with smooth coefficients. From now forward the following
spectral problem will play an important role
Q(ω, t, λ,Dω)v(ω) = 0, ω ∈ G, (2.2)
Dj
ωv(ω) = 0, ω ∈ ∂G, j = 0, . . . ,m− 1. (2.3)
It is well known [5, p. 146] that for every t ∈ [0,∞) its spectrum is discrete.
Theorem 2.2 [16]. Let u(x, t) be a generalized solution of the problem (1.1) – (1.3)
in the space
◦
Hm,0(e−γt,Ω∞) and let ftk ∈ L∞(0,∞;H l
0(Ω)) for k 6 2m + l + 1,
ftk(x, 0) = 0 for k 6 2m+ l. In addition supppose that the strip
m− n
2
6 Imλ 6 2m+ l − n
2
does not contain points of spectrum of the problem (2.2), (2.3) for every t ∈ [0,∞).
Then u(x, t) ∈ H2m+l
0 (e−γ2m+lt,Ω∞) and the following estimate holds:
‖u‖2
H2m+l
0 (e−γ2m+lt,Ω∞)
6 C
2m+l+1∑
k=0
‖ftk‖2L∞(0,∞;Hl
0(Ω)),
where C = const.
3. Asymptotic expansions of generalized solutions. In this section we will derive
asymptotic expansions of generalized solutions of the problem (1.1) – (1.3) in a nei-
ghbourhood of the conical point, in the case that the condition imposed on the spectrum
of the problem (2.2), (2.3) in Theorem 2.2 is not satisfied. The following result was
obtained in [17].
Theorem 3.1. Let u(x, t) be a generalized solution of the problem (1.1) – (1.3)
in the spaces
◦
Hm,0(e−γt,Ω∞), and let ftk ∈ L∞(0,∞;H l
0(Ω)) for k 6 l + 2m + 1,
ftk(x, 0) = 0 for k 6 l + 2m. Assume that in the strip m− n
2
6 Imλ 6 2m+ l − n
2
,
there exists only one simple eigenvalue λ(t) of the problem (2.2), (2.3) such that
2m+ l − 1− n
2
< Imλ(t) < 2m+ l − n
2
.
Then the following representation holds:
u(x, t) = c(x, t)r−iλ(t) + u1(x, t),
where c(x, t) ∈ V 2m+l
Im λ(t)(e
−γ2m+lt, Ω∞), u1 ∈ H2m+l
0 (e−γ2m+lt,Ω∞).
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 12
ASYMPTOTIC EXPANSIONS OF SOLUTIONS OF THE FIRST INITIAL BOUNDARY-VALUE ... 1645
Attention is now turned to the case that the strip m − n
2
6 Imλ 6 2m + l − n
2
contains finite simple eigenvalues λ1(t), . . . , λN0(t) of the spectral problem (2.2), (2.3).
Consider in K the Dirichlet problem for the system
L0(0, t,D)u = r−iλ0(t)−2m
M∑
s=0
lns rfs(ω, t), (3.1)
where ω is a local coordinate system on Sn−1.
Lemma 3.1 [4, p. 17]. Let fs(ω, t), s = 0, . . . ,M, be infinitely differentiable functi-
ons of ω. Then there exists a solution of the Dirichlet problem for the system (3.1) having
the form
u(x, t) = r−iλ0(t)
M+µ∑
s=0
lns rf̃s(ω, t),
where f̃s, s = 0, . . . ,M + µ, are the infinitely differentiable functions of ω, µ = 1 if λ0
is a simple eigenvalue of the problem (2.2), (2.3) and µ = 0 if λ0 is not an eigenvalue
of this problem.
From now forward, we denote
L2,loc[0,∞) =
{
c(t) : c(t) ∈ L2[0, T ] for all T > 0
}
.
Lemma 3.2. Let u(x, t) be a generalized solution of the Dirichlet problem for
the system (2.1) for a.e. t ∈ [0,∞) such that u ≡ 0 whenever |x| > R = const, and
let utk ∈ H2m+l,0
β (e−γkt,K∞), Ftk ∈ H l,0
β′ (e
−γkt,K∞) for k 6 h, β′ < β 6 m + l.
Assume that the straight lines
Imλ = −β + 2m+ l − n
2
and Imλ = −β′ + 2m+ l − n
2
do not contain points of spectrum of the problem (2.2), (2.3) for every t ∈ [0,∞), and
in the strip
−β + 2m+ l − n
2
< Imλ < −β′ + 2m+ l − n
2
there exists only one simple eigenvalue λ(t) of the problem (2.2), (2.3). Then the following
representation holds:
u(x, t) = c(t)r−iλ(t)φ(ω, t) + u1(x, t), (3.2)
where φ is an infinitely differentiable function of (ω, t) which does not depend on the
solution, ctk ∈ L2,loc[0,∞) and (u1)tk ∈ H2m+l,0
β′ (e−γkt,K∞) for k 6 h.
Proof. From Theorem 3.2 in [20], it follows that
u(x, t) = c(t)r−iλ(t)φ(ω, t) + u1(x, t), (3.3)
where φ(ω, t) is the eigenfunction of the problem (2.2), (2.3) which correspond to the
eigenvalue λ(t), u1 ∈ H2m+l,0
β′ (e−γt,K∞), and
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 12
1646 NGUYEN MANH HUNG, CUNG THE ANH
c(t) = i
∫
K
F (x, t)r−iλ(t)+2m−nψ(x, t)dx,
where ψ is the eigenfunction of the problem conjugating to the problem (2.2), (2.3)
and which corresponds to the eigenvalue λ(t). Since Imλ(t) > β′ − 2m − l +
n
2
and
F ∈ H l,0
β′ (e
−γt,K∞), so c(t) ∈ L2,loc[0,∞). Hence the assertion is proved for h = 0.
