Asymptotic Solutions of the Dirichlet Problem for the Heat Equation at a Characteristic Point
The Dirichlet problem for the heat equation in a bounded domain $G ⊂ ℝ^{n+1}$ is characteristic because there are boundary points at which the boundary touches a characteristic hyperplane $t = c$, where c is a constant. For the first time, necessary and sufficient conditions on the boundary guarante...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508177099915264 |
|---|---|
| author | Antoniouk, A. Vict. Kiselev, O. M. Tarkhanov, N. N. Антонюк, О. Вік. Кисельов, О. М. Тарханов, Н. Н. |
| author_facet | Antoniouk, A. Vict. Kiselev, O. M. Tarkhanov, N. N. Антонюк, О. Вік. Кисельов, О. М. Тарханов, Н. Н. |
| author_sort | Antoniouk, A. Vict. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T10:26:46Z |
| description | The Dirichlet problem for the heat equation in a bounded domain $G ⊂ ℝ^{n+1}$ is characteristic because there are boundary points at which the boundary touches a characteristic hyperplane $t = c$, where c is a constant. For the first time, necessary and sufficient conditions on the boundary guaranteeing that the solution is continuous up to the characteristic point were established by Petrovskii (1934) under the assumption that the Dirichlet data are continuous. The appearance of Petrovskii’s paper was stimulated by the existing interest to the investigation of general boundary-value problems for parabolic equations in bounded domains. We contribute to the study of this problem by finding a formal solution of the Dirichlet problem for the heat equation in a neighborhood of a cuspidal characteristic boundary point and analyzing its asymptotic behavior. |
| first_indexed | 2026-03-24T02:21:03Z |
| format | Article |
| fulltext |
UDC 517.951, 517.953, 514.954
A. Vict. Antoniouk (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv),
O. M. Kiselev (Inst. Math. Ufim. Sci. Center Rus. Acad. Sci., Russia),
N. N. Tarkhanov (Univ. Potsdam Inst. Math., Germany)
ASYMPTOTIC SOLUTIONS OF THE DIRICHLET PROBLEM
FOR THE HEAT EQUATION AT A CHARACTERISTIC POINT*
АСИМПТОТИЧНI РОЗВ’ЯЗКИ ЗАДАЧI ДIРIХЛЕ ДЛЯ РIВНЯННЯ
ТЕПЛОПРОВIДНОСТI В ХАРАКТЕРИСТИЧНIЙ ТОЧЦI
The Dirichlet problem for the heat equation in a bounded domain G ⊂ Rn+1 is characteristic because there are boundary
points at which the boundary touches a characteristic hyperplane t = c, where c is a constant. It was I.G. Petrovskii (1934)
who first established necessary and sufficient conditions on the boundary guaranteeing that the solution is continuous up to
the characteristic point provided that the Dirichlet data are continuous. The appearance of the paper was stimulated by the
existing interest in studying general boundary-value problems for parabolic equations in bounded domains. We contribute
to the study by constructing a formal solution of the Dirichlet problem for the heat equation in a neighborhood of a cuspidal
characteristic boundary point and showing its asymptotic character.
Задача Дiрiхле для рiвняння теплопровiдностi в обмеженiй областi G ⊂ Rn+1 є характеристичною, оскiльки iснують
граничнi точки, в яких границя є дотичною до характеристичної гiперплощини t = c, де c є сталою. I. Г. Петров-
ський (1934) уперше встановив необхiднi та достатнi умови на границю, що гарантують неперервнiсть розв’язку
аж до характеристичної точки за умови, що данi Дiрiхле є неперервними. Поява даної роботи була викликана
постiйним iнтересом до вивчення загальних граничних задач для рiвнянь параболiчного типу в обмежених облас-
тях. Наш внесок у вивчення цiєї проблеми полягає в побудовi формального розв’язку задачi Дiрiхле для рiвняння
теплопровiдностi в околi гострокiнцевої характеристичної граничної точки та дослiдженнi його асимптотичного
характеру.
Introduction. The problem we consider in this paper goes back at least as far as [8] who proved
the existence of a classical solution to the first boundary-value problem for the heat equation in a
noncylindrical plane domain. By classical is meant “continuous up to the boundary,” and a boundary
point is called regular if any weak solution of the problem is continuous up to the point, provided
the boundary data are continuous. The domain is assumed to be bounded by an interval [a, b] of the
x -axis and two curves x = X1(t) and x = X2(t) in the upper half-plane through (a, 0) and (b, 0),
respectively. The Dirichlet data are posed on the interval and both lateral curves. All points of the
interval [a, b] are characteristic. The interval [a, b] may shrink up to a point (say (0, 0)) in which case
the origin is the only characteristic point.
The theory of [8] applies in particular to the plane domains G consisting of all (x, t) ∈ R2, such
that |x| < 1 and f(|x|) < t < f(1), where f(r) is a C1 function on (0, 1] satisfying f(r) > 0, f′(r) 6= 0
* The research of first author was supported by the Alexander von Humboldt Foundation and grant No. 01-01-12 of
National Academy of Sciences of Ukraine (under the joint Ukrainian-Russian project of National Academy of Sciences of
Ukraine and Russian Foundation of Basic Research); second and third authors were supported by the Russian Foundation
for Basic Research (grant 11-01-91330-NNIO_a) and German Research Society (DFG) (grant TA 289/4-2).
c© A. VICT. ANTONIOUK, O. M. KISELEV, N. N. TARKHANOV, 2014
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 10 1299
1300 A. VICT. ANTONIOUK, O. M. KISELEV, N. N. TARKHANOV
for all r ∈ (0, 1] and f(0+) = 0. The boundary point (0, 0) proves to be regular if f−1(t) satisfies the
Hölder condition of exponent larger than 1/2. When applied to the function f(r) = rp, this obviously
implies 0 < p < 2. Note that for 1 < p < 2 the origin is a true (i.e., smooth) characteristic point at
the boundary while for 0 < p < 1 this is a cuspidal (i.e., singular) boundary point.
The paper [8] exploited the fundamental solution of the heat equation and integral equations of
potential theory. A more careful analysis led Petrovskii in [16] to an explicit necessary and sufficient
condition for a boundary point to be regular. This latter paper initiated an extensive literature devoted
to general boundary-value problems for parabolic equations (see, for example, [10, 14]). Mention
that the classical paper [19] was actually motivated by the first boundary-value problem for the heat
equation in a bounded domain G ⊂ Rn. On the other hand, [10] made essential use of function spaces
of Slobodetskii [19]. Unfortunately, [10] suffers several drawbacks which, however, do not affect the
main result of this seminal paper.
