On the Solvability of a Problem Nonlocal in Time for a Semilinear Multidimensional Wave Equation
We study a nonlocal (in time) problem for semilinear multidimensional wave equations. The theorems on existence and uniqueness of solutions of this problem are proved. Вивчається нелокальна за часом задача для напівлінійних багатовимiрних хвильових рівнянь. Доведено теореми про існування та єдиність...
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Kharibegashvili, S. Midodashvili, B. 2020-02-13T12:58:21Z 2020-02-13T12:58:21Z 2015 On the Solvability of a Problem Nonlocal in Time for a Semilinear Multidimensional Wave Equation / S. Kharibegashvili, B. Midodashvili // Український математичний журнал. — 2015. — Т. 67, № 1. — С. 88–105. — Бібліогр.: 23 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/165435 517.9 We study a nonlocal (in time) problem for semilinear multidimensional wave equations. The theorems on existence and uniqueness of solutions of this problem are proved. Вивчається нелокальна за часом задача для напівлінійних багатовимiрних хвильових рівнянь. Доведено теореми про існування та єдиність розв'язків цієї задачі. en Інститут математики НАН України Український математичний журнал Статті On the Solvability of a Problem Nonlocal in Time for a Semilinear Multidimensional Wave Equation Про розв'язність нелокальної за часом задачі для напівлінійного багатовимірного хвильового рівняння Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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On the Solvability of a Problem Nonlocal in Time for a Semilinear Multidimensional Wave Equation |
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On the Solvability of a Problem Nonlocal in Time for a Semilinear Multidimensional Wave Equation Kharibegashvili, S. Midodashvili, B. Статті |
| title_short |
On the Solvability of a Problem Nonlocal in Time for a Semilinear Multidimensional Wave Equation |
| title_full |
On the Solvability of a Problem Nonlocal in Time for a Semilinear Multidimensional Wave Equation |
| title_fullStr |
On the Solvability of a Problem Nonlocal in Time for a Semilinear Multidimensional Wave Equation |
| title_full_unstemmed |
On the Solvability of a Problem Nonlocal in Time for a Semilinear Multidimensional Wave Equation |
| title_sort |
on the solvability of a problem nonlocal in time for a semilinear multidimensional wave equation |
| author |
Kharibegashvili, S. Midodashvili, B. |
| author_facet |
Kharibegashvili, S. Midodashvili, B. |
| topic |
Статті |
| topic_facet |
Статті |
| publishDate |
2015 |
| language |
English |
| container_title |
Український математичний журнал |
| publisher |
Інститут математики НАН України |
| format |
Article |
| title_alt |
Про розв'язність нелокальної за часом задачі для напівлінійного багатовимірного хвильового рівняння |
| description |
We study a nonlocal (in time) problem for semilinear multidimensional wave equations. The theorems on existence and uniqueness of solutions of this problem are proved.
Вивчається нелокальна за часом задача для напівлінійних багатовимiрних хвильових рівнянь. Доведено теореми про існування та єдиність розв'язків цієї задачі.
|
| issn |
1027-3190 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/165435 |
| citation_txt |
On the Solvability of a Problem Nonlocal in Time for a Semilinear Multidimensional Wave Equation / S. Kharibegashvili, B. Midodashvili // Український математичний журнал. — 2015. — Т. 67, № 1. — С. 88–105. — Бібліогр.: 23 назв. — англ. |
| work_keys_str_mv |
AT kharibegashvilis onthesolvabilityofaproblemnonlocalintimeforasemilinearmultidimensionalwaveequation AT midodashvilib onthesolvabilityofaproblemnonlocalintimeforasemilinearmultidimensionalwaveequation AT kharibegashvilis prorozvâznístʹnelokalʹnoízačasomzadačídlânapívlíníinogobagatovimírnogohvilʹovogorívnânnâ AT midodashvilib prorozvâznístʹnelokalʹnoízačasomzadačídlânapívlíníinogobagatovimírnogohvilʹovogorívnânnâ |
| first_indexed |
2025-11-25T22:34:34Z |
| last_indexed |
2025-11-25T22:34:34Z |
| _version_ |
1850567818007805952 |
| fulltext |
UDC 517.9
S. Kharibegashvili (I. Javakhishvili Tbilisi State Univ. and Georg. Techn. Univ., Georgia),
B. Midodashvili (I. Javakhishvili Tbilisi State Univ. and Gori Teaching Univ., Georgia)
ON THE SOLVABILITY OF A PROBLEM NONLOCAL IN TIME
FOR A SEMILINEAR MULTIDIMENSIONAL WAVE EQUATION
ПРО РОЗВ’ЯЗНIСТЬ НЕЛОКАЛЬНОЇ ЗА ЧАСОМ ЗАДАЧI
ДЛЯ НАПIВЛIНIЙНОГО БАГАТОВИМIРНОГО ХВИЛЬОВОГО РIВНЯННЯ
We study a nonlocal (in time) problem for semilinear multidimensional wave equations. The theorems on existence and
uniqueness of solutions of this problem are proved.
Вивчається нелокальна за часом задача для напiвлiнiйних багатовимiрних хвильових рiвнянь. Доведено теореми
про iснування та єдинiсть розв’язкiв цiєї задачi.
1. Introduction. In the space Rn+1 of variables x = (x1, . . . , xn) and t, in the cylindrical domain
DT = Ω × (0, T ), where Ω is a Lipschitz domain in Rn, consider a nonlocal problem of finding a
solution u(x, t) of the following equation:
Lλu :=
∂2u
∂t2
−
n∑
i=1
∂2u
∂x2
i
+ λf(x, t, u) = F (x, t), (x, t) ∈ DT , (1.1)
satisfying the homogeneous boundary condition on the part of the boundary Γ := ∂Ω× (0, T ) of the
cylinder DT
u
∣∣
Γ
= 0, (1.2)
the initial condition
u(x, 0) = ϕ(x), x ∈ Ω, (1.3)
and the nonlocal condition
Kµut := ut(x, 0)− µut(x, T ) = ψ(x), x ∈ Ω, (1.4)
where f, F, ϕ and ψ are given functions; λ and µ are given nonzero constants and n ≥ 2.
To the study of nonlocal problems for partial differential equations there are devoted many papers.
When a nonlocal problem is posed for abstract evolution equations and hyperbolic partial differential
equations we would suggest the reader refer to works [1 – 15] and the references therein.
Note that the problem (1.1) – (1.4) in the work [15] is studied in the class of continuous functions
for the case of one spatial variable, i.e., for n = 1. The method of investigation given in the work [15],
based on the integral representation of the solution of corresponding linear problem, is useless for
multidimensional case, i.e., for n > 1. In this work the problem (1.1) – (1.4) in the multidimensional
case is studied in the Sobolev space W 1
2 (DT ), basing on expansions of the functions from the space
0
W 1
2(Ω) in the basis, consisting of eigenfunctions of spectral problem ∆w = λ̃w, w |∂Ω = 0 and
using embedding theorems in the Sobolev spaces. It must be noted also that if for n = 1 there is no
need of any restriction on the behavior of function f(x, t, u) with respect to variable u when u→∞,
c© S. KHARIBEGASHVILI, B. MIDODASHVILI, 2015
88 ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 1
ON THE SOLVABILITY OF A PROBLEM NONLOCAL IN TIME . . . 89
while in the case n > 1, we require of function f(x, t, u) that for u → ∞ it must have a growth
not exceeding polynomial. Moreover, for using the embedding theorems in the Sobolev spaces we
additionally require that the order of polynomial growth must be less than a certain value, which
depends of the dimension of the space.