Assume that the assertion is true for 0, 1, . . . , h− 1. Denoting uth by v. From (2.1)
we obtain
(−1)m−1L0(0, t,D)v = Fth + (−1)m
h∑
k=1
(
h
k
)
L0tk(0, t,D)uth−k , (3.4)
where
L0tk =
∑
|p|=|q|=m
∂kapq(0, t)
∂tk
DpDq.
Putting S0(ω, t) = r−iλ(t)φ(ω, t). Since φ(ω, t) ∈ C∞(ω, t) [21], from (3.3) it follows
that
h∑
k=1
(
h
k
)
L0tk(0, t,D)uth−k =
h∑
k=1
(
h
k
)
L0tk(0, t,D)
[
(cS0)th−k
]
+
+
h∑
k=1
(
h
k
)
L0tk(0, t,D)(u1)th−k .
Using the induction hypothesis, we obtain
h∑
k=1
(
h
k
)
L0tk(0, t,D)uth−k = F1 −
h∑
k=1
(
h
k
)
cth−kL0(0, t,D)(S0)tk , (3.5)
where F1 ∈ H l,0
β′ (e
−γh−1t,K∞). From (3.4) and (3.5) we see that
(−1)m−1L0(0, t,D)v = F2 − (−1)m
h∑
k=1
(
h
k
)
cth−kL0(0, t,D)(S0)tk , (3.6)
where F2 ∈ H l,0
β′ (e
−γht,K∞). Hence by arguments used in the proof of case h = 0 we
can find
uth = v =
h∑
k=1
(
h
k
)
cth−k(S0)tk + d(t)S0 + u2, (3.7)
where d(t) ∈ L2,loc[0,∞), u2 ∈ H2m+l,0
β′ (e−γht,K∞). From this equality it follows
that
S0,1 = uth −
h∑
k=2
(
h
k
)
cth−k(S0)tk − (h− 1)cth−1(S0)t = cth−1(S0)t + dS0 + u2.
(3.8)
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ASYMPTOTIC EXPANSIONS OF SOLUTIONS OF THE FIRST INITIAL BOUNDARY-VALUE ... 1647
Now differentiating the equality (3.3) (h− 1) times by t. As a result we obtain
uth−1 =
h−1∑
k=0
(
h− 1
k
)
cth−k−1(S0)tk + (u1)th−1 . (3.9)
We rewrite (3.9) in the form
S0,2 = uth−1 −
h−1∑
k=1
(
h− 1
k
)
cth−k−1(S0)tk = cth−1S0 + (u1)th−1 . (3.10)
Then
(S0,2)t = uth −
h−1∑
k=1
(
h− 1
k
)[
cth−k(S0)tk + cth−k−1(S0)tk+1
]
=
= uth −
h∑
k=1
(
h
k
)
cth−k(S0)tk + cth−1(S0)t.
From this equality and (3.7) we obtain
(S0,2)t = cth−1(S0)t + dS0 + u2.
Putting S1 = S−1
0 (u1)th−1 , S2 = S−1
0 u2 − S−2
0 (S0)t(u1)th−1 . It is easy to check
that
S−1
0 S0,2 = cth−1 + S1, (S−1
0 S0,2)t = d+ S2.
It follows that
I(t) = cth−1(t)− cth−1(0)−
t∫
0
d(τ)dτ =
=
t∫
0
S2(x, τ)dτ − S1(x, t) + S1(x, 0).
Since (u1)th−1 ∈ H2m+l,0
β′ (e−γh−1t,K∞), u2 ∈ H2m+l,0
β′ (e−γht,K∞), so S1, S2 ∈
∈ H0,0
−n/2(e
−γht,K∞). Therefore I(t) ∈ H0
−n
2
(K), i.e., I(t) ≡ 0. Hence cth = d ∈
∈ L2,loc[0,∞) and (u1)th = u2 ∈ H2m+l,0
β′ (e−γht,K∞).
This completes the proof.
Proposition 3.1. Let u(x, t) be a generalized solution of the problem (1.1) – (1.3)
in the spaces
◦
Hm,0(e−γt,Ω∞) such that u ≡ 0 whenever |x| > R = const, and let
ftk ∈ L∞(0,∞;L2(K)) for k 6 h + 2, ftk(x, 0) = 0 for k 6 h + 1. Assume that the
straight lines
Imλ = m− n
2
and Imλ = 2m− n
2
do not contain points of spectrum of the problem (2.2), (2.3) for every t ∈ [0,∞), and
in the strip
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1648 NGUYEN MANH HUNG, CUNG THE ANH
m− n
2
< Imλ < 2m− n
2
there exists only one simple eigenvalue λ(t) of the problem (2.2), (2.3). Then the following
representation holds:
u(x, t) =
m−1∑
s=0
cs(t)r−iλ(t)+sPm−1,s(ln r) + u1(x, t), (3.11)
where Pm−1,s is a polynomial having order less than m and its coefficients are infinitely
differentiable functions of (ω, t), (cs)tk ∈ L2,loc[0,∞), (u1)tk ∈ H2m,0
0 (e−γk+1t,K∞)
for k 6 h.
Proof. First we will prove that if
m− n
2
< Imλ(t) < m+m0 −
n
2
, 1 6 m0 6 m,
then
u(x, t) =
m0−1∑
s=0
cs(t)r−iλ(t)+sPm0−1,s(ln r) + u1(x, t), (3.12)
where Pm0−1,s is a polynomial having order less thanm0 and its coefficients are infinitely
differentiable functions of (ω, t), (cs)tk ∈ L2,loc[0,∞) and (u1)tk ∈ H2m,0
m−m0
(e−γk+1t,
K∞) for k 6 h.
We introduce the notation: L1 = (−1)m−1
[
L0(0, t,D) − L(x, t,D)
]
. From the
system (1.1) we get
(−1)m−1L0(0, t,D)u = F, (3.13)
where F = −i(ut + f) + L1u.