The most cited paper of V. A. Kondrat’ev is [11] studying boundary-value problems for elliptic
equations in domains with conical points on the boundary. Asymptotics of solutions of general
boundary-value problems for elliptic equations in domains with cusps remains still a challenge for
mathematicians (see, for example, [6, 7, 9, 12] and reference therein).
According to the MathSciNet of the AMS there has been merely 8 citations to the paper [10]
while this latter already contains all of the techniques of [11], especially the asymptotics of solutions
at conical points. At the end of the 90’s Kondrat’ev called the last author’s attention to the paper [10]
saying “Here are cusps.” In spite of the fact that [10] deals with C∞ boundaries the analysis near
characteristic boundary points reveals Fuchs-type operators typical for conical singularities, provided
that the contact degree of the boundary and characteristic plane is at least the anisotropy quotient (2
for the heat equation). If the contact degree is less than the anisotropy quotient, the analysis close
to the characteristic point requires pseudodifferential operators typical for cuspidal points on the
boundary (cf. [8] discussed above).
The structure of asymptotics at a conical point is completely determined by the spectrum of
the problem frozen at the singular point. To an eigenvalue λn of multiplicity µn there correspond
eigenfunctions |x|−ıλn(log |x|)j with j = 0, 1, . . . , µn−1. Each horizontal strip of finite width in the
complex plane contains finitely many values λn, and the set of all λn is infinite. The expansions of
solutions over these basic functions fail usually to converge, and so the series should be thought of
as asymptotic.
Moreover, in the absence of embedding theorems the concept of asymptotic in the sense of
Poincaré does not apply. We are thus led to asymptotic expansions related to certain filtrations on
function spaces, a purely algebraic concept, which is a true substitution for Poincaré’s asymptotics,
see for instance [13] and elsewhere.
In mathematics, a (descending) filtration is an indexed set Fn of subspaces of a given vector
space F , with the index n running over entire numbers, subject to the condition that Fn+1 ⊂ Fn for
all n. Let F−∞ be the union of the Fn. Given any f ∈ F−∞, by
f ∼
∞∑
n=nf
fn (0.1)
with fn ∈ Fn is meant that
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 10
ASYMPTOTIC SOLUTIONS OF THE DIRICHLET PROBLEM FOR THE HEAT EQUATION . . . 1301
f −
N∑
n=nf
fn ∈ FN+1
holds for every N ≥ nf . We intend to develop this generalisation of Poincaré’s asymptotics in a
forthcoming publication.
As filtration Kondrat’ev used in [10] weighted Slobodetskii spaces, where the weight functions
are powers of the distance to the characteristic point. Analysis on manifolds with point and more
general singularities has since exploited weighted function spaces.
In [3, 4] the first boundary-value problem is studied for the heat equation in a bounded plane
domain with cuspidal points at the boundary at which the tangent coincides with a characteristic
t = c, where c is a constant. The paper [2] contributed to the study of the first boundary-value
problem for the 1D heat equation in a bounded plane domain by evaluating the first term of the
asymptotic of a solution at the characteristic point. The goal of the present paper is to explicitly
compute full asymptotic expansions.
Our scheme of construction of asymptotic series for a solution near a characteristic point consists
in the following. In Section 1 we resolve singularities at the characteristic point by blowing-up this
point to a segment of the x -axis containing the origin. The domain G close to the origin blows up to
a half-strip. In Section 2 we construct a formal solution of the transformed problem in the half-strip.
This is actually a formal Puiseux series in fractional powers of t unless p = 2. In Section 3 we
construct a formal solution in the case p = 2, which reveals immediately asymptotic expansions
on manifolds with conical points. To prove the asymptotic character of formal solution we need
an existence theorem which is a part of Fredholm theory for the first boundary-value problem for
the heat equation. To this end we describe in Section 4 a change of variables which transforms the
characteristic point to the point at infinity along the t -axis. In Section 5 we discuss the Fredholm
property of the first boundary-value problem for the heat equation. When the Fredholm property has
been proved one obtains real solutions of the problem which expand as formal series. In this case
one introduces the difference between the real solution and a partial sum of the formal series and
substitutes this remainder to the equations. This yields a nonhomogeneous problem for the remainder,
and the formal solvability might testify to the possibility of estimating the remainder. We follow this
way to show in Section 6 the asymptotic character of formal solution in the sense (0.1).
Needless to say that our results go far beyond the first boundary-value problem for the heat
equation and extend to general boundary-value problems for parabolic equations in bounded domains.
1. Blow-up techniques. Consider the first boundary-value problem for the heat equation in a
bounded domain G ⊂ R2. The boundary of G is assumed to be C∞ except for a finite number
of singular points. A boundary point is called characteristic if the boundary is smooth at this point
and the tangent is orthogonal to the t -axis. Since G is bounded, there are at least two characteristic
points on the boundary unless it has a singularity at a characteristic point. In this paper we restrict
our discussion to characteristic points which may moreover bear boundary singularities. By the
local principle of [18] it is sufficient to study the problem only in a small neighbourhood of any
characteristic (singular) point. Thus, the domain G looks like that of Figure 1 with n = 1, i.e., it is
bounded by a curve t = |x|p, with p > 0 an arbitrary real number, from below and by a horizontal
segment from above. This is a typical domain for problems of such a type. As usual, no conditions
are posed on the upper segment see [20].
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 10
1302 A. VICT. ANTONIOUK, O. M. KISELEV, N. N. TARKHANOV
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����
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t = f(x)
0
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�
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xn−1
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t = −∞
Fig. 1. Resolution of singularities at a characteristic point.
If p > 1, then the origin is a characteristic point of the boundary. If 0 < p ≤ 1, then the
boundary has a singularity at the origin, which is a conical point for p = 1 and a cusp for p < 1. As
mentioned, the case p ≥ 2 was treated in [10] in the framework of analysis of Fuchs-type operators.
The paper [2] demonstrates rather strikingly that, for 0 < p < 2, the problem to be considered
is specified in analysis on manifolds with cusps. A modern approach to studying boundary-value
problems in domains with cuspidal boundary points is based on the so-called blow-up techniques,
cf. [17]. While giving a complete characterisation of Fredholm problems, the approach falls short of
providing asymptotics of solutions at singular points.
The first boundary-value problem for the heat equation in the domain G is formulated as follows:
Write Σ for the set of all characteristic points 0, . . . on the boundary of G. Given functions f in G
and u0 at ∂G \ Σ, find a function u on G \ Σ which satisfies
u′t − u′′x,x = f in G, u = u0 at ∂G \ Σ. (1.1)
By the local principle of Simonenko [18], the Fredholm property of problem (1.1) in suitable
function spaces is equivalent to the local invertibility of this problem at each point of the closure of
G. Here we focus upon the characteristic points like the origin P = (0, 0).
Suppose the domain G is described in a neighbourhood of the origin by the inequality
t > |x|p, (1.2)
where p is a positive real number. There is no loss of generality in assuming that |x| ≤ 1.