Below, on the function f = f(x, t, u) we impose the following requirements:
f ∈ C(DT × R), |f(x, t, u)| ≤M1 +M2|u|α, (x, t, u) ∈ DT × R, (1.5)
where
0 ≤ α = const <
n+ 1
n− 1
. (1.6)
Remark 1.1. The embedding operator I : W 1
2 (DT ) → Lq(DT ) represents a linear continuous
compact operator for 1 < q <
2(n+ 1)
n− 1
,when n > 1 [16]. At the same time the Nemitski operatorN :
Lq(DT )→ L2(DT ), acting by the formula Nu = f(x, t, u) due to (1.5) is continuous and bounded
if q ≥ 2α [17]. Thus, since due to (1.6) we have 2α <
2(n+ 1)
n− 1
, then there exists the number q
such that 1 < q <
2(n+ 1)
n− 1
and q ≥ 2α. Therefore, in this case the operator
N0 = NI :
0
W 1
2(DT ,Γ)→ L2(DT ), (1.7)
where
0
W 1
2(DT ,Γ) := {w ∈ W 1
2 (DT ) : w |Γ = 0}, will be continuous and compact. Besides, from
u ∈
0
W 1
2(DT ,Γ) it follows that f(x, t, u) ∈ L2(DT ) and, if um → u in the space
0
W 1
2(DT ,Γ), then
f(x, t, um)→ f(x, t, u) in the space L2(DT ).
Definition 1.1. Let function f satisfy conditions (1.5) and (1.6), F ∈ L2(DT ), ϕ ∈
0
W 1
2(Ω) :=
:= {v ∈ W 1
2 (Ω) : v |∂Ω = 0}, ψ ∈ L2(Ω). We call a function u a generalized solution of prob-
lem (1.1) – (1.4), if u ∈
0
W 1
2(DT ,Γ) and there exists a sequence of functions um ∈
0
C 2(DT ,Γ) :=
:=
{
w ∈ C2(DT ) : w |Γ = 0
}
such that
lim
m→∞
‖um − u‖ 0
W 1
2(DT ,Γ)
= 0, lim
m→∞
‖Lλum − F‖L2(DT ) = 0, (1.8)
lim
m→∞
‖um |t=0 − ϕ‖ 0
W 1
2(Ω)
= 0, lim
m→∞
‖Kµumt − ψ‖L2(Ω) = 0. (1.9)
It is obvious that a classical solution u ∈ C2(DT ) of problem (1.1) – (1.4) represents a generalized
solution of this problem. It is easy to verify that a generalized solution of problem (1.1) – (1.4) is
a solution of problem (1.1) in the sense of the theory of distributions. Indeed, let Fm := Lλum,
ϕm := um |t=0, ψm := Kµumt. Multiplying the both sides of the equality Lλum = Fm by test
function w ∈ V :=
{
v ∈
0
W 1
2(DT ,Γ) : v(x, T )−µv(x, 0) = 0, x ∈ Ω
}
and integrating in the domain
DT , after simple transformations, connected with integration by parts and the equality w |Γ = 0, we
get ∫
Ω
[umt(x, T )w(x, T )− umt(x, 0)w(x, 0)]dx+
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 1
90 S. KHARIBEGASHVILI, B. MIDODASHVILI
+
∫
DT
[
−umtwt +
n∑
i=1
umxiwxi + λf(x, t, um)w
]
dx dt =
∫
DT
Fmw dxdt ∀w ∈ V. (1.10)
Due toKµumt = ψm(x) andw(x, T )−µw(x, 0) = 0, x ∈ Ω, it is easy to see that umt(x, T )w(x, T )−
− umt(x, 0)w(x, 0) = umt(x, T )(w(x, T ) − µw(x, 0)) − ψm(x)w(x, 0) = −ψm(x)w(x, 0), x ∈ Ω.
Therefore, equality (1.10) takes the form
−
∫
Ω
ψm(x)w(x, 0)dx+
+
∫
DT
[
−umtwt +
n∑
i=1
umxiwxi + λf(x, t, um)w
]
dx dt =
∫
DT
Fmw dxdt ∀w ∈ V. (1.11)
In view of (1.5), (1.6) according to Remark 1.1 we have f(x, t, um) → f(x, t, u) in the space
L2(DT ), when um → u in the space
0
W 1
2(DT ,Γ). Therefore, due to (1.8) and (1.9), passing to the
limit in equality (1.11) for m→∞, we get
−
∫
Ω
ψw(x, 0)dx+
∫
DT
[
−utwt +
n∑
i=1
uxiwxi + λf(x, t, u)w
]
dx dt =
∫
DT
Fwdx dt ∀w ∈ V.
(1.12)
Since C∞0 (DT ) ⊂ V, then from (1.12), integrating by parts, we have∫
DT
[
u2w + λf(x, t, u)w
]
dx dt =
∫
DT
Fw dx dt ∀w ∈ C∞0 (DT ), (1.13)
where 2 := ∂2/∂t2 −
∑n
i=1
∂2/∂x2
i , and C∞0 (DT ) is a space of finite infinitely differentiable
functions in DT . Equality (1.13), which is valid for any w ∈ C∞0 (DT ), means that a generalized
solution u of problem (1.1) – (1.4) is a solution of equation (1.1) in the sense of the theory of
distributions, besides, since the trace operator u → u |t=0 is well defined in the space
0
W 1
2(DT ,Γ),
and, particularly, is a continuous operator from the space
0
W 1
2(DT ,Γ) into the space L2(Ω×{t = 0}),
then due to (1.8) and (1.9) we receive that the initial condition (1.3) is fulfilled in the sense of the
trace theory, while the nonlocal condition (1.4) in the integral sense is taken into account in equality
(1.12), which is valid for all w ∈ V. Note also that if a generalized solution u belongs to the class
C2(DT ), then due to the standard reasoning, connected with the integral equality (1.12), which is
valid for any w ∈ V [16], we have that u is a classical solution of problem (1.1) – (1.4), satisfying
the equation (1.1), the boundary condition (1.2), the initial condition (1.3) and the nonlocal condition
(1.4) pointwisely.
Note that even in the linear case, i.e., for λ = 0, problem (1.1) – (1.4) is not always well-posed.
For example, when λ = 0 and |µ| = 1, the corresponding to (1.1) – (1.4) homogeneous problem may
have infinite number of linearly independent solutions (see Remark 3.2).
The work is organized in the following way. In Section 2 we single out the class of semilinear
equations (1.1), when for |µ| < 1 a priori estimate is valid for the generalized solution of prob-
lem (1.1) – (1.4). In Section 3 on the basis of a priori estimate, received in the previous section, the
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 1
ON THE SOLVABILITY OF A PROBLEM NONLOCAL IN TIME . . . 91
solvability of problem (1.1) – (1.4) is proved. Finally, in Section 4 we give the conditions imposed
on the data of the problem, which provide the uniqueness of the solution of this problem.
2. A priori estimate of the solution of problem (1.1) – (1.4). Let
g(x, t, u) =
u∫
0
f(x, t, s)ds, (x, t, u) ∈ DT × R. (2.1)
Consider the following conditions imposed on function g = g(x, t, u):
g(x, t, u) ≥ −M3, (x, t, u) ∈ DT × R, (2.2)
gt ∈ C(DT × R), gt(x, t, u) ≤M4, (x, t, u) ∈ DT × R, (2.3)
where Mi = const ≥ 0, i = 3, 4.