From Theorem 2.1 and Lemma 2.1 it follows that utk ∈ H2m,0
m (e−γk+1t,K∞),
k 6 h. On the other hand, utk+1 ∈ Hm,0(e−γk+1t,K∞), ftk ∈ L∞(0,∞;L2(K)),
k 6 h. Therefore Ftk ∈ H0,0
m−1(e
−γk+1t,K∞), k 6 h.
Let m− n
2
< Imλ(t) < m+ 1− n
2
. From Lemma 3.2 it follows that
u(x, t) = c(t)r−iλ(t)φ(ω, t) + u1(x, t), (3.14)
where φ is an infinitely differentiable function of (ω, t) what does not depend on the
solution, ctk ∈ L2,loc[0,∞), (u1)tk ∈ H2m,0
m−1 (e−γk+1t,K∞) for k 6 h. Hence (3.12) is
proved for m0 = 1.
Assume that (3.12) holds for m0 6 m− 1. We distinguish the following cases.
Case 1: m− n
2
< Imλ(t) < m+m0−
n
2
. Using the induction hypothesis we obtain
(3.12). Putting
Sm0 = (−1)m
m0−1∑
s=0
cs(t)r−iλ(t)+sPm0−1,s(ln r). (3.15)
Then
LSm0 = F1(x, t) +
∑
j+s6m0
m0−1∑
s=0
cs(t)r−iλ(t)−2m+s+jP̃m0−1,s,j(ln r), (3.16)
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ASYMPTOTIC EXPANSIONS OF SOLUTIONS OF THE FIRST INITIAL BOUNDARY-VALUE ... 1649
where (F1)tk ∈ H0,0
m−m0−1(e
−γt,K∞) for k 6 h, and P̃m0−1,s,j is a polynomial having
order less than m0 and its coefficients are infinitely differentiable functions of (ω, t).
From (3.12), (3.13), and (3.16) we obtain
(−1)m−1L0(0, t,D)u1 = F2(x, t)+
+
∑
j+s6m0
m0−1∑
s=0
cs(t)r−iλ(t)−2m+s+jP̃m0−1,s,j(ln r), (3.17)
where F2 = −i(ut + f) + L1u1 + F1 ∈ H0,0
m−m0−1(e
−γ1t,K∞).
Since (u1)tk ∈ H2m,0
m−m0
(e−γk+1t,K∞) for k 6 h, we have
(F2)tk ∈ H0,0
m−m0−1(e
−γk+1t,K∞),
for k 6 h.
By Lemma 3.1 there exists a function
ω1 =
∑
j+s6m0
m0−1∑
s=0
cs(t)r−iλ(t)+s+jPm0,s,j(ln r) (3.18)
such that
(−1)m−1L0(0, t,D)ω1 =
∑
j+s6m0
m0−1∑
s=0
cs(t)r−iλ(t)−2m+s+jP̃m0−1,s,j(ln r), (3.19)
where Pm0,s,j is a polynomial having order less than m0 + 1 and its coefficients are
infinitely differentiable functions of (ω, t).
Putting v1 = u1 − ω1. From (3.17) and (3.19) it follows that
(−1)m−1L0(0, t,D)v1 = F2(x, t).
By Lemma 3.2 we obtain
v1(x, t) = c(t)r−iλ(t)ϕ(ω, t) + u2(x, t), (3.20)
where ϕ is an infinitely differentiable function of (ω, t) which does not depend on the
solution, ctk ∈ L2,loc[0,∞), (u2)tk ∈ H2m,0
m−m0−1(e
−γk+1t,K∞) for k 6 h.
From (3.18) and (3.20) it follows that
u1(x, t) =
∑
j+s6m0
m0−1∑
s=0
cs(t)r−iλ(t)+s+jPm0,s,j(ln r)+
+ c(t)r−iλ(t)ϕ(ω, t) + u2(x, t).
Hence and from (3.12) we get
u(x, t) =
m0∑
s=0
c̃s(t)r−iλ(t)+sP̃m0,s(ln r) + u2(x, t), (3.21)
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1650 NGUYEN MANH HUNG, CUNG THE ANH
where P̃m0,s is a polynomial having order less than m0 +1 and its coefficients are infini-
tely differentiable functions of (ω, t), (c̃s)tk ∈L2,loc[0,∞), (u2)tk ∈H2m,0
m−m0−1(e
−γk+1t,
K∞) for 0 6 k 6 h.
Case 2: m + m0 −
n
2
< Imλ(t) < m + m0 + 1 − n
2
. Since the strip m − n
2
6
6 Imλ 6 m+m0 −
n
2
does not contain points of spectrum of the problem (2.2), (2.3)
so utk ∈ H2m,0
m−m0
(e−γk+1t,K∞), k 6 h. Therefore, from Lemma 3.2 it follows that
u(x, t) = c(t)r−iλ(t)ϕ(ω, t) + u1(x, t),
where ϕ is an infinitely differentiable function of (ω, t) which does not depend on the
solution , ctk ∈ L2,loc[0,∞), (u1)tk ∈ H2m,0
m−m0−1(e
−γk+1t,K∞) for k 6 h.
Case 3: There exists t0 such that Imλ(t0) = m + m0 −
n
2
. Dividing the interval
[0,∞) by points T0 = 0 < T1 < . . . < Ts < . . . such that one of following cases
happens in each interval [Ts−1, Ts], s = 1, 2, . . .:
(i) m− n
2
< Imλ(t) < m+m0 −
n
2
,
(ii) m+m0 −
n
2
< Imλ(t) < m+m0 + 1− n
2
,
(iii) m+m0 − µ− n
2
< Imλ(t) < m+m0 − µ+ 1− n
2
, 0 < µ < 1.