We now blow up the domain G at P by introducing new coordinates (ω, r) with the aid of
x = r1/p ω, t = r, (1.3)
where |ω| < 1 and r ∈ (0, 1). It is clear that the new coordinates are singular at r = 0, for the entire
segment [−1, 1] on the ω -axis is blown down into the origin by (1.3). The rectangle (−1, 1)× (0, 1)
transforms under the change of coordinates (1.3) into the part of the domain G nearby (0, 0) lying
below the line t = 1.
In the domain of coordinates (ω, r) problem (1.1) reduces to an ordinary differential equation
with respect to the variable r with operator-valued coefficients. More precisely, under transformation
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 10
ASYMPTOTIC SOLUTIONS OF THE DIRICHLET PROBLEM FOR THE HEAT EQUATION . . . 1303
(1.3) the derivatives in t and x change by the formulas
∂u
∂t
=
∂u
∂r
− 1
r
ω
p
∂u
∂ω
,
∂u
∂x
=
1
r1/p
∂u
∂ω
,
and so (1.1) transforms into
rQ U ′r − U ′′ω,ω − rQ−1
ω
p
U ′ω = rQF in (−1, 1)× (0, 1), U = U0 at {±1} × (0, 1),
(1.4)
where U(ω, r) and F (ω, r) are pullbacks of u(x, t) and f(x, t) under transformation (1.3), respec-
tively, and Q =
2
p
.
We are now interested in the local solvability of problem (1.4) near the edge r = 0 in the rectangle
(−1, 1)× (0, 1). Note that the equation degenerates at r = 0, since the coefficient r2/p of the higher
order derivative in r vanishes at r = 0. The exponent Q is of crucial importance for specifying the
ordinary differential equation. If p = 2 then it is a Fuchs-type equation, these are also called regular
singular equations. The Fuchs-type equations fit well into an algebra of pseudodifferential operators
based on the Mellin transform. If p > 2, then the singularity of the equation at r = 0 is weak and so
regular theory of finite smoothness applies. In the case p < 2 the degeneracy at r = 0 is strong and
the equation can not be treated except by the theory of slowly varying coefficients [17].
2. Formal series expansion for homogeneous problem. To determine appropriate function
spaces in which a solution of problem (1.4) is sought, one constructs formal series expansions for the
solutions of the corresponding homogeneous problem. That is
rQ U ′r − U ′′ω,ω − rQ−1
ω
p
U ′ω = 0 in (−1, 1)× (0,∞), U(±1, r) = 0 on (0,∞). (2.1)
We first consider the case p 6= 2. We look for a formal solution to (2.1) of the form
U(ω, r) = eS(r) V (ω, r), (2.2)
where S is a differentiable function of r > 0 and V expands as a formal Puiseux series with nontrivial
principal part
V (ω, r) =
1
reN
∞∑
j=0
Vj(ω) rej ,
the (possibly) complex exponentN and real exponent e have to be determined. In fact, the factor r−eN
might be included into the definition of expS as exp(−eN ln r), however, we prefer to highlight the
key role of Puiseux series.
Substituting (2.2) into (2.1) yields
rQ
(
S′V + V ′r
)
− V ′′ω,ω − rQ−1
ω
p
V ′ω = 0 in (−1, 1)× (0,∞), V (±1, r) = 0 on (0,∞).
In order to reduce this boundary-value problem to an eigenvalue problem we require the function
S to satisfy the eikonal equation rQS′ = λ with a complex constant λ. When Q 6= 1, i.e., p 6= 2, this
implies
S(r) = λ
r1−Q
1−Q
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 10
1304 A. VICT. ANTONIOUK, O. M. KISELEV, N. N. TARKHANOV
up to an inessential constant to be included into a factor of expS. In this manner the problem reduces
to
rQ V ′r − V ′′ω,ω − rQ−1
ω
p
V ′ω = −λV in (−1, 1)× (0,∞), V (±1, r) = 0 on (0,∞).
(2.3)
If e =
Q− 1
k
for some natural number k, then
rQ V ′r =
∞∑
j=k
e(j −N − k)Vj−kr
e(j−N),
V ′′ω,ω =
∞∑
j=0
V ′′j r
e(j−N),
rQ−1 V ′ω =
∞∑
j=k
V ′j−kr
e(j−N),
as is easy to check. On substituting these equalities into (2.3) and equating the coefficients of the
same powers of r we get two collections of Sturm – Liouville problems
−V ′′j + λVj = 0 in (−1, 1), Vj = 0 at ∓ 1, (2.4)
for j = 0, 1, . . . , k − 1, and
−V ′′j + λVj =
ω
p
V ′j−k − e(j −N − k)Vj−k in (−1, 1), Vj = 0 at ∓ 1, (2.5)
for j = mk,mk + 1, . . . ,mk + (k − 1), where m takes on all natural values.
Given any j = 0, 1, . . . , k−1, the Sturm – Liouville problem (2.4) considered in space L2(−1, 1)
has obviously simple eigenvalues λn = −
(π
2
n
)2
for n ≥ 1, a nonzero eigenfunction corresponding
to λn being sin
π
2
n(ω + 1). It follows that
Vn,j(ω) = cn,j sin
π
2
n(ω + 1), (2.6)
for j = 0, 1, . . . , k − 1, where cn,j are some constants. Without restriction of generality we can
assume that the first coefficient Vn,0 in the Puiseux expansion of V is different from zero. Hence,
Vn,j = cn,jVn,0 for j = 1, . . . , k − 1.
On having determined the functions Vn,0, . . . , Vn,k−1 belonging to the standard Sobolev space
H2(−1, 1), we turn our attention to problems (2.5) with j = k, . . . , 2k − 1. Set
fn,j =
ω
p
V ′n,j−k − e(j −N − k)Vn,j−k,
then for the inhomogeneous problem (2.5) to possess a nonzero solution Vn,j it is necessary and suf-
ficient that the right-hand side fn,j be orthogonal to all solutions of the corresponding homogeneous
problem, to wit Vn,0. The orthogonality refers to the scalar product in L2(−1, 1):
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 10
ASYMPTOTIC SOLUTIONS OF THE DIRICHLET PROBLEM FOR THE HEAT EQUATION . . . 1305(
fn,j , Vn,0
)
= 0 for j = k, . . . , 2k − 1.