Let us consider some classes of functions f = f(x, t, u) frequently encountered in applications
and which satisfy conditions (1.5), (2.2) and (2.3) :
1. f(x, t, u) = f0(x, t)β(u), where f0,
∂
∂t
f0 ∈ C(DT ) and β ∈ C(R), |β(u)| ≤ M̃1 + M̃2|u|α,
M̃i = const ≥ 0, α = const ≥ 0. In this case g(x, t, u) = f0(x, t)
∫ u
0
β(s)ds and when f0 ≥ 0,
∂
∂t
f0 ≤ 0,
∫ u
0
β(s)ds ≥ −M, M = const ≥ 0, conditions (1.5), (2.2) and (2.3) will be fulfilled.
2. f(x, t, u) = f0(x, t)|u|α signu, where f0,
∂
∂t
f0 ∈ C(DT ) and α > 1. In this case g(x, t, u) =
= f0(x, t)
|u|α+1
α+ 1
and when f0 ≥ 0,
∂
∂t
f0 ≤ 0, conditions (1.5), (2.2) and (2.3) will be also fulfilled.
Lemma 2.1. Let λ > 0, |µ| < 1, F ∈ L2(DT ), ϕ ∈
0
W 1
2(Ω), ψ ∈ L2(Ω) and conditions (1.5),
(1.6), (2.2), (2.3) be fulfilled. Then for a generalized solution u of problem (1.1) – (1.4) the following
a priori estimate:
‖u‖ 0
W 1
2(DT ,Γ)
≤ c1‖F‖L2(DT ) + c2‖ϕ‖ 0
W 1
2(Ω)
+ c3‖ψ‖L2(Ω) + c4‖ϕ‖
α+1
2
0
W 1
2(Ω)
+ c5 (2.4)
is valid with nonnegative constants ci = ci(λ, µ,Ω, T,M1,M2,M3,M4) not depending on u, F, ϕ,
ψ, and ci > 0 for i < 4, whereas in the linear case, i.e., when λ = 0 the constants c4 = c5 = 0 and
due to (2.4) in this case we have the uniqueness of the solution of problem (1.1) – (1.4).
Proof. Let u be a generalized solution of problem (1.1) – (1.4). In view of Definition 1.1 there
exists a sequence of the functions um ∈
0
C 2(DT ,Γ) such that limit equalities (1.8), (1.9) are fulfilled.
Set
Lλum = Fm, (x, t) ∈ DT , (2.5)
um |Γ = 0, (2.6)
um(x, 0) = ϕm(x), x ∈ Ω, (2.7)
Kµumt = ψm(x), x ∈ Ω. (2.8)
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 1
92 S. KHARIBEGASHVILI, B. MIDODASHVILI
Multiplying both sides of equation (2.5) by 2umt and integrating in the domain Dτ := DT ∩{t <
< τ}, 0 < τ ≤ T, due to (2.1), we obtain∫
Dτ
∂
∂t
(∂um
∂t
)2
dx dt− 2
∫
Dτ
n∑
i=1
∂2um
∂x2
i
∂um
∂t
dx dt+ 2λ
∫
Dτ
d
dt
(
g(x, t, um(x, t)
)
dx dt−
−2λ
∫
Dτ
gt(x, t, um(x, t))dx dt = 2
∫
Dτ
Fm
∂um
∂t
dx dt. (2.9)
Let ωτ := {(x, t) ∈ DT : x ∈ Ω, t = τ}, 0 ≤ τ ≤ T. Denote by ν := (νx1 , νx2 , . . . , νxn , νt)
the unit vector of the outer normal to ∂Dτ . Since νxi
∣∣
ωτ∪ω0
= 0, i = 1, . . . , n, νt
∣∣
Γτ=Γ∩{t≤τ} = 0,
νt
∣∣
ωτ
= 1, νt
∣∣
ω0
= −1, then, taking into account equalities (2.6) and integrating by parts, we have∫
Dτ
∂
∂t
(∂um
∂t
)2
dx dt =
∫
∂Dτ
(∂um
∂t
)2
νtds =
∫
ωτ
u2
mtdx−
∫
ω0
u2
mtdx, (2.10)
−2
∫
Dτ
∂2um
∂x2
i
∂um
∂t
dx dt =
∫
Dτ
[(u2
mxi)t − 2(umxiumt)xi ]dx dt =
=
∫
ωτ
u2
mxidx−
∫
ω0
u2
mxidx, i = 1, . . . , n, (2.11)
2λ
∫
Dτ
d
dt
(
g(x, t, um(x, t)
)
dxdt = 2λ
∫
∂Dτ
g(x, t, um(x, t))νtds =
= 2λ
∫
ωτ
g(x, t, um(x, t))dx− 2λ
∫
ω0
g(x, t, um(x, t))dx. (2.12)
In view of (2.10), (2.11), (2.12) from (2.9) we get∫
ωτ
[
u2
mt +
n∑
i=1
u2
mxi
]
dx =
∫
ω0
[
u2
mt +
n∑
i=1
u2
mxi
]
dx− 2λ
∫
ωτ
g(x, t, um(x, t))dx+
+2λ
∫
ω0
g(x, t, um(x, t))dx+ 2λ
∫
Dτ
gt(x, t, um(x, t))dx dt+ 2
∫
Dτ
Fmumtdx dt. (2.13)
Let
wm(τ) :=
∫
ωτ
[
u2
mt +
n∑
i=1
u2
mxi
]
dx. (2.14)
Since 2Fmumt ≤ ε−1F 2
m + εu2
mt for any ε = const > 0, then due to (2.2), (2.3) and (2.14) from
(2.13) it follows that
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 1
ON THE SOLVABILITY OF A PROBLEM NONLOCAL IN TIME . . . 93
wm(τ) ≤ wm(0) + 2λM3 mes Ω + 2λ
∫
ω0
|g(x, t, um(x, t))|dx+
+2λM4τ mes Ω + ε
∫
DT
u2
mtdx dt+ ε−1
∫
DT
F 2
mdx dt. (2.15)
Taking into account that
∫
Dτ
u2
mtdx dt =
τ∫
0
∫
ωs
u2
mtdx
ds ≤ τ∫
0
∫
ωs
[
u2
mt +
n∑
i=1
u2
mxi
]
dx
ds =
τ∫
0
wm(s)ds,
from (2.15) we obtain
wm(τ) ≤ ε
τ∫
0
wm(s)ds+ wm(0) + 2λ(M3 +M4τ) mes Ω+
+2λ
∫
ω0
|g(x, t, um(x, t))|dx+ ε−1
∫
Dτ
F 2
mdx dt, 0 < τ ≤ T. (2.16)
Because of Dτ ⊂ DT , 0 < τ ≤ T, then according to the Gronwall’s lemma [18] from (2.16) it
follows that
wm(τ) ≤
[
wm(0) + 2λ(M3 +M4T ) mes Ω+
+2λ
∫
ω0
|g(x, t, um(x, t))|dx+ ε−1
∫
DT
F 2
mdx dt
]
eετ , 0 < τ ≤ T. (2.17)
Using obvious inequality
|a+ b|2 = a2 + b2 + 2ab ≤ a2 + b2 + ε1a
2 + ε−1
1 b2 = (1 + ε1)a2 + (1 + ε−1