If (i) (or (ii)) happens in the interval [Ts−1, Ts], then by repeating the proof of Case 1
(resp. Case 2), we obtain
u(x, t) = c(s)(t)r−iλ(t)ϕ(ω, t) + u
(s)
1 (x, t), t ∈ [Ts−1, Ts], (3.22)
where ϕ is an infinitely differentiable function of (ω, t) what does not depend on the
solution, c(s)
tk ∈ L2[Ts−1, Ts], (u1)
(s)
tk ∈ H2m,0
m−m0−1(K × [Ts−1, Ts]) for k 6 h. If
(iii) happens then repeating the argument in the proof of Case 2, we obtain (3.22) for
(u(s)
1 )tk ∈ H2m,0
m−m0−1+µ(K × [Ts−1, Ts]), k 6 h. Hence and from arguments analogous
to the proof of Case 1, after that set c(t) = c(s)(t), u1(x, t) = u
(s)
1 (x, t), whenever
t ∈ [Ts−1, Ts], we obtain (3.21).
From above arguments follow (3.12). For m0 = m, from (3.12) we obtain (3.11).
Proposition 3.1 is proved.
Proposition 3.2. Let u(x, t) be a generalized solution of the problem (1.1) – (1.3)
in the spaces
◦
Hm,0(e−γt,Ω∞) such that u ≡ 0 whenever |x| > R = const, and let
ftk ∈ L∞(0,∞;H l
0(K)) for k 6 2l+ h+ 2, ftk(x, 0) = 0 for k 6 2l+ h+ 1. Assume
that the straight lines
Imλ = m− n
2
and Imλ = 2m+ l − n
2
do not contain points of spectrum of the problem (2.2), (2.3) for every t ∈ [0,∞), and
in the strip
m− n
2
< Imλ < 2m+ l − n
2
there exists only one simple eigenvalue λ(t) of the problem (2.2), (2.3). Then the following
representation holds:
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ASYMPTOTIC EXPANSIONS OF SOLUTIONS OF THE FIRST INITIAL BOUNDARY-VALUE ... 1651
u(x, t) =
l+m−1∑
s=0
cs(t)r−iλ(t)+sP3l+m−1,s(ln r) + u1(x, t), (3.23)
where P3l+m−1,s is a polynomial having order less than 3l + m and its coeffici-
ents are infinitely differentiable functions of (ω, t), (cs)tk ∈ L2,loc[0,∞), (u1)tk ∈
∈ H2m+l,0
0 (e−γk+1t,K∞) for k 6 h+ l.
Proof. We will use the induction on l. If l = 0 the statement follows from Proposi-
tion 3.1. Let the statement be true for l − 1. We distinguish the following cases:
Case 1: m− n
2
< Imλ(t) < 2m+ l− 1− n
2
. From inductive hypothesis we obtain
u(x, t) =
l+m−2∑
s=0
cs(t)r−iλ(t)+sP3l+m−4,s(ln r) + u1(x, t), (3.24)
where P3l+m−4,s is a polynomial having order less than 3l + m − 3 and its coeffi-
cients are infinitely differentiable functions of (ω, t), (cs)tk ∈ L2,loc[0,∞), (u1)tk ∈
∈ H2m+l−1,0
0 (e−γk+1t,K∞) for k 6 h+ l − 1.
From (3.13) and (3.24) we find
(−1)m−1L0(0, t,D)u1 = F3 + (−1)mLS − iSt, (3.25)
where F3 = −i[(u1)t + f ] + L1u1, and
S =
l+m−2∑
s=0
cs(t)r−iλ(t)+sP3l+m−4,s(ln r).
Since ftk ∈ L∞(0,∞;H l
0(K)) for k 6 2l + h + 2 and ftk(x, 0) = 0 for k 6
6 2l + h + 1, so ftk ∈ L∞(0,∞;H l−1
0 (K)), k 6 2(l − 1) + (h + 2) + 2, and
ftk(x, 0) = 0, k 6 2(l − 1) + (h + 2) + 1. Therefore, (cs)tk ∈ L2,loc[0,∞) and
(u1)tk ∈ H2m+l−1,0
0 (e−γk+1t,K∞) for k 6 h + l + 1. Hence it follows that (F3)tk ∈
∈ H l,0
0 (e−γk+1t,K∞) for k 6 h+ l. On the other hand
(−1)mLS − iSt = F4 +
l−1+m∑
s=0
c̃s(t)r−iλ(t)−2m+sP̃3l+m−2,s(ln r),
where P̃3l+m−2,s is a polynomial having order less than 3l+m−1 and its coefficients are
infinitely differentiable functions of (ω, t), (F4)tk ∈ H l,0
0 (e−γk+1t,K∞), and (c̃s)tk ∈
∈ L2,loc[0,∞) for k 6 h+ l. Therefore from (3.25) we obtain
(−1)m−1L0(0, t,D)u1 = F5 +
l+m−1∑
s=0
c̃s(t)r−iλ(t)−2m+sP̃3l+m−2,s(ln r), (3.26)
where F5 = F3 + F4 ∈ H l,0
0 (e−γ1t,K∞) ⊆ H l−1,0
−1 (e−γ1t,K∞).
By Lemma 3.2 and by arguments used in the proof of Proposition 3.1 we can find
u1(x, t) =
l+m−1∑
s=0
c̃s(t)r−iλ(t)+sP̃3l+m−1,s(ln r) + u2(x, t), (3.27)
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1652 NGUYEN MANH HUNG, CUNG THE ANH
where P̃3l+m−1,s is a polynomial having order less than 3l + m and its coefficients
are infinitely differentiable functions of (ω, t), (u2)tk ∈ H2m+l−1,0
−1 (e−γk+1t,K∞) for
k 6 h + l. By Lemma 2.2 we have (u2)tk ∈ H2m+l,0
0 (e−γk+1t,K∞) for k 6 h + l.
Hence and from (3.24) it follows that
u(x, t) =
l+m−1∑
s=0
cs(t)r−iλ(t)+sP3l+m−1,s(ln r) + u2(x, t), (3.28)
where P3l+m−1,s is a polynomial having order less than 3l + m and its coefficients
are infinitely differentiable functions of (ω, t), (cs)tk ∈ L2,loc[0,∞), and (u2)tk ∈
∈ H2m+l,0
0 (e−γk+1t,K∞) for k 6 h+ l.