Let us evaluate the scalar product (fn,j , Vn,0) for j = k, . . . , 2k − 1. We get
(fn,j , Vn,0) = cn,j−k
(
1
p
(ωV ′n,0, Vn,0)− e(j −N − k)(Vn,0, Vn,0)
)
and
(ωV ′n,0, Vn,0) = ω |Vn,0|2
∣∣∣ 1
−1
−(Vn,0, Vn,0)− (Vn,0, ωV
′
n,0) = −(Vn,0, Vn,0)− (ωV ′n,0, Vn,0),
the latter equality being due to the fact that Vn,0 is real-valued and vanishes at ±1. Hence,
(ωV ′n,0, Vn,0) = −1
2
(Vn,0, Vn,0)
and
(fn,j , Vn,0) = −cn,j−k
(
1
2p
+ e(j −N − k)
)
(Vn,0, Vn,0) (2.7)
for j = k, . . . , 2k − 1.
Since Vn,0 6= 0, the condition (fn,j , Vn,0) = 0 fulfills for j = k if and only if
eN =
1
2p
. (2.8)
Under this condition, problem (2.5) with j = k is solvable and its general solution has the form
Vn,k = Wn,k + cn,kVn,0 ∈ H2(−1, 1),
where Wn,k is a particular solution of (2.5) and cn,k is an arbitrary constant. Moreover, for (fn,j ,
Vn,0) = 0 to fulfill for j = k+1, . . . , 2k−1 it is necessary and sufficient that cn,1 = . . . = cn,k−1 = 0,
i.e., all of Vn,1, . . . , Vn,k−1 vanish. This in turn implies that fn,k+1 = . . . = fn,2k−1 = 0, whence
Vn,j = cn,jVn,0 for all j = k + 1, . . . , 2k − 1, where cn,j are arbitrary constants. We choose the
constants cn,k+1, . . . , cn,2k−1 in such a way that the solvability conditions of the next k problems are
fulfilled.
More precisely, we consider the problem (2.5) for j = 2k, the right-hand side being
fn,2k =
(
ω
p
W ′n,k − e(k −N)Wn,k
)
+ cn,k
(
ω
p
V ′n,0 − e(k −N)Vn,0
)
=
=
(
ω
p
W ′n,k − e(k −N)Wn,k
)
+ cn,k (fn,k − ek Vn,0) .
Combining (2.7) and (2.8) we conclude that
(fn,k − ek Vn,0, Vn,0) = −ek (Vn,0, Vn,0) = (1−Q) (Vn,0, Vn,0)
is different from zero. Hence, the constant cn,k can be uniquely defined in such a way that (fn,2k,
Vn,0) = 0. Moreover, the functions fn,2k+1, . . . , fn,3k−1 are orthogonal to Vn,0 if and only if
cn,k+1 = . . . = cn,2k−1 = 0. It follows that Vn,j vanishes for each j = k + 1, . . . , 2k − 1.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 10
1306 A. VICT. ANTONIOUK, O. M. KISELEV, N. N. TARKHANOV
Continuing in this fashion we construct a sequence of functions Vn,j ∈ H2(−1, 1), for j =
= 0, 1, . . . , satisfying equations (2.4) and (2.5). The functions Vn,j(ω) are defined uniquely up to
a common constant factor cn,0 = cn. Moreover, Vn,j vanishes identically unless j = mk with
m = 0, 1, . . . . Therefore,
Vn(ω, r) =
1
reN
∞∑
m=0
Vn,mk(ω) remk =
1
rQ/4
∞∑
m=0
Vn,mk(ω) r(Q−1)m (2.9)
is a unique (up to a constant factor) formal series expansions of the solution to the problem (2.3)
corresponding to λ = λn.
From (2.9) it is seen that the powers in the formal series expansions don’t depend on the parameter
k, and the corresponding functions Vn,mk(ω) recovered from the system (2.4) with the right-hand
side constructed from the functions Vn,m(k−1). Thus, without loss of generality we may assume that
k = 1 and write formal series expansions in the form
Vn(ω, r) =
1
rQ/4
∞∑
m=0
Vn,m(ω) r(Q−1)m.
Moreover, due to (2.5) we have the following reccurent relation for the functions Vn,m:
−V ′′n,0 + λn Vn,0 = 0 in (−1, 1), Vn,0 = 0 at ∓ 1, (2.10)
and
−V ′′n,m + λn Vn,m =
ω
p
V ′n,m−1 +
(
Q
4
+ (m− 1)(1−Q)
)
Vn,m−1 in (−1, 1),
Vn,m = 0 at ∓ 1,
(2.11)
for m ≥ 1.
Theorem 2.1. Let p 6= 2. Then an arbitrary solution of homogeneous problem (2.1) has formal
series expansions of the form
U(ω, r) =
∞∑
n=1
cn
rQ/4
exp
(
λn
r1−Q
1−Q
) ∞∑
m=0
Vn,m(ω)
r(1−Q)m
,
where λn = −
(π
2
n
)2
are the eigenvalues of the Sturm – Liouville problem (2.4).
Proof. The theorem follows readily from (2.2).
In the original coordinates (x, t) close to the point P = (0, 0) in G the formal series expansions
for the solution of problem (1.1) looks like
u(x, t) =
∞∑
n=0
cn
tQ/4
exp
(
λn
t1−Q
1−Q
) ∞∑
m=0
Vn,m
( x
t1/p
)(1
t
)(1−Q)m
. (2.12)
If 1 − Q > 0, i.e., p > 2, expansion (2.12) behaves in much the same way as boundary layer
expansion in singular perturbation problems, since the eigenvalues are all negative. The threshold
value p = 2 is a turning contact order under which the boundary layer degenerates.
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ASYMPTOTIC SOLUTIONS OF THE DIRICHLET PROBLEM FOR THE HEAT EQUATION . . . 1307
3. The exceptional case p = 2. In this section we consider the case p = 2 in detail. For p = 2,
problem (2.1) takes the form
r U ′r − U ′′ω,ω −
ω
2
U ′ω = 0 in (−1, 1)× (0,∞), U(±1, r) = 0 on (0,∞). (3.1)
The problem is specified as Fuchs-type equation on the half-axis with coefficients in boundary-value
problems on the interval [−1, 1]. Such equations have been well understood, see [5] and elsewhere.
If one searches for a formal solution to (3.1) of the form U(ω, r) = eS(r) V (ω, r), then the
eikonal equation rS′ = λ gives S(r) = λ ln r, and so eS(r) = rλ, where λ is a complex number. It
makes therefore no sense to looking for V (ω, r) being a formal Puiseux series in fractional powers
of r. The choice e = (Q− 1)/k no longer works, and so a good substitute for a fractional power of
r is the function 1/ ln r. Thus,
V (ω, r) =
∞∑
j=0
Vj(ω)
(
1
ln r
)j−N
has to be a formal series expansions for the solution of
r V ′r − V ′′ω,ω −
ω
2
V ′ω = −λV in (−1, 1)× (0,∞), V (±1, r) = 0 on (0,∞),
N being a nonnegative integer. Substituting the series for V (ω, r) into these equations and equating
the coefficients of the same powers of ln r yields two collections of Sturm – Liouville problems
−V ′′0 −
ω
2
V ′0 + λV0 = 0 in (−1, 1), V0 = 0 at ∓ 1, (3.2)
for j = 0, and
−V ′′j −
ω
2
V ′j + λVj = (j −N − 1)Vj−1 in (−1, 1), Vj = 0 at ∓ 1, (3.3)
for j ≥ 1.