1 )b2,
which is valid for any ε1 > 0, from (2.8) we have
|umt(x, 0)|2 = |µumt(x, T ) + ψm(x)|2 ≤ |µ|2(1 + ε1)u2
mt(x, T ) + (1 + ε−1
1 )ψ2
m(x). (2.18)
From (2.18) we obtain ∫
ω0
u2
mtdx =
∫
Ω
|umt(x, 0)|2 dx ≤
≤ |µ|2(1 + ε1)
∫
Ω
u2
mt(x, T )dx+ (1 + ε−1
1 )
∫
Ω
ψ2
m(x) dx =
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94 S. KHARIBEGASHVILI, B. MIDODASHVILI
= |µ|2(1 + ε1)
∫
ωT
u2
mtdx+ (1 + ε−1
1 )‖ψm‖2L2(Ω). (2.19)
In view of (2.7), (2.14) from (2.17) we get∫
ωT
u2
mtdx ≤ wm(T ) ≤
∫
ω0
n∑
i=1
ϕ2
mxidx+
∫
ω0
u2
mtdx+M5
eεT , (2.20)
where
M5 = 2λ(M3 +M4T ) mes Ω + 2λ
∫
ω0
|g(x, t, um(x, t))|dx+ ε−1
∫
DT
F 2
m dx dt. (2.21)
From (2.19) and (2.20) it follows that∫
ω0
u2
mtdx ≤ |µ|2(1 + ε1)
∫
ω0
n∑
i=1
ϕ2
mxidx+
∫
ω0
u2
mtdx+M5
eεT + (1 + ε−1
1 )‖ψm‖2L2(Ω). (2.22)
Because |µ| < 1, then positive constants ε and ε1 can be chosen so small that
µ1 = |µ|2(1 + ε1)eεT < 1. (2.23)
Due to (2.23) from (2.22) we obtain∫
ω0
u2
mtdx ≤ (1− µ1)−1
|µ|2(1 + ε1)
∫
ω0
n∑
i=1
ϕ2
mxidx+M5
eεT + (1 + ε−1
1 )‖ψm‖2L2(Ω)
≤
≤ (1− µ1)−1
[
|µ|2(1 + ε1)
(
‖ϕm‖20
W 1
2(Ω)
+M5
)
eεT + (1 + ε−1
1 )‖ψm‖2L2(Ω)
]
. (2.24)
From (2.7), (2.14) and (2.24) it follows that
wm(0) =
∫
ω0
[
u2
mt +
n∑
i=1
ϕ2
mxi
]
dx ≤ ‖ϕm‖20
W 1
2(Ω)
+
+(1− µ1)−1
[
|µ|2(1 + ε1)
(
‖ϕm‖20
W 1
2(Ω)
+M5
)
eεT + (1 + ε−1
1 )‖ψm‖2L2(Ω)
]
. (2.25)
In view of (2.21), (2.25) from (2.17) we get
wm(τ) ≤
‖ϕm‖20W 1
2(Ω)
+ (1− µ1)−1 ×
×
|µ|2(1 + ε1)
‖ϕm‖20
W 1
2(Ω)
+ 2λ(M3 +M4T ) mes Ω +
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ON THE SOLVABILITY OF A PROBLEM NONLOCAL IN TIME . . . 95
+ 2λ
∫
ω0
|g(x, t, um(x, t))|dx+ ε−1
∫
DT
F 2
mdx dt
eεT + (1 + ε−1
1 )‖ψm‖2L2(Ω
+
+ 2λ(M3 +M4T ) mes Ω + 2λ
∫
ω0
|g(x, t, um(x, t))|dx+ ε−1
∫
DT
F 2
mdx dt
eεT =
= γ̃1‖Fm‖2L2(DT ) + γ̃2‖ϕm‖20
W 1
2(Ω)
+ γ̃3‖ψm‖2L2(Ω) + γ̃4
∫
ω0
|g(x, t, um(x, t))|dx+ γ̃5. (2.26)
Here
γ̃1 = ε−1eεT
[
(1− µ1)−1(1 + ε1)eεT + 1
]
,
γ̃2 = eεT
[
1 + (1− µ1)−1|µ|2(1 + ε1)
]
,
γ̃3 = (1− µ1)−1(1 + ε−1
1 )eεT ,
γ̃4 = 2λ[(1− µ1)−1|µ|2(1 + ε1) + 1]eεT ,
γ̃5 = 2λ(M3 +M4T ) mes Ω[(1− µ1)−1|µ|2(1 + ε1)eεT + 1]eεT .
(2.27)
Since for fixed τ the function um(x, τ) ∈
0
W 1
2(Ω), then due to the Friedrichs inequality [16] we
have ∫
ωτ
[
u2
m + u2
mt +
n∑
i=1
u2
mxi
]
dx ≤ c0wm(τ) = c0
∫
ωτ
[
u2
mt +
n∑
i=1
u2
mxi
]
dx, (2.28)
where positive constant c0 = c0(Ω) does not depend on um.
From (2.26) and (2.28) it follows
‖um‖20
W 1
2(DT ,Γ)
=
T∫
0
∫
ωτ
(
u2
m + u2
mt +
n∑
i=1
u2
mxi
)
dx
dτ ≤
≤
T∫
0
c0wm(τ)dτ ≤ c0T γ̃1‖Fm‖2L2(DT ) + c0T γ̃2‖ϕm‖20
W 1
2(Ω)
+ c0T γ̃3‖ψm‖2L2(Ω)+
+c0T γ̃4
∫
Ω
∣∣g(x, 0, um(x, 0))
∣∣dx+ c0T γ̃5. (2.29)
Due to (2.1), (1.5) we have
|g(x, 0, s)| ≤M6 +M7|s|α+1, (2.30)
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96 S. KHARIBEGASHVILI, B. MIDODASHVILI
where M6 and M7 are some nonnegative constants. Taking into account (2.30) from (2.29) we get
‖um‖20
W 1
2(DT ,Γ)
≤ c0T γ̃1‖Fm‖2L2(DT ) + c0T γ̃2‖ϕm‖20
W 1
2(Ω)
+ c0T γ̃3‖ψm‖2L2(Ω)+
+c0T γ̃4M6 mes Ω + c0T γ̃4M7
∫
Ω
|um(x, 0)|α+1dx+ c0T γ̃5. (2.31)
Reasoning from Remark 1.1, concerning the space W 1
2 (Ω), in view of the equality dim Ω =
= dimDT − 1 = n show that the embedding operator I : W 1
2 (Ω) → Lq(Ω) is a linear continuous
compact operator for 1 < q <
2n
n− 2
, when n > 2 and for any q > 1 when n = 2 [16]. At the same
time the Nemitski operator N1 : Lq(Ω)→ L2(Ω), acting by the formula N1u = |u|
α+1
2 is continuous
and bounded if q ≥ 2
α+ 1
2
= α + 1 [17]. Thus, if α + 1 <
2n
n− 2
, i.e., α <
n+ 2
n− 2
, which is
fulfilled due to (1.6) since
n+ 1
n− 1
<
n+ 2
n− 2
, then there exists number q such that 1 < q <
2n
n− 2
and
q ≥ α+ 1. Therefore, in this case the operator
N2 = N1I : W 1
2 (Ω)→ L2(Ω)
will be continuous and compact. Thus due to (1.9), (2.7) it follows that
lim
m→∞
∫
Ω
|um(x, 0)|α+1dx =
∫
Ω
|ϕ(x)|α+1dx, (2.32)
and also [16] ∫
Ω
|ϕ(x)|α+1dx ≤ C1‖ϕ‖α+1
0
W 1
2(Ω)
(2.33)
with positive constant C1, not dependent on ϕ ∈
0
W 1
2(Ω).