Case 2: 2m+ l− 1− n
2
< Imλ(t) < 2m+ l− n
2
. It follows from Theorem 2.1 and
Lemma 2.1 that utk ∈ H2m,0
m (e−γk+1t,K∞) for k 6 h+2l. On the other hand, the strip
m − n
2
6 Imλ 6 2m − m
2
does not contain points of spectrum of the problem (2.2),
(2.3) for every t ∈ [0,∞). Hence and from theorems on the smoothness of solutions of
elliptic problems in domains with conical points (see, e.g., [4, 5, 8, 22]) it follows that
utk ∈ H2m,0
0 (e−γk+1t,K∞) for k 6 h+ 2l.
We will prove that if ftk ∈ L∞(0,∞;Hj
0(K)) for k 6 2j+h+2 and ftk(x, 0) = 0
for k 6 2j + h+ 1, then utk ∈ H2m+j,0
0 (e−γk+1t,K∞), k 6 h+ 2l− j. This assertion
was proved for j = 0. Assume that it is true for j−1. Since ftk ∈ L∞(0,∞;Hj−1
0 (K))
for k 6 2(j−1)+(h+2)+2 and ftk(x, 0) = 0 for k 6 2(j−1)+(h+2)+1, then from
inductive hypothesis it follows that utk ∈ H2m+j−1,0
0 (e−γk+1t,K∞), k 6 h+2l−j+3.
Therefore utk+1 ∈ Hj−1,0
−1 (e−γk+2t,K∞) for k 6 h + 2l − j. Hence and from the fact
that the strip
2m+ j − 1− n
2
6 Imλ 6 2m+ j − n
2
does not contain points of spectrum of the problem (2.2), (2.3) for every t ∈ [0,∞), we
obtain utk ∈ H2m+j−1,0
−1 (e−γk+1t,K∞), k 6 h + 2l − j. It follows from Lemma 2.2
that utk ∈ H2m+j,0
0 (e−γk+1t,K∞) for k 6 h+ 2l − j.
By Lemma 3.2 and from above arguments we obtain
u(x, t) = c(t)r−iλ(t)ϕ(ω, t) + u1(x, t), (3.29)
where ϕ is an infinitely differentiable function of (ω, t) which does not depend on the
solution, ctk ∈ L2,loc[0,∞), and (u1)tk ∈ H2m+l,0
0 (e−γk+1t,K∞) for k 6 h+ l.
Case 3: There exists t0 such that Imλ(t0) = 2m+ l − 1− n
2
· By arguments used
in Case 3 in the proof of Proposition 3.1, this case can be managed.
Proposition 3.2 is proved.
Proposition 3.3. Let u(x, t) be a generalized solution of the problem (1.1) – (1.3)
in the spaces
◦
Hm,0(e−γt,Ω∞) such that u ≡ 0 whenever |x| > R = const, and let
ftk ∈ L∞(0,∞;H l
0(K)) for k 6 2l+ h+ 2, ftk(x, 0) = 0 for k 6 2l+ h+ 1. Assume
that the straight lines
Imλ = m− n
2
and Imλ = 2m+ l − n
2
do not contain points of spectrum of the problem (2.2), (2.3) for every t ∈ [0,∞), and
in the strip
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ASYMPTOTIC EXPANSIONS OF SOLUTIONS OF THE FIRST INITIAL BOUNDARY-VALUE ... 1653
m− n
2
< Imλ < 2m+ l − n
2
there exist only simple eigenvalues λ1(t), λ2(t), . . . , λN0(t) of the problem (2.2), (2.3)
such that
Imλ1(t) < Imλ2(t) < . . . < ImλN0(t), t ∈ [0,∞),
Imλj(t) 6= Imλk(t) +N, j 6= k, N ∈ Z, j, k = 1, . . . , N0.
Then the following representation holds:
u(x, t) =
N0∑
j=1
l+m−1∑
s=0
cs,j(t)r−iλj(t)+sP3l+m−1,s,j(ln r) + u1(x, t), (3.30)
where P3l+m−1,s,j is a polynomial having order less than 3l + m and its coeffici-
ents are infinitely differentiable functions of (ω, t), (cs,j)tk ∈ L2,loc[0,∞), (u1)tk ∈
∈ H2m+l,0
0 (e−γk+1t,K∞) for k 6 h+ l.
Proof. For each t0 ∈ (0,∞) there exists ε > 0 such thatm+µj−1−
n
2
< Imλj(t) <
< m + µj −
n
2
, t ∈ [t0 − ε, t0 + ε], µj = const ≥ 0, j = 1, . . . , N0. Therefore, there
exist the numbers T0 = 0 < T1 < T2 < . . . such that m + µj−1,s −
n
2
< Imλj(t) <
< m+µj,s−
n
2
, t ∈ [Ts−1, Ts], µj,s = const, j = 1, . . . , N0, s = 1, 2, . . . . By arguments
used in case 3 in the proof of Proposition 3.1 if necessary, we can assume that
m− n
2
< Imλ1(t) < m+ µ1 −
n
2
< Imλ2(t) < . . .
. . . < m+ µN0−1 −
n
2
< ImλN0(t) < 2m+ l − n
2
, t ∈ [0,∞).