Problem (3.2) has a nonzero solution V0 if and only if λ is an eigenvalue of the operator Lv =
= v′′+
1
2
ω v′ whose domain consists of all functions v from the Sobolev space H2(−1, 1) vanishing
at ∓1. Then, equalities (3.3) for j = 1, . . . , N mean that V1, . . . , VN are actually root functions of
the operator corresponding to the eigenvalue λ. In other words, Vn,0, . . . , Vn,N is a Jordan chain of
length N + 1 corresponding to the eigenvalue λn. Note that for j = N + 1 the right-hand side of
(3.3) vanishes, and so Vn,N+1, Vn,N+2, . . . is also a Jordan chain corresponding to the eigenvalue λn.
This suggests that the series breaks beginning at j = N+1. Furthermore, it follows from the Sturm –
Liouville theory that problem (3.2) has a discrete sequence {λn}n=1,2,... of real eigenvalues. If
−v′′ − 1
2
ω v′ + λv = 0
on (−1, 1) for some function v ∈ H2(−1, 1) vanishing at ∓1, then
‖v′‖2 + λ‖v‖2 =
1
2
(ωv′, v), (3.4)
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1308 A. VICT. ANTONIOUK, O. M. KISELEV, N. N. TARKHANOV
where the scalar product and norm are those of L2(−1, 1). By the Schwarz inequality, we get
|(ωv′, v)| ≤ ‖v′‖ ‖v‖. Since
‖v′‖2 + λ‖v‖2 =
1
2
‖v′‖ ‖v‖+
(
‖v′‖ − 1
4
‖v‖
)2
+
(
λ− 1
16
)
‖v‖2 ≥
≥ 1
2
‖v′‖ ‖v‖+
(
λ− 1
16
)
‖v‖2,
we conclude that equality (3.4) fulfills only for the function v = 0 unless λ ≤ 1
16
. Hence, λn ≤
1
16
for all n = 1, 2, . . . . Each eigenvalue λn is simple whence N = 0.
Theorem 3.1. Suppose p = 2. Then an arbitrary formal series expansions for the solution of
homogeneous problem (2.1) has the form U(ω, r) =
∑∞
n=1
rλn Vn,0(ω), where λn is the eigenvalues
of the problem (3.2).
Proof. The theorem follows immediately from the above discussion.
In the original coordinates (x, t) near the point P = (0, 0) in G the formal series expansions for
the solution proves to be
u(x, t) =
∞∑
n=1
cn t
λn Vn,0
( x
t1/2
)
.
Of course, Theorem 3.1 can be proved immediately, for the homogeneous problem (2.1) admits a
separation of variables. Namely, set U(ω, r) = R(r)Ω(ω). Substituting this into equation (3.1) yields
rR′Ω− Ω′′ − ω
2
Ω′R = 0,
which is equivalent to
rR′ = λR, Ω′′ − ω
2
Ω′ = λΩ.
Then R(r) = rλn , where the parameter λn is determined from the boundary-value problem for Ω.
The function Ω can be described in terms of parabolic cylinder functions, see [1]. To transform the
equation for Ω to the equation of parabolic cylinder, set
Ω(ω) = exp
(
ω2
8
)
y(ω).
Then y satisfies
y′′ +
((ω
4
)2
+ λn −
1
4
)
y = 0.
Two linearly independent solutions of this equation are called functions of parabolic cylinder.
4. Resolution of singularities at infinity. Throughout this part we will assume that 0 < p < 2,
i.e., Q = 2/p is greater than 1. As mentioned in the Introduction, this case is not included in the
treatise [10] and it was first studied in [2]. For 1 < p < 2, the origin is a characteristic boundary
point of the domain G. For 0 < p < 1, the origin is a cuspidal point at the boundary.
We are actually interested in the local solvability of problem (1.4) near the edge r = 0 in the
rectangle (−1, 1) × (0, 1). Note that the equation degenerates at r = 0, since the coefficient rQ of
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ASYMPTOTIC SOLUTIONS OF THE DIRICHLET PROBLEM FOR THE HEAT EQUATION . . . 1309
the higher order derivative in r vanishes at r = 0. If Q = 1, the equation is of Fuchs type and is
studied within the framework of Mellin calculus. In order to handle this degeneration in an orderly
fashion for Q > 1, we find a change of coordinates s = δ(r) in the interval (0, 1), such that
rQ
d
dr
=
d
ds
.
Such a function δ is determined uniquely up to some constant from the equation δ′(r) = r−Q and is
given by
δ(r) =
r1−Q
1−Q
(4.1)
for r > 0. Note that δ(0+) = −∞. Problem (1.4) becomes
U ′s − U ′′ω,ω +
1
2− p
1
s
ωU ′ω =
(
δ(1)
s
) 2
2−p
F in (−1, 1)× (−∞, δ(1)),
U = U0 at {±1} × (−∞, δ(1)),
(4.2)
where we use the same letter to designate U and the push-forward of U under the transformation
s = δ(r), and similarly for F. Above δ(1) =
1
1−Q
< 0.
Thus we have transformed boundary-value problem (1.1) to the boundary-value problem (4.2)
with the operator
A(s)U = U ′s − U ′′ω,ω +
1
2− p
ω
s
U ′ω (4.3)
considered in the spaceH1
0(−∞, δ(1)) of functions U such that U = U0 for ω = ±1, s ∈ (−∞, δ(1))
and
‖U‖2H1
0(−∞,δ(1))
=
δ(1)∫
−∞
(
‖U ′(s)‖2L2(−1,1) + ‖U(s)‖2H2(−1,1)
)
|s|
3
p−2 dω ds <∞. (4.4)
Factor s3/(p−2) arise due to the chage of Lebesgue measure dx dt under the change of coordi-
nates (1.3) and (4.1).
We now rewrite formal series expansions for the solution to homogeneous problem (2.1) in the
new coordinates (ω, s). On substituting (4.1) into Theorem 2.1 we get immediately
U(ω, s) =
∞∑
n=1
cn ((1−Q)s)
1
4
Q
Q−1 exp(λns)
∞∑
m=0
Vn,m(ω)
((1−Q)s)m
(4.5)
for s in a neighbourhood of −∞, where λn = −
(π
2
n
)2
.