In view of (1.8), (1.9), (2.5) – (2.8), (2.32) and (2.33), passing in (2.31) to the limit for m → ∞
we obtain
‖u‖20
W 1
2(DT ,Γ)
≤ c0T γ̃1‖F‖2L2(DT ) + c0T γ̃2‖ϕ‖20
W 1
2(Ω)
+ c0T γ̃3‖ψ‖2L2(Ω)+
+c0T γ̃4M7C1‖ϕ‖α+1
0
W 1
2(Ω)
+ c0T (γ̃5 + γ̃4M6 mes Ω). (2.34)
Taking the square root from the both sides of inequality (2.34) and using the obvious inequality(∑k
i=1
a2
i
)1/2
≤
∑k
i=1
|ai| we finally get
‖u‖ 0
W 1
2(DT ,Γ)
≤ c1‖F‖L2(DT ) + c2‖ϕ‖ 0
W 1
2(Ω)
+ c3‖ψ‖L2(Ω) + c4‖ϕ‖
α+1
2
0
W 1
2(Ω)
+ c5. (2.35)
Here
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c1 = (c0T γ̃1)1/2, c2 = (c0T γ̃2)1/2, c3 = (c0T γ̃3)1/2,
c4 = (c0T γ̃4M7C1)1/2, c5 = [c0T (γ̃5 + γ̃4M6 mes Ω)]1/2,
(2.36)
where γ̃i, 1 ≤ i ≤ 5, are defined in (2.27). In the linear case, i.e., for λ = 0, due to (2.27) the
constants γ̃4 = γ̃5 = 0 and from (2.36) it follows that in estimate (2.4) the constants c4 = c5 = 0.
Whence it follows the uniqueness of the solution of problem (1.1) – (1.4) in the linear case.
Lemma 2.1 is proved.
3. The existence of the solution of problem (1.1) – (1.4). For the existence of the solution of
problem (1.1) – (1.4) in the case |µ| < 1 we will use well known facts about the solvability of the
following linear mixed problem [16]:
L0u :=
∂2u
∂t2
−
n∑
i=1
∂2u
∂x2
i
= F (x, t), (x, t) ∈ DT , (3.1)
u
∣∣
Γ
= 0, u(x, 0) = ϕ(x), ut(x, 0) = ψ̃(x), x ∈ Ω, (3.2)
where F, ϕ and ψ̃ are given functions.
For F ∈ L2(DT ), ϕ ∈
0
W 1
2(Ω), ψ̃ ∈ L2(Ω) the unique generalized solution u of problem (3.1),
(3.2) (in the sense of equality (1.12) where f = 0, and the number µ = 0 in the definition of the
space V and u |t=0 = ϕ) from the class E2,1(DT ) with the norm [16]
‖v‖2E2,1(DT ) = sup
0≤τ≤T
∫
ωτ
[
u2 + u2
t +
n∑
i=1
u2
xi
]
dx
is given by formula [16]
u =
∞∑
k=1
ak cosµkt+ bk sinµkt+
1
µk
t∫
0
Fk(τ) sinµk(t− τ)dτ
ϕk(x), (3.3)
where λ̃k = −µ2
k, 0 < µ1 ≤ µ2 ≤ . . . , limk→∞ µk = ∞ are the eigenvalues, while ϕk ∈
0
W 1
2(Ω)
are the corresponding eigenfunctions of the spectral problem ∆w = λ̃w, w |∂Ω = 0 in the domain
Ω
(
∆ :=
∑n
i=1
∂2
∂x2
i
)
, simultaneously forming orthonormal basis in L2(Ω) and orthogonal basis in
0
W 1
2(Ω) in the sense of scalar product (v, w) 0
W 1
2(Ω)
=
∫
Ω
∑n
i=1
vxiwxidx [16], i.e.,
(ϕk, ϕl)L2(Ω) = δlk, (ϕk, ϕl) 0
W 1
2(Ω)
= −λkδlk, δlk =
1, l = k,
0, l 6= k.
(3.4)
Here
ak = (ϕ,ϕk)L2(Ω), bk = µ−1
k (ψ̃, ϕk)L2(Ω), k = 1, 2, . . . , (3.5)
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98 S. KHARIBEGASHVILI, B. MIDODASHVILI
F (x, t) =
∞∑
k=1
Fk(t)ϕk(x), Fk(t) = (F,ϕk)L2(ωt), ωτ := DT ∩ {t = τ}, (3.6)
besides, for the solution u from (3.3) it is valid the following estimate [16, 19]:
‖u‖E2,1(DT ) ≤ γ(‖F‖L2(DT ) + ‖ϕ‖ 0
W 1
2(Ω)
+ ‖ψ̃‖L2(Ω)) (3.7)
with positive constant γ, not dependent on F, ϕ and ψ̃.
Let us consider the linear problem corresponding to (1.1) – (1.4), i.e., the case when λ = 0 :
L0u :=
∂2u
∂t2
−
n∑
i=1
∂2u
∂x2
i
= F (x, t), (x, t) ∈ DT , (3.8)
u|Γ = 0, u(x, 0) = ϕ(x), Kµut = ψ(x), x ∈ Ω. (3.9)
Let us show that when |µ| < 1 for any F ∈ L2(DT ), ϕ ∈
0
W 1
2(Ω) and ψ ∈ L2(Ω) there exists a
unique generalized solution of problem (3.8), (3.9) in the sense of Definition 1.1 for λ = 0. Indeed, for
ϕ ∈
0
W 1
2(Ω) and ψ ∈ L2(Ω) there are valid the expansions ϕ =
∑∞
k=1
akϕk and ψ =
∑∞
k=1
dkϕk
in the spaces
0
W 1
2(Ω) and L2(Ω), respectively, where ak = (ϕ,ϕk)L2(Ω) and dk = (ψ,ϕk)L2(Ω) [16].
Therefore, setting
ϕm =
m∑
k=1
akϕk, ψm =
m∑
k=1
dkϕk, (3.10)
we have
lim
m→∞
‖ϕm − ϕ‖ 0
W 1
2(Ω)
= 0, lim
m→∞
‖ψm − ψ‖L2(Ω) = 0. (3.11)
Since the space of finite infinitely differentiable functions C∞0 (DT ) is dense in the space L2(DT ),
then for F ∈ L2(DT ) and any natural number m there exists a function Fm ∈ C∞0 (DT ) such that
‖Fm − F‖L2(DT ) <
1
m
. (3.12)
On the other hand, for function Fm in the space L2(DT ) there is valid the following expansion [16]:
Fm(x, t) =
∞∑
k=1
Fm,k(t)ϕk(x), Fm,k(t) = (Fm, ϕk)L2(Ω). (3.13)
Therefore, there exists a natural number lm such that limm→∞ lm =∞ and for
F̃m(x, t) =
lm∑
k=1
Fm,k(t)ϕk(x) (3.14)
the inequality
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ON THE SOLVABILITY OF A PROBLEM NONLOCAL IN TIME . . . 99
‖F̃m − Fm‖L2(DT ) <
1
m
(3.15)
is valid. From (3.12) and (3.15) it follows
lim
m→∞
‖F̃m − F‖L2(DT ) = 0. (3.16)
The solution u = um of problem (3.1), (3.2) for ϕ = ϕlm , ψ̃ =
∑lm
k=1
d̃kϕk and F = F̃m, where
ϕlm and F̃m are defined in (3.10) and (3.14), is given by formula (3.3) which, due to (3.4) – (3.6),
takes the form
um =
lm∑
k=1
ak cosµkt+
d̃k
µk
sinµkt+
1
µk
t∫
0
Fm,k(τ) sinµk(t− τ)dτ
ϕk(x). (3.17)
For determination of the coefficients d̃k let us substitute the right-hand side of expression (3.17) into
the equality Kµumt = ψlm(x), where ψlm is defined in (3.10). Consequently, taking into account
that the system of functions {ϕk(x)} represents a basis in L2(Ω) and 1− µ cosµkT 6= 0 for |µ| < 1,
we obtain the following formulas:
d̃k =
1
1− µ cosµkT
(ϕlm , ϕk)L2(Ω) − akµµk sinµkT +
+ µ
T∫
0
Fm,k(τ) cosµk(T − τ)dτ
, k = 1, . . . , lm. (3.18)
Below we assume that the Lipschitz domain Ω is such that eigenfunctions ϕk ∈ C2(Ω), k ≥ 1. For
example, this will take place if ∂Ω ∈ C [n/2]+3 [19]. This fact will also take place in the case of a piece-
wisely smooth Lipschitz domain, e.g., for the parallelepiped Ω = {x ∈ Rn : |xi| < ai, i = 1, . . . , n}
the correspondent eigenfunctions ϕk ∈ C∞(Ω) [20]. Therefore, since Fm ∈ C∞0 (DT ), then due
to (3.13) the function Fm,k ∈ C2([0, T ]), and consequently the function um from (3.17) belongs to
the space C2(DT ). Further, since ϕk|∂Ω = 0, then due to (3.17) we have um |Γ = 0, and thereby
um ∈
0
C 2(DT ,Γ), m = 1, 2, . . . .