In order to prove the statement we will use induction on N0. If N0 = 1 the statement
follows from Proposition 3.2. Let the statement be true for N0 − 1. First, consider the
case µN0−1 ≥ m. For simplicity we assume that µN0−1 = m+l0, l0 < l. From inductive
hypothesis we obtain
u(x, t) =
N0−1∑
j=1
l0+m−1∑
s=0
cs,j(t)r−iλj(t)+sP3l0+m−1,s,j(ln r) + u1(x, t), (3.31)
where P3l0+m−1,s,j is a polynomial having order less than 3l0 + m and its coeffici-
ents are infinitely differentiable functions of (ω, t), (cs,j)tk ∈ L2,loc[0,∞), (u1)tk ∈
∈ H2m+l0,0
0 (e−γk+1t,K∞) for k 6 h+ l0. Repeating arguments analogous to the proof
of (3.26) we have
(−1)m−1L0(0, t,D)u1 = F̃ +
N0−1∑
j=1
l0+m∑
s=0
c̃s,j(t)r−iλj(t)−2m+sP̃3l0+m+1,s,j(ln r),
(3.32)
where F̃ ∈ H l0+1,0
0 (e−γ1t,K∞), P̃3l0+m+1,s,j is a polynomial having order less than
3l0 +m+2 and its coefficients are infinitely differentiable functions of (ω, t), (c̃s,j)tk ∈
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1654 NGUYEN MANH HUNG, CUNG THE ANH
∈ L2,loc[0,∞), k 6 h. Hence it follows that if 2m+ l1 −
n
2
< ImλN0(t) < 2m+ l1 +
+ 1− n
2
for l1 ≥ l0, then
u1(x, t) =
N0∑
j=1
l1+m∑
s=0
c̃s,j(t)r−iλj(t)+sP̃3l1+m+2,s(ln r) + u2(x, t), (3.33)
where P̃3l1+m+2,s is a polynomial having order less than 3l1 +m+3 and its coefficient
are infinitely differentiable functions of (ω, t), (u2)tk ∈ H2m+l1+1,0
0 (e−γk+1t,K∞), for
k 6 h+ l1.
Since the strip
2m+ l1 + 1− n
2
6 Imλ 6 2m+ l − n
2
does not contain points of spectrum of the problem (2.2), (2.3) so from (3.31), (3.32)
and (3.33) we obtain (3.30).
If there exists t0 such that ImλN0(t0) = 2m+ l1 −
n
2
then by Lemma 3.2 and from
arguments used in case 3 in the proof of Proposition 3.1 we obtain (3.30).
Finally, if µN0−1 < m, for simplicity we assume that µN0−1 = m0, 0 6 m0 < m,
then repeating the proofs of Proposition 3.1, Proposition 3.2, using above arguments and
Lemma 3.2 we obtain the conclusion.
Proposition 3.3 is proved.
We can now state the main theorem on the asymptotic expansion of the generalized
solution of problem (1.1) – (1.3) in a neighbourhood of the conical point.
Theorem 3.2. Let u(x, t) be a generalized solution of the problem (1.1) – (1.3)
in the spaces
◦
Hm,0(e−γt,Ω∞), and let ftk ∈ L∞(0,∞;H l
0(Ω)) for k 6 2l + h + 2,
ftk(x, 0) = 0 for k 6 2l + h+ 1. Assume that the straight lines
Imλ = m− n
2
and Imλ = 2m+ l − n
2
do not contain points of spectrum of the problem (2.2), (2.3) for every t ∈ [0,∞), and
in the strip
m− n
2
< Imλ < 2m+ l − n
2
there exist only simple eigenvalues λ1(t), . . . , λN0(t) of the problem (2.2), (2.3) such
that
Imλ1(t) < Imλ2(t) < . . . < ImλN0(t), t ∈ [0,∞),
Imλj(t) 6= Imλk(t) +N, j 6= k, N ∈ Z, j, k = 1, . . . , N0.
Then the following representation holds in a neighbourhood of the conical point
u(x, t) =
N0∑
j=1
l+m−1∑
s=0
cs,j(t)r−iλj(t)+sP3l+m−1,s,j(ln r) + u1(x, t), (3.34)
where P3l+m−1,s,j is a polynomial having order less than 3l + m and its coefficients
are infinitely differentiable functions of (ω, t), (cs,j)tk ∈ L2,loc[0,∞) and (u1)tk ∈
∈ H2m+l,0
0 (e−γk+1t,Ω∞) for k 6 h+ l.
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ASYMPTOTIC EXPANSIONS OF SOLUTIONS OF THE FIRST INITIAL BOUNDARY-VALUE ... 1655
Proof. Surrounding the point 0 by a neighbourhood U0 with so small diameter that
the intersection of Ω and U0 coincides with K. Consider a function u0 = ϕ0u, where
ϕ0 ∈
◦
C∞(U0) and ϕ0 ≡ 1 in some neighbourhood of 0. The function u0 satisfies the
system
(−1)m−1iL(x, t,D)u0 − (u0)t = ϕ0f + L′(x, t,D)u,
where L′(x, t,D) is a linear differential operator having order less than 2m. Coefficients
of this operator depend on the choice of the function ϕ0 and equal to 0 outside U0.
Hence and from arguments analogous to the proof of Proposition 3.3, we obtain
ϕ0u(x, t) =
N0∑
j=1
l+m−1∑
s=0
cs,j(t)r−iλj(t)+sP3l+m−1,s,j(ln r) + u2(x, t), (3.35)
where P3l+m−1,s,j is a polynomial having order less than 3l + m and its coeffici-
ents are infinitely differentiable functions of (ω, t), (cs,j)tk ∈ L2,loc[0,∞), (u2)tk ∈
∈ H2m+l,0
0 (e−γk+1t,K∞) for k 6 h+ l.
The function ϕ1u = (1 − ϕ0)u equals to 0 in some neighbourhood of the conical
point. We can apply the known theorem on the smoothness of solutions of elliptic
problems in a smooth domain to this function and obtain ϕ1u ∈ H2m+l
0 (Ω) for a.e.
t ∈ [0,∞). Hence we have (ϕ1u)tk ∈ H2m+l,0
0 (e−γk+1t,Ω∞) for k 6 h+ l.
Since u = ϕ0u+ ϕ1u so from (3.35) we obtain (3.34).
Theorem 3.2 is proved.