5. Fredholm property of the first boundary-value problem. In this section we state the solv-
ability of the transformated boundary-value problem (4.2). For this we need to introduce the scale of
spaces Hkγ,µ(−∞, T ) of functions with values in standard Sobolev spaces H2k(−1, 1). In particular
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1310 A. VICT. ANTONIOUK, O. M. KISELEV, N. N. TARKHANOV
case, when k = 1, γ = 0, µ = µ0 ≡
3
2(p− 2)
and T = δ(1) these spaces coincide with the spaces
H1
0(−∞, δ(1)) introduces in the previous part, so that H1
0(−∞, δ(1)) = H1
0,µ0
(−∞, δ(1)). We say
that function u with values in space H2k(−1, 1) belongs to the space Hkγ,µ(−∞, T ), T ≤ ∞ for
k ∈ N, γ ≤ 0, and µ > µ0 if the following norm is finite
‖U‖Hkγ,µ(−∞,T ) :=
T∫
−∞
e−2γs s2µ
k∑
j=0
‖U (j)(s)‖2
H2(k−j)(−1,1)ds
1/2 . (5.1)
When k = 0 and µ = 0 we denote the space H0
γ,0(−∞, T ) by L2γ(−∞, T ) with the corresponding
norm. Main statement of this section is given by the following theorem.
Theorem 5.1. Let γ < 0, γ 6= λn, n ≥ 0, where λn are the eigenvalues of operator ∆ in space
L2(−1, 1). Then for any µ > −1 there exists T0(µ) ∈ R such that for all T < T0 operator (4.3) of
problem (4.2) acting in the spaces
A(s) : H1
γ,µ(−∞, T ) 7→ L2γ(−∞, T ) (5.2)
is invertible and the following estimate holds:
‖U‖H1
γ,µ(−∞,T ) ≤ C ‖A(s)U‖L2γ(−∞,T ). (5.3)
The proof of this theorem breaks into a sequence of lemmas.
Lemma 5.1. Let γ < 0, γ 6= λn, n ≥ 0, µ ∈ R and T < 0. Then the operator
(∂s −∆)−1 : L2γ(−∞, T ) 7→ H1
γ,µ(−∞, T )
is bounded and the following estimate holds:
‖U‖H1
γ,µ(−∞,T ) ≤ C ‖F‖L2γ(−∞,T ), (5.4)
U(s) =
1
2π
∫
=mσ=γ
eiσs(−iσ −∆)−1F̂ (σ) dσ. (5.5)
The statement of this lemma is also true for γ = 0. In this case µ should be less than −1
2
.
Proof. Consider F ∈ L2γ(−∞, T ). For |s| > |T | let us continue function F by zero, then
F ∈ L2γ(R). Applying Fourier transform with respect to the variable s to the equation
∂sU(s)−∆U(s) = F (s) (5.6)
we have
−iσÛ −∆Û = F̂ . (5.7)
Since the line =mσ = γ for γ 6= λn consists of regular points of operator ∆ and F̂ ∈ L2(−1, 1) it
follows that operator
(−iσ −∆)−1 : L2(−1, 1) 7→ H2(−1, 1) is bounded (5.8)
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ASYMPTOTIC SOLUTIONS OF THE DIRICHLET PROBLEM FOR THE HEAT EQUATION . . . 1311
and for all v ∈ L2(−1, 1):
|σ| · ‖(iσ + ∆)−1v‖L2(−1,1) ≤ C ′‖v‖L2(−1,1), (5.9)
where C ′ does not depend on σ.
Therefore solution Û of equation (5.7) is given by
Û =
(
−iσ −∆
)−1
F̂ ∈ H2(−1, 1). (5.10)
Applying inverse Fourier transform, to the solution of equation (5.6) we get representation (5.5).
Denote H1 = H2(−1, 1) and H0 = L2(−1, 1). For γ < 0 we obtain
‖U‖2H1
γ,µ(−∞,T ) =
1
2π
T∫
−∞
e−2γs s2µ
∥∥∥∥∥∥∥
∫
=mσ=γ
(iσ) eiσs(−iσ −∆)−1F̂ (σ)dσ
∥∥∥∥∥∥∥
2
H0
ds+
+
1
2π
T∫
−∞
e−2γs s2µ
∥∥∥∥∥∥∥
∫
=mσ=γ
eiσs(−iσ −∆)−1F̂ (σ)dσ
∥∥∥∥∥∥∥
2
H1
ds ≤
≤ 1
2π
T∫
−∞
e−4γs s2µ
∫
=mσ=γ
|σ|2
∥∥(iσ + ∆)−1F̂ (σ)
∥∥2
H0
dσ ds+
+
1
2π
T∫
−∞
e−4γs s2µ
∫
=mσ=γ
∥∥∥(iσ + ∆)−1F̂ (σ)
∥∥∥2
H1
dσ ds. (5.11)
Due to (5.8), (5.9) and the Parseval theorem, for
‖F‖2L2γ(R) = 2π
∫
=mσ=γ
∥∥F̂ (σ)
∥∥2
L2(−1,1) dσ
we have
‖U‖2H1
γ,µ(−∞,T ) ≤ C
T∫
−∞
e−4γs s2µ
∫
=mσ=γ
∥∥F̂ (σ)
∥∥2
H0
dσ ds =
= C ′
T∫
−∞
e−4γs s2µ ‖F‖2L2γ(R)ds = C ′′ ‖F‖2L2γ(−∞,T ). (5.12)
To estimate expression in (5.12) in the case γ = 0 we only remark that operator ∆ with zero Dirichlet
boundary conditions has no eigenfunctions corresponding to zero eigenvalue, therefore it is invertible.
Thus estimate (5.12) also holds for γ = 0. In this case µ should be less than −1
2
to make the integral
above convergent.
Lemma 5.1 is proved.
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1312 A. VICT. ANTONIOUK, O. M. KISELEV, N. N. TARKHANOV
Lemma 5.2. Let T < 0 be fixed. Then for all γ ≤ 0, γ 6= λn, n ≥ 0 and µ > −1 operator
B(s) =
1
2− p
ω
s
∂
∂ω
: H1
γ,µ(−∞, T ) 7→ L2γ(−∞, T )
is bounded and for all ϕ ∈ H1
γ,µ(−∞, T )
‖B(·)ϕ‖L2γ(−∞,T ) ≤
C
|T |µ+1
‖ϕ‖H1
γ,µ(−∞,T ). (5.13)
Proof. As in Lemma 5.1 set H1 = H2(−1, 1) and H0 = L2(−1, 1). For ϕ ∈ H1
γ,µ(−∞, T ) we
have
‖B(·)ϕ‖2L2γ(−∞,T ) =
T∫
−∞
e−2γs ‖B(s)ϕ(s)‖2H0
ds =
= C
T∫
−∞
e−2γs s−2
∥∥∥∥ω∂ϕ(s)
∂ω
∥∥∥∥2
H0
ds ≤
≤ C
T∫
−∞
e−2γs s−2
(
‖ϕ′s‖2H0
+ ‖ϕ‖2H1
)
ds =
= C
T∫
−∞
e−2γs
s2µ
s2(µ+1)
(
‖ϕ′s‖2H0
+ ‖ϕ‖2H1
)
ds ≤
≤ C
|T |2(µ+1)
T∫
−∞
e−2γs s2µ
(
‖ϕ′s‖2H0
+ ‖ϕ‖2H1
)
ds =
=
C
|T |2(µ+1)
‖ϕ‖2H1
γ,µ
,
since |s| > |T |.
