Georgian Mathematical Journal
Volume 9 (2002), Number 1, 57–70
ESTIMATES OF A STABILIZATION RATE AS t → ∞ OF
SOLUTIONS OF A NONLINEAR INTEGRO-DIFFERENTIAL
EQUATION
T. JANGVELADZE 1 AND Z. KIGURADZE
Abstract. The asymptotic behavior as t → ∞ of solutions of a nonlinear
integro-differential equation is studied. The equation arises as a model describing the penetration of the electromagnetic field in to a substance.
2000 Mathematics Subject Classification: 35K55, 45K05.
Key words and phrases: Nonlinear integro-differential parabolic equation,
asymptotic behavior.
1. Introduction. The main result. This paper is devoted