4. An example. The Schrödinger equation is the fundamental equation of nonrelati-
vistic quantum mechanics. In the simplest case for a particle without spin in an external
field it has the form
i~
∂ψ
∂t
= − ~
2m
4ψ + V (x)ψ,
where x ∈ R3, ψ = ψ(x, t) is the wave function of a quantum particle, giving the
complex amplitude characterizing the presence of the particle at each point x (in parti-
cular |ψ(x, t)|2 is interpreted as the probability density for the particle to be at the point
x at the instant t), m is the mass of the particle, ~ is Planck’s constant, and V (x) is the
external field potential (a real-value function).
We consider in Ω∞ the mathematical model of Schrödinger equation
i4u− ut = f, (4.1)
with an initial condition
u|t=0 = 0, (4.2)
and a boundary condition
u|S∞ = 0. (4.3)
Let the cone K = {x : x/|x| ∈ G}, where G is a smooth domain on the unit sphere
Sn−1. The Laplacian in polar coordinate (r, ω) in Rn is given by
(4u)(r, ω) =
1
rn−1
∂
∂r
(
rn−1 ∂
∂r
)
u(r, ω) +
1
r2
4ωu(r, ω),
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 12
1656 NGUYEN MANH HUNG, CUNG THE ANH
where 4ω is the Laplace – Beltrami operator on the unit sphere Sn−1. Therefore the
corresponding spectral problem has the form
4ωv + [(iλ)2 + i(2− n)λ]v = 0, ω ∈ G, (4.4)
v|∂G = 0. (4.5)
4.1. Case n = 2. Assume in a neighbourhood of the coordinate origin, ∂Ω coincides
with a rectilinear angle having measure is β. Then the spectral problem (4.4), (4.5) has
the form
vωω − λ2v = 0, 0 < ω < β, (4.6)
v(0) = v(β) = 0. (4.7)
Eigenvalues of the problem (4.6), (4.7) are λk = ± iπk
β
, k ∈ N∗. From Theorems 2.2
and 3.1 we obtain the following proposition.
Proposition 4.1. Let u(x, t) be a generalized solution of the problem (4.1) – (4.3)
in the space
◦
H1,0(e−γt,Ω∞), and let f, ft, ftt, fttt ∈ L∞(0,∞;L2(Ω)), f(x, 0) =
= ft(x, 0) = ftt(x, 0) = 0. Then
(i) if β < π, then u ∈ H2
0 (e−γ2t,Ω∞),
(ii) if β > π, then
u(x, t) = c(x, t)rπ/β + u1(x, t),
where c(x, t) ∈ V 2
π/β(e−γ2t,Ω∞), u1 ∈ H2
0 (e−γ2t,Ω∞).
4.2. Case n = 3. Let kj , j = 1, 2, . . . , are eigenvalues of the Dirichlet problem for
the equation
4ωv + kv = 0, ω ∈ G, (4.8)
with 0 < k1 6 k2 6 k3 6 . . . . Then
λj = i
(
−1
2
±
√
1
4
+ kj
)
, j = 1, 2, . . . , (4.9)
are eigenvalues of the spectral problem (4.4), (4.5).
Let vj be the eigenfunction corresponding to the eigenvalue λj .
We will prove that λj is simple. Indeed, let ϕ1, ϕ2, . . . , ϕr, r ≥ 1, are connecting
functions of the eigenfunction vj . From (4.4) and definition of the connecting function,
we have
4ω v̄j + [(iλj)2 − iλj ]v̄j = 0, (4.10)
4ωϕl + [(iλj)2 − iλj ]ϕl − (2λj + i)vj = 0. (4.11)
Then ∫
G
4ωϕlv̄jdω =
∫
G
ϕl4ω v̄jdω. (4.12)
From (4.10), (4.11) and (4.12) it follows that
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 12
ASYMPTOTIC EXPANSIONS OF SOLUTIONS OF THE FIRST INITIAL BOUNDARY-VALUE ... 1657
(2λj + i)
∫
G
|vj |2dω = 0. (4.13)
Since Im (2λj + i) 6= 0, so vj = 0. It is a contradiction. Thus, λj is simple.
Let f, ft, ftt, fttt ∈ L∞(0,∞;L2(Ω), f(x, 0) = ft(x, 0) = ftt(x, 0) = 0, and
u(x, t) be a generalized solution of the problem (4.1) – (4.3) in the space
◦
H1,0(e−γt,Ω∞).
We distinguish the following cases.
Case 1: Imλ1 >
1
2
. Since the strip −1
2
6 Imλ 6
1
2
does not contain eigen-
values of the spectral problem (4.4), (4.5), from Theorem 2.2 we obtain that u(x, t) ∈
∈ H2
0 (e−γ2t,Ω∞).
Case 2: Imλ1 6
1
2
. Let λ1, . . . , λN0 , are eigenvalues of the spectral problem (4.4),
(4.5) satisfying
−1
2
< Imλ1 < . . . < ImλN0 6
1
2
.
(i) If the straight line Imλ =
1
2
does not contain eigenvalues of the spectral
problem (4.4), (4.5), then from Theorem 3.2 we obtain
u(x, t) =
N0∑
j=1
cj(t)rIm λjφj(ω) + u0(x, t), (4.14)
where φj are infinitely differetiable functions of ω, and cj ∈ L2,loc[0,∞), j = 1, . . . , N0,
and u0 ∈ H2,0
0 (e−γ1t,Ω∞).
Consider the domain
Ωρ =
{
x ∈ Ω:
1
2
ρ < |x| < 2ρ
}
, ρ = const > 0.
Let ρ be small enough such that the boundary of Ωρ coincides with the cone K.
Putting v(x′, t) = u0(ρx′, t). Since u0 ∈ H2,0
0 (e−γ1t,Ω∞), from embedding theorems
for the domain K ′ =
{
x′ ∈ K :
1
2
< |x′| < 2
}
, we obtain
|v(x′, t)|2 6 C1
∫
K′
v2 + |grad v|2 +
∑
|α|=2
|Dαv|2
dx′, C1 = const.