Lemma 5.2 is proved.
Proof of Theorem 5.1. For F ∈ L2γ(−∞, T ) let us represent the problem
A(s) ≡ ∂sU −∆U +B(s)U = F
in the form
U − (∂s −∆)−1B(s)U = (∂s −∆)−1F. (5.14)
Due to Lemmas 5.1 and 5.2 we may guarantee that for µ > −1 there exists such T0 that for any
T > T0 the norm of operator
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ASYMPTOTIC SOLUTIONS OF THE DIRICHLET PROBLEM FOR THE HEAT EQUATION . . . 1313
(∂s −∆)−1B(s) : H1
γ,µ(−∞, T ) 7→ H1
γ,µ(−∞, T )
is less than some δ < 1:
‖(∂s −∆)−1B(s)‖H1
γ,µ(−∞,T )→H1
γ,µ(−∞,T ) < δ. (5.15)
This imply that equation (5.14) has unique solution for F ∈ L2γ(−∞, T ).
Theorem 5.1 is proved.
On changing the coordinates by
ω =
x
t1/p
,
s =
t1−Q
1−Q
,
we pull back the function spaces H1
γ,µ(−∞, δ(1)) to the original domain G. Theorem 5.1 then yields
a condition of local solvability of problem (1.1) at the characteristic point, see [2].
6. Asymptotic property of formal solution. We now turn to the proof of asymptotic property
of formal series expansions of the solution of the first boundary-value problem for the heat equation
at a characteristic point. To do this we denote by Hkγ,m(−∞, T ) the spaces Hkγ,µ for µ = µ0 + m,
m ∈ N with µ0 =
3
2(p− 2)
and Hkγ,µ0 for m = 0. Let us remark that for any k,m ≥ 1, γ ∈ R,
T < δ(1)
Hkγ,m+1(−∞, T ) ⊂ Hkγ,m(−∞, T ) ⊂ H1
0(−∞, δ(1)) ≡ H1
0,µ0(−∞, δ(1)).
The main result of this paper reads as follows.
Theorem 6.1. Suppose that λK+1 < γ < λK . Then the formal series expansion (4.5) of the
solution U ∈ H1
0(−∞, δ(1)) of problem (4.2) is actually asymptotic in the sense (0.1).
Proof. Due to (4.5) the solution to the homogeneous boundary-value problem (4.2) in space
H1
0(−∞, δ(1)) has the form
U(ω, s) =
∞∑
n=1
Un(ω, s), (6.1)
where
Un(ω, s) = cn,Q s
1
4
Q
Q−1 exp (λns)
∞∑
m=0
Vn,m(ω)
((1−Q)s)m
and λn = −
(
n
π
2
)
, cn,Q = cn(1−Q)
1
4
Q
Q−1 .
For each M ≥ 0 and K ≥ 0 we introduce the function
UK,M (ω, s) =
K∑
n=0
cn,Q s
1
4
Q
Q−1 exp (λns)
M∑
m=0
Vn,m(ω)
(1−Q)msm
on (−1, 1) × (−∞, S). Direct calculations show that for any finite K these functions belong to the
space H1
γ,M (−∞, S) for M > µ0.
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1314 A. VICT. ANTONIOUK, O. M. KISELEV, N. N. TARKHANOV
Given any nonnegative integers M and K, set
RK+1,M+1(ω, s) = U(ω, s)−
K∑
n=0
cn,Q s
1
4
Q
Q−1 eλns
M∑
m=0
Vn,m(ω)
(1−Q)msm
for (ω, s) ∈ [−1, 1]× (−∞, S). Then we have
U(ω, s) = UK,M (ω, s) +RK+1,M+1(ω, s).
Due to the Theorem 5.1 solution U of the problem (4.2) belongs to the space H1
γ,M , therefore
RK+1,M+1 ∈ H1
γ,M , too. The theorem will be proved if we will establish that RK+1,M+1(ω, s) ∈
∈ H1
γ,M+1.
Let us first calculate operator A(s) on UK,M . We have
(UK,M (ω, s))′s =
K∑
n=0
cn,Q s
1
4
Q
Q−1 eλns
(
M∑
m=0
λnVn,m(ω)
(1−Q)msm
−
−
M+1∑
m=1
(
Q
4
+ (m− 1)(Q− 1)
)
Vm−1,n(ω)
(1−Q)msm
)
,
(UK,M (ω, s))′′ωω =
K∑
n=0
cn,Q s
1
4
Q
Q−1 eλns
M∑
m=0
V ′′n,m(ω)
(1−Q)msm
,
1
2− p
ω
s
(UK,M (ω, s))′ω =
K∑
n=0
cn,Q
2− p
s
1
4
Q
Q−1 eλns
M∑
m=0
ωV ′n,m(ω)
(1−Q)msm+1
=
= −
K∑
n=0
cn,Qs
1
4
Q
Q−1 eλns
M+1∑
m=1
ω
p
V ′m−1,n(ω)
(1−Q)msm
,
where we used that
1−Q
2− p
= −1
p
.
Therefore for the operator A(s) (4.3) of boundary-value problem (4.2) we get
A(s)UK,M =
K∑
n=0
cn,Q s
1
4
Q
Q−1 eλns
(
M∑
m=0
−V ′′n,m + λnVn,m
(1−Q)msm
−
−
M+1∑
m=1
ωV ′n,m−1
p
+
(
Q
4
+ (m− 1)(1−Q)
)
Vn,m−1
(1−Q)msm
)
=
= −
K∑
n=0
cn,Q s
1
4
Q
Q−1 eλns
ω
p
V ′n,M +
(
Q
4
+M(1−Q)
)
Vn,M
(1−Q)M+1sM+1
. (6.2)
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 10
ASYMPTOTIC SOLUTIONS OF THE DIRICHLET PROBLEM FOR THE HEAT EQUATION . . . 1315
Now we define a function XK+1,M+1(ω, s) from the equality
RK+1,M+1(ω, s) = cK,Q s
1
4
Q
Q−1 eλKs
XK+1,M+1(ω, s)
(1−Q)M+1sM+1
. (6.3)
Then we obtain
A(s)RK+1,M+1 =
cK,Q s
1
4
Q
Q−1 eλKs
(1−Q)M+1sM+1
YK+1,M+1(ω, s), (6.4)
where
YK+1,M+1(ω, s) =
=
(
XK+1,M+1
)′′
ωω
− 1
2− p
ω
s
(
XK+1,M+1
)′
ω
+
+
(
λK −
(
M + 1− 1
4
Q
Q− 1
)1
s
)
XK+1,M+1.