According to the construction the function um from (3.17) satisfies the following equalities:
um |Γ = 0, L0um = F̃m, um(x, 0) = ϕlm(x), Kµumt = ψlm(x), x ∈ Ω, (3.19)
and thereby
(um − uk)|Γ = 0, L0(um − uk) = F̃m − F̃k, (um − uk)(x, 0) = (ϕlm − ϕlk)(x),
Kµ(umt − ukt) = (ψlm − ψlk)(x), x ∈ Ω.
Therefore, from a priori estimate (2.4), where for λ = 0 the coefficients c4 = c5 = 0, we obtain
‖um − uk‖ 0
W 1
2(DT ,Γ)
≤ c1‖F̃m − F̃k‖L2(DT ) + c2‖ϕlm − ϕlk‖ 0
W 1
2(Ω)
+ c3‖ψlm − ψlk‖L2(Ω). (3.20)
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100 S. KHARIBEGASHVILI, B. MIDODASHVILI
In view of (3.11) and (3.16) from (3.20) it follows that the sequence um ∈
0
C 2(DT ,Γ) is
fundamental in the complete space
0
W 1
2(DT ,Γ). Therefore, there exists a function u ∈
0
W 1
2(DT ,Γ)
such that due to (3.11), (3.16) and (3.19) there are valid the limit equalities (1.8), (1.9) for λ = 0.
The last means that the function u is a generalized solution of problem (3.8), (3.9). The uniqueness
of this solution follows from a priori estimate (2.4) where the constants c4 = c5 = 0 for λ = 0.
Therefore, for the solution u of problem (3.8), (3.9) we have u = L−1
0 (F,ϕ, ψ), where L−1
0 :
L2(DT )×
0
W 1
2(Ω)× L2(Ω)→
0
W 1
2(DT ,Γ), which norm due to (2.4) can be estimated as follows:
‖L−1
0 ‖
L2(DT )×
0
W 1
2(Ω)×L2(Ω)→
0
W 1
2(DT ,Γ)
≤ γ0 = max(c1, c2, c3). (3.21)
Due to the linearity of the operator L−1
0 : L2(DT )×
0
W 1
2(Ω)× L2(Ω)→
0
W 1
2(DT ,Γ) we have a
representation
L−1
0 (F,ϕ, ψ) = L−1
0 (F, 0, 0) + L−1
0 (0, ϕ, 0) + L−1
0 (0, 0, ψ) = L−1
01 (F ) + L−1
02 (ϕ) + L−1
03 (ψ),
(3.22)
where L−1
01 : L2(DT )→
0
W 1
2(DT ,Γ), L−1
02 :
0
W 1
2(Ω)→
0
W 1
2(DT ,Γ) and L−1
03 : L2(Ω)→
0
W 1
2(DT ,Γ)
are linear continuous operators, besides, according to (3.21)
‖L−1
01 ‖
L2(DT )→
0
W 1
2(DT ,Γ)
≤ γ0, ‖L−1
02 ‖ 0
W 1
2(Ω)→
0
W 1
2(DT ,Γ)
≤ γ0,
‖L−1
03 ‖
L2(Ω)→
0
W 1
2(DT ,Γ)
≤ γ0.
(3.23)
Remark 3.1. Note, that for F ∈ L2(DT ), ϕ ∈
0
W 1
2(Ω), ψ ∈ L2(Ω), due to (1.5), (1.6), (3.21) –
(3.23) and Remark 1.1 the function u ∈
0
W 1
2(DT ,Γ) is a generalized solution of problem (1.1) – (1.4)
if and only if, when u is a solution of the functional equation
u = L−1
01 (−λf(x, t, u)) + L−1
01 (F ) + L−1
02 (ϕ) + L−1
03 (ψ) (3.24)
in the space
0
W 1
2(DT ,Γ).
Rewrite equation (3.24) in the form
u = A0u := −λL−1
01 (N0u) + L−1
01 (F ) + L−1
02 (ϕ) + L−1
03 (ψ), (3.25)
where the operator N0 :
0
W 1
2(DT ,Γ) → L2(DT ) from (1.7), according to Remark 1.1 is continuous
and compact operator. Therefore, due to (3.23) the operator A0 :
0
W 1
2(DT ,Γ) →
0
W 1
2(DT ,Γ) from
(3.25) is also continuous and compact. At the same time, according to Lemma 2.1 and (2.36) for
any parameter τ ∈ [0, 1] and for any solution u of the equation u = τA0u with the parameter τ it
is valid the same a priori estimate (2.4) with nonnegative constants ci, not dependent on u, F, ϕ, ψ
and τ. Therefore, due to the Schaefer’s fixed point theorem [21], equation (3.25), and therefore, due
to Remark 3.1 problem (1.1) – (1.4) has at least one solution u ∈
0
W 1
2(DT ,Γ). Thus, we have proved
the following theorem.
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ON THE SOLVABILITY OF A PROBLEM NONLOCAL IN TIME . . . 101
Theorem 3.1. Let λ > 0, |µ| < 1, F ∈ L2(DT ), ϕ ∈
0
W 1
2(Ω), ψ ∈ L2(Ω), conditions (1.5),
(1.6), (2.2), (2.3) be fulfilled. Then problem (1.1) – (1.4) has at least one generalized solution.
Remark 3.2. Note that for |µ| = 1, even in the linear case, i.e., for f = 0, the homogeneous
problem corresponding to (1.1) – (1.4) may have finite or even infinite number of linearly independent
solutions. Indeed, in the case µ = 1 denote by Λ(1) the set of points µk from (3.3), for which the ratio
µkT
2π
is a natural number, i.e., Λ(1) =
{
µk :
µkT
2π
∈ N
}
. If we search for a solution of problem (3.8),
(3.9) in the form of representation (3.3), then for determination of unknown coefficients bk,contained
in it, let us substitute the right-hand side of this representation into the equality Kµut = ψ(x). As a
result we have
µk(1− µ cosµkT )bk = (ψ,ϕk)L2(Ω) − akµk sinµkT +
T∫
0
Fk(τ) cosµk(T − τ)dτ. (3.26)
It is obvious, that when Λ(1) 6= ∅ and µk ∈ Λ(1), µ = 1 we have 1− cosµkT = 0 and for F = 0,
ϕ = ψ = 0 and thereby for ak = 0, Fk(τ) = 0 equality (3.26) will be satisfied by any number
bk. Therefore, in accordance with (3.3) the function uk(x, t) = C sinµktϕk(x), C = const 6= 0,
satisfies the homogeneous problem corresponding to (3.8), (3.9). Analogously, in the case µ = −1
denote by Λ(−1) the set of points µk from (3.3) for which the ratio
µkT
π
is odd integer number. In
this case 1 − µ cosµkT = 0 for µk ∈ Λ(−1), µ = −1 and the function uk(x, t) = C sinµktϕk(x),
C = const 6= 0, is a nontrivial solution of the homogeneous problem corresponding to (3.8), (3.9).