Substituting x = ρx′ in this inequality, we obtain
|u0(x, t)|2 6 C1
∫
Ωρ
ρ−3u2
0 + ρ−1|gradu0|2 + ρ
∑
|α|=2
|Dαu0|2
dx.
Hence
ρ−1|u0(x, t)|2 6 C1
∫
Ωρ
ρ−4u2
0 + ρ−2|gradu0|2 +
∑
|α|=2
|Dαu0|2
dx 6
6 C2
∫
Ωρ
r−4u2
0 + r−2|gradu0|2 +
∑
|α|=2
|Dαu0|2
dx 6
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 12
1658 NGUYEN MANH HUNG, CUNG THE ANH
6 C3‖u0(x, t)‖2H2
0 (Ω) 6 C4‖f(x, t)‖2L2(Ω),
where Ci = const, i = 1, 2, 3, 4. For |x| = ρ we obtain
|u0(x, t)| 6 Cr1/2, C = const. (4.15)
From (4.12) and (4.13) we have
|u(x, t)| 6 CrIm λ1 , C = const .
(ii) If ImλN0 =
1
2
, we choose ε > 0 such that the straight line ImλN0 =
1
2
+ ε
does not contain eigenvalues of the spectral problem (4.4), (4.5) and −1
2
< Imλ1 < . . .
. . . < ImλN0 <
1
2
+ ε. Repeating arguments analogous to the proof of Proposition 3.3,
we obtain
u(x, t) =
N0∑
j=1
cj(t)rIm λjφj(ω) + u0(x, t),
where φj are infinitely differetiable functions of ω, and cj ∈ L2,loc[0,∞), j = 1, . . . , N0,
and u0 ∈ H2,0
0 (e−γ1t,Ω∞).
By using arguments analogous to the proof of case (i), we obtain
|u(x, t)| 6 CrIm λ1 , C = const .
If Ω is convex in a neighbourhood of the the coordinate origin, then the strip
−1
2
6 Imλ < 1 does not contain eigenvalues of the spectral problem (4.4), (4.5) (see
[3, p. 290]). It follows from Theorem 2.2 that u(x, t) ∈ H2
0 (e−γ2t,Ω∞).
From above arguments we have the following proposition.
Proposition 4.2. Let u(x,t) be a generalized solution of the problem (4.1) – (4.3)
in the space
◦
H1,0(e−γt,Ω∞), and let f, ft, ftt, fttt ∈ L∞(0,∞;L2(Ω)), f(x, 0) =
= ft(x, 0) = ftt(x, 0) = 0. Then
|u(x, t)| 6 C|x|Im λ1 , C = const .
Moreover, if Ω is convex in a neighbourhood of the the coordinate origin then u(x, t) ∈
∈ H2
0 (e−γ2t,Ω∞).
4.3. Case n > 3. In this case, the strip
1− n
2
6 Imλ 6 2− n
2
does not contain eigenvalues of the spectral problem (4.4), (4.5) (see [3, p. 289]).
From Theorem 2.2, we obtain the following proposition.
Proposition 4.3. Let u(x, t) be a generalized solution of the problem (4.1) – (4.3)
in the space
◦
H1,0(e−γt,Ω∞), and let f, ft, ftt, fttt ∈ L∞(0,∞;L2(Ω)), f(x, 0) =
= ft(x, 0) = ftt(x, 0) = 0. Then u ∈ H2
0 (e−γ2t,Ω∞).
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 12
ASYMPTOTIC EXPANSIONS OF SOLUTIONS OF THE FIRST INITIAL BOUNDARY-VALUE ... 1659
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Received 07.07.08
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 12
|
| id | umjimathkievua-article-3128 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:36:46Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/b7/45dfcf10892104f6b2778eeb8e0803b7.pdf |
| spelling | umjimathkievua-article-31282020-03-18T19:45:55Z Asymptotic expansions of solutions of the first initial boundary-value problem for Schrödinger systems in domains with conical points. II Асимптотичн розклади розв'язків першої початкової крайової задачі для систем Шредінгера в областях з конічними точками. II Cung, The Anh Nguen, Van Hung Кунг, Тхе Анх Нгуєн, Ван Хунг We consider asymptotic expansions of solutions of the first initial boundary-value problem for strong Schrödinger systems near a conical point of the boundary of a domain. Розглянуто асимптотичні розклади розв'язків першої початкової крайової задачi для сильно шредінгерових систем біля конічної точки межі області. Institute of Mathematics, NAS of Ukraine 2009-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3128 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 12 (2009); 1640-1659 Український математичний журнал; Том 61 № 12 (2009); 1640-1659 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3128/3004 https://umj.imath.kiev.ua/index.php/umj/article/view/3128/3005 Copyright (c) 2009 Cung The Anh; Nguen Van Hung |
| spellingShingle | Cung, The Anh Nguen, Van Hung Кунг, Тхе Анх Нгуєн, Ван Хунг Asymptotic expansions of solutions of the first initial boundary-value problem for Schrödinger systems in domains with conical points. II |
| title | Asymptotic expansions of solutions of the first initial boundary-value problem for Schrödinger systems in domains with conical points. II |
| title_alt | Асимптотичн розклади розв'язків першої початкової крайової задачі для систем Шредінгера в областях з конічними точками. II |
| title_full | Asymptotic expansions of solutions of the first initial boundary-value problem for Schrödinger systems in domains with conical points. II |
| title_fullStr | Asymptotic expansions of solutions of the first initial boundary-value problem for Schrödinger systems in domains with conical points. II |
| title_full_unstemmed | Asymptotic expansions of solutions of the first initial boundary-value problem for Schrödinger systems in domains with conical points. II |
| title_short | Asymptotic expansions of solutions of the first initial boundary-value problem for Schrödinger systems in domains with conical points. II |
| title_sort | asymptotic expansions of solutions of the first initial boundary-value problem for schrödinger systems in domains with conical points. ii |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3128 |
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