Thus, for homogeneous boundary-value problem (4.2), due to (6.2), we have
A(s)U = A(s)UK,M +A(s)RK+1,M+1 =
= −
K∑
n=0
cn,Q s
1
4
Q
Q−1 eλns
(1−Q)M+1sM+1
(
ω
p
V ′n,M +
(
Q
4
+M(1−Q)
)
Vn,M
)
+
+
cK,Q s
1
4
Q
Q−1 eλKs
(1−Q)M+1sM+1
YK+1,M+1(ω, s) = 0.
Therefore
YK+1,M+1(ω, s) =
=
K∑
n=0
c′n,Qe
(λn−λK)s
(
ω
p
V ′n,M +
(
Q
4
+M(1−Q)
)
Vn,M
)
(2.11)
=
(2.11)
=
K∑
n=0
c′n,Qe
(λn−λK)s
(
−V ′′n,M+1 + λnVn,M+1
)
.
Since Vn,M+1 ∈ H2(−1, 1) we get that YK+1,M+1 ∈ H0
0,µ0
(−∞, T ). Thus, using representa-
tion (6.4), it is easy to see that
A(s)RK+1,M+1 ∈ H0
γ,M+1(−∞, T ) ⊂ L2γ(−∞, T ). (6.5)
Indeed, due to the representation (6.4)
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 10
1316 A. VICT. ANTONIOUK, O. M. KISELEV, N. N. TARKHANOV
A(s)RK+1,M+1 = C sαeλKsYK+1,M+1,
where α =
1
4
Q
Q− 1
− (M + 1). By the definition of space H0
γ,M+1(−∞, T ) we obtain
‖A(s)RK+1,M+1‖H0
γ,M+1(−∞,T )
=
=
T∫
−∞
e−2γss2(µ0+M+1)‖A(s)RK+1,M+1‖2L2(−1,1)ds
1/2 =
= C
T∫
−∞
e−2s(γ−λK)s2(µ0+M+1)s2α‖YK+1,M+1‖2L2(−1,1)ds
1/2
which is finite, for γ < λK and integration runs in the negative half-axis. This implies (6.5).
Therefore, by Theorem 5.1, there exists T0 = T0(M + 1) such that RK+1,M+1 ∈ H1
γ,M+1, as
desired.
Theorem 6.1 is proved.
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ASYMPTOTIC SOLUTIONS OF THE DIRICHLET PROBLEM FOR THE HEAT EQUATION . . . 1317
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ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 10
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| id | umjimathkievua-article-2223 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:21:03Z |
| publishDate | 2014 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/d2/672e9a9be51f6e9c12e2bd770c79c4d2.pdf |
| spelling | umjimathkievua-article-22232019-12-05T10:26:46Z Asymptotic Solutions of the Dirichlet Problem for the Heat Equation at a Characteristic Point Асимптотичні розв'язки задачі Діріхле для рівняння теплопроводності в характеристичній точці Antoniouk, A. Vict. Kiselev, O. M. Tarkhanov, N. N. Антонюк, О. Вік. Кисельов, О. М. Тарханов, Н. Н. The Dirichlet problem for the heat equation in a bounded domain $G ⊂ ℝ^{n+1}$ is characteristic because there are boundary points at which the boundary touches a characteristic hyperplane $t = c$, where c is a constant. For the first time, necessary and sufficient conditions on the boundary guaranteeing that the solution is continuous up to the characteristic point were established by Petrovskii (1934) under the assumption that the Dirichlet data are continuous. The appearance of Petrovskii’s paper was stimulated by the existing interest to the investigation of general boundary-value problems for parabolic equations in bounded domains. We contribute to the study of this problem by finding a formal solution of the Dirichlet problem for the heat equation in a neighborhood of a cuspidal characteristic boundary point and analyzing its asymptotic behavior. Задача Діріхлє для рівняння тєплопровідності в обмеженій області $G ⊂ ℝ^{n+1}$ є характеристичною, оскільки існують граничні точки, в яких границя є дотичною до характеристичної гіперплощини $t = c$, де c є сталою. I. Г. Петров-ський (1934) уперше встановив необхідні та достатні умови на границю, що гарантують неперервність розв'язку аж до характеристичної точки за умови, що дані Діріхле є неперервними. Поява даної роботи була викликана постійним інтересом до вивчення загальних граничних задач для рівнянь параболічного типу в обмежених областях. Наш внесок у вивчення цієї проблеми полягає в побудові формального розв'язку задачі Діріхле для рівняння теплопровідності в околі гострокінцевої характеристичної граничної точки та дослідженні його асимптотичного характеру. Institute of Mathematics, NAS of Ukraine 2014-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2223 Ukrains’kyi Matematychnyi Zhurnal; Vol. 66 No. 10 (2014); 1299–1317 Український математичний журнал; Том 66 № 10 (2014); 1299–1317 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2223/1447 https://umj.imath.kiev.ua/index.php/umj/article/view/2223/1448 Copyright (c) 2014 Antoniouk A. Vict.; Kiselev O. M.; Tarkhanov N. N. |
| spellingShingle | Antoniouk, A. Vict. Kiselev, O. M. Tarkhanov, N. N. Антонюк, О. Вік. Кисельов, О. М. Тарханов, Н. Н. Asymptotic Solutions of the Dirichlet Problem for the Heat Equation at a Characteristic Point |
| title | Asymptotic Solutions of the Dirichlet Problem for the Heat Equation at a Characteristic Point |
| title_alt | Асимптотичні розв'язки задачі Діріхле для рівняння теплопроводності в характеристичній точці |
| title_full | Asymptotic Solutions of the Dirichlet Problem for the Heat Equation at a Characteristic Point |
| title_fullStr | Asymptotic Solutions of the Dirichlet Problem for the Heat Equation at a Characteristic Point |
| title_full_unstemmed | Asymptotic Solutions of the Dirichlet Problem for the Heat Equation at a Characteristic Point |
| title_short | Asymptotic Solutions of the Dirichlet Problem for the Heat Equation at a Characteristic Point |
| title_sort | asymptotic solutions of the dirichlet problem for the heat equation at a characteristic point |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2223 |
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