For example, when n = 2, Ω = (0, 1) × (0, 1) the eigenvalues and eigenfunctions of the Laplace
operator ∆ are [20]
λk = −π2(k2
1 + k2
2), ϕk(x1, x2) = sin k1π1x1 sin k2πx2, k = (k1, k2),
i.e., µk = π
√
k2
1 + k2
2. For k1 = p2−q2, k2 = 2pq, where p and q are any integer numbers we obtain
µk = π(p2 + q2) [22]. In this case for
T
2
∈ N we have
µkT
2π
= (p2 + q2)
T
2
∈ N and according to the
said above, when µ = 1 the homogeneous problem corresponding to (3.8), (3.9) has infinite number
of linearly independent solutions
up,q(x, t) = sinπ(p2 + q2)t sinπ(p2 − q2)x1 sin 2πpqx2 ∀p, q ∈ N. (3.27)
Analogously, when µ = −1 the solutions of the homogeneous problem corresponding to (3.8), (3.9)
in the case when p is an even number, while q and T odd numbers, are the functions from (3.27).
4. The uniqueness of the solution of problem (1.1) – (1.4). On the function f in equation (1.1)
let us impose the following additional requirements:
f, f ′u ∈ C(DT × R), |f ′u(x, t, u)| ≤ a+ b|u|γ , (x, t, u) ∈ DT × R, (4.1)
where a, b, γ = const ≥ 0.
It is obvious that from (4.1) we have condition (1.5) for α = γ + 1 and when γ <
2
n− 1
we get
α = γ + 1 <
n+ 1
n− 1
and, therefore, condition (1.6) is fulfilled.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 1
102 S. KHARIBEGASHVILI, B. MIDODASHVILI
Theorem 4.1. Let |µ| < 1, F ∈ L2(DT ), ϕ ∈
0
W 1
2(Ω), ψ ∈ L2(Ω) and condition (4.1) be
fulfilled for γ <
2
n− 1
, and also hold conditions (2.2), (2.3). Then there exists a positive number
λ0 = λ0(F, f, ϕ, ψ, µ,DT ) such that for 0 < λ < λ0 problem (1.1) – (1.4) can not have more than
one generalized solution.
Proof. Indeed, suppose that problem (1.1) – (1.4) has two different generalized solutions u1 and
u2. According to Definition 1.1 there exist sequences of functions ujk ∈
0
C 2(DT ,Γ), j = 1, 2, such
that
lim
k→∞
‖ujk − uj‖ 0
W 1
2(DT ,Γ)
= 0, lim
k→∞
‖Lλujk − F‖L2(DT ) = 0, (4.2)
lim
k→∞
‖ujk|t=0 − ϕ‖ 0
W 1
2(Ω)
= 0, lim
k→∞
‖Kµujkt − ψ‖L2(Ω) = 0, j = 1, 2. (4.3)
Let
w := u2 − u1, wk := u2k − u1k, Fk := Lλu2k − Lλu1k, (4.4)
gk := λ(f(x, t, u1k)− f(x, t, u2k)). (4.5)
In view of (4.2), (4.3) and (4.4) it is easy to see that
lim
k→∞
‖wk − w‖ 0
W 1
2(DT ,Γ)
= 0, lim
k→∞
‖Fk‖L2(DT ) = 0, (4.6)
lim
k→∞
‖wk|t=0‖ 0
W 1
2(Ω)
= 0, lim
k→∞
‖Kµwkt‖L2(Ω) = 0. (4.7)
In view of (4.4), (4.5) the function wk ∈
0
C 2(DT ,Γ) satisfies the following equalities:
∂2wk
∂t2
−
n∑
i=1
∂2wk
∂x2
i
=
(
Fk + gk
)
(x, t), (x, t) ∈ DT , (4.8)
wk |Γ = 0, (4.9)
wk(x, 0) = ϕ̃k(x), x ∈ Ω, (4.10)
Kµwkt := wkt(x, 0)− µwkt(x, T ) = ψ̃k(x), x ∈ Ω, (4.11)
where ϕ̃k(x) := u2k(x, 0)− u1k(x, 0), ψ̃k(x) := Kµu2kt −Kµu1kt.
First let us estimate the function gk from (4.5). Taking into account the obvious inequality
|d1 + d2|γ ≤ 2γ max(|d1|γ , |d2|γ) ≤ 2γ(|d1|γ + |d2|γ) for γ ≥ 0, due to (4.1) we have
|f(x, t, u2k)− f(x, t, u1k)| =
∣∣∣∣∣∣(u2k − u1k)
1∫
0
f ′u(x, t, u1k + τ(u2k − u1k))dτ
∣∣∣∣∣∣ ≤
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 1
ON THE SOLVABILITY OF A PROBLEM NONLOCAL IN TIME . . . 103
≤ |u2k − u1k|
1∫
0
(a+ b|(1− τ)u1k + τu2k|γ)dτ ≤ a|u2k − u1k|+
+2γb|u2k − u1k|(|u1k|γ + |u2k|γ) = a|wk|+ 2γb|wk|(|u1k|γ + |u2k|γ). (4.12)
In view of (4.5) from (4.12) we obtain
‖gk‖L2(DT ) ≤ λa‖wk‖L2(DT ) + λ2γb‖ |wk|(|u1k|γ + |u2k|γ)‖L2(DT ) ≤
≤ λa‖wk‖L2(DT ) + λ2γb‖wk‖Lp(DT )‖(|u1k|γ + |u2k|γ)‖Lq(DT ). (4.13)
Here we used the Hölder’s inequality [23]
‖v1v2‖Lr(DT ) ≤ ‖v1‖Lp(DT )‖v2‖Lq(DT ),
where
1
p
+
1
q
=
1
r
and in the capacity of p, q and r we take
p = 2
n+ 1
n− 1
, q = n+ 1, r = 2. (4.14)
Since dimDT = n + 1, then according to the Sobolev embedding theorem [17] for 1 ≤ p ≤
≤ 2(n+ 1)
n− 1
we get
‖v‖Lp(DT ) ≤ Cp‖v‖W 1
2 (DT ) ∀v ∈W 1
2 (DT ) (4.15)
with positive constant Cp, not dependent on v ∈W 1
2 (DT ).
Due to the condition of the theorem γ <
2
n− 1
and, therefore, γ(n+ 1) <
2(n+ 1)
n− 1
. Thus, due
to (4.14) from (4.15) we have
‖wk‖Lp(DT ) ≤ Cp‖wk‖W 1
2 (DT ), p =
2(n+ 1)
n− 1
, k ≥ 1, (4.16)
‖(|u1k|γ + |u2k|γ)‖Lq(DT ) ≤ ‖ |u1k|γ‖Lq(DT ) + ‖ |u2k|γ‖Lq(DT ) =
= ‖u1k‖γLγ(n+1)(DT ) + ‖u2k‖γLγ(n+1)(DT ) ≤ C
γ
γ(n+1)
(
‖u1k‖γW 1
2 (DT )
+ ‖u2k‖γW 1
2 (DT )
)
. (4.17)
In view of the first equality of (4.2) there exists a natural number k0 such that for k ≥ k0 we
obtain
‖uik‖γW 1
2 (DT )
≤ ‖ui‖γW 1
2 (DT )
+ 1, i = 1, 2, k ≥ k0. (4.18)
Further, in view of (4.16), (4.17) and (4.18) from (4.13) we get
‖gk‖L2(DT ) ≤ λa‖wk‖L2(DT ) + λ2γbCpC
γ
γ(n+1)(‖u1‖γW 1
2 (DT )
+
+‖u2‖γW 1
2 (DT )
+ 2)‖wk‖W 1
2 (DT ) ≤ λM8‖wk‖W 1
2 (DT ), (4.19)
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 1
104 S. KHARIBEGASHVILI, B. MIDODASHVILI
where we have used the inequality ‖wk‖L2(DT ) ≤ ‖wk‖W 1
2 (DT ),
M8 = a+ 2γbCpC
γ
γ(n+1)
(
‖u1‖γW 1
2 (DT )
+ ‖u2‖γW 1
2 (DT )
+ 2
)
, p = 2
n+ 1
n− 1
. (4.20)
Since a priori estimate (2.4) is valid for λ = 0, then due to (2.27) and (2.36) in this estimate
c4 = c5 = 0, and thereby for the solution wk of problem (4.8) – (4.11) the following estimate:
‖wk‖ 0
W 1
2(DT ,Γ)
≤ c0
1‖Fk + gk‖L2(DT ) + c0
2‖ϕ̃k‖ 0
W 1
2(Ω)
+ c0
3‖ψ̃k‖L2(Ω) (4.21)
is valid, where the constants c0
1, c
0
2, c
0
3 do not depend on λ.
Because of ‖wk‖ 0
W 1
2(DT ,Γ)
= ‖wk‖W 1
2 (DT ) and due to (4.19) from (4.21) we have
‖wk‖ 0
W 1
2(DT ,Γ)
≤ c0
1‖Fk‖L2(DT ) + λc0
1M8‖wk‖ 0
W 1
2(DT ,Γ)
+ c0
2‖ϕ̃k‖ 0
W 1
2(Ω)
+ c0
3‖ψ̃k‖L2(Ω). (4.22)
Note that since for u1 and u2 it is valid a priori estimate (2.4), then the constant M8 from (4.20)
will depend on λ, F, f, ϕ, ψ, DT , besides, due to (2.27) and (2.36) the value of M8 continuously
depends on λ for λ ≥ 0 and
0 ≤ lim
λ→0+
M8 = M0
8 < +∞. (4.23)
Due to (4.23) there exists a positive number λ0 = λ0(F, f, ϕ, ψ, µ,DT ) such that for
0 < λ < λ0 (4.24)
we obtain λc0
1M8 < 1. Indeed, let us fix arbitrarily a positive number ε1. Then, due to (4.23), there
exists a positive number λ1, such that 0 ≤ M8 < M0
8 + ε1 for 0 ≤ λ < λ1. It is obvious that
for λ0 = min
(
λ1, (c
0
1(M0
8 + ε1))−1
)
the condition λc0
1M8 < 1 will be fulfilled. Therefore, in the
case (4.24) from (4.22) we get
‖wk‖ 0
W 1
2(DT ,Γ)
≤ (1− λc0
1M8)−1
[
c0
1‖Fk‖L2(DT ) + c0
2‖ϕ̃k‖ 0
W 1
2(Ω)
+ c0
3‖ψ̃k‖L2(Ω)
]
, k ≥ k0.
(4.25)
From (4.2) and (4.4) it follows that limk→∞ ‖wk‖ 0
W 1
2(DT ,Γ)
= ‖u2 − u1‖ 0
W 1
2(DT ,Γ)
. On the other
hand due to (4.6), (4.7) and (4.10), (4.11) from (4.25) we have limk→∞ ‖wk‖ 0
W 1
2(DT ,Γ)
= 0. Thus
‖u2 − u1‖ 0
W 1
2(DT ,Γ)
= 0, i.e., u2 = u1, which leads to contradiction.
Theorem 4.1 is proved.
1. Byszewski L., Lakshmikantham V. Theorem about the existence and uniqueness of a solution of a nonlocal abstract
Cauchy problem in a Banach space // Appl. Anal. – 1991. – 4, № 1. – P. 11 – 19.
2. Kiguradze T. Some boundary value problems for systems of linear partial differential equations of hyperbolic type //
Mem. Different. Equat. Math. Phys. – 1994. – 1. – P. 1 – 144.
3. Kiguradze T. I. Some nonlocal problems for linear hyperbolic systems (in Russian) // Dokl. Akad. Nauk. – 1995. –
345, № 3. – P. 300 – 302 (English transl.: Dokl. Math. – 1995. – 52, № 3. – P. 376 – 378).
4. Aizicovici S., McKibben M. Existence results for a class of abstract nonlocal Cauchy problems // Nonlinear Anal. –
2000. – 39, № 5. – P. 649 – 668.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 1
ON THE SOLVABILITY OF A PROBLEM NONLOCAL IN TIME . . . 105
5. Gordeziani D. G., Avalishvili G. A. Investigation of the nonlocal initial boundary value problems for some hyperbolic
equations // Hirosima Math. J. – 2001. – 31. – P. 345 – 366.
6. Avalishvili G. A. Nonlocal in time problems for evolution equations of second order // J. Appl. Anal. – 2002. – 8,
№ 2. – P. 245 – 259.
7. Midodashvili B. A nonlocal problem for fourth order hyperbolic equations with multiple characteristics // Electron.
J. Different. Equat. – 2002. – 2002, № 85. – P. 1 – 7.
8. Bouziani A. On a class of nonclassical hyperbolic equations with nonlocal conditions // J. Appl. Math. Stochast.
Anal. – 2002. – 15, № 2. – P. 135 – 153.
9. Kharibegashvili S. S. On the well-posedness of some nonlocal problems for the wave equation (in Russian) //
Differents. Uravneniya. – 2003. – 39, № 4. – P. 539 – 553 (English transl.: Different. Equat. – 2003. – 39, № 4. –
P. 577 – 592).
10. Bogveradze G., Kharibegashvili S. On some nonlocal problems for a hyperbolic equation of second order on a plane
// Proc. A. Razmadze Math. Inst. – 2004. – 136. – P. 1 – 36.
11. Xue X. Existence of solutions for semilinear nonlocal Cauchy problems in Banach spaces // Electron. J. Different.
Equat. – 2005. – 2005, № 64. – P. 1 – 7.
12. Beilin S. A. On a mixed nonlocal problem for a wave equation // Electron. J. Different. Equat. – 2006. – 2006,
№ 103. – P. 1 – 10.
13. Hernández E. Existence of solutions for an abstract second-order differential equation with nonlocal conditions //
Electron. J. Different. Equat. – 2009. – 2009, № 96. – P. 1 – 10.
14. Kharibegashvili S., Midodashvili B. Some nonlocal problems for second order strictly hyperbolic systems on the
plane // Georg. Math. J. – 2010. – 17, № 2. – P. 287 – 303.
15. Kharibegashvili S., Midodashvili B. Solvability of nonlocal problems for semilinear one-dimensional wave equations
// Electron. J. Different. Equat. – 2012. – 2012, № 28. – P. 1 – 16.
16. Ladyzhenskaya O. A. The boundary value problems of mathematical physics. – New York: Springer-Verlag, 1985.
17. Kufner A., Fučik S. Nonlinear differential equations. – Amsterdam; New York: Elsevier, 1980.
18. Beckenbach E., Bellman R. Inequalities. – Berlin: Springer-Verlag, 1961.
19. Mikhailov V. P. Partial differential equations. – Moscow: Mir, 1978.
20. Mikhlin S. G. Mathematical physics. An advanced course. – Amsterdam: North-Holland, 1970.
21. Evans L. C. Partial differential equations // Grad. Stud. Math. – Providence, RI: Amer. Math. Soc., 1998. – 19.
22. Jones G. A., Jones J. M. Elementary number theory. – Springer, 1998.
23. Reed M., Simon B. Methods of modern mathematical physics. II: Fourier analysis. Selfadjointness. – New York etc.:
Acad. Press, 1975.
Received 20.11.13,
after revision — 16.08.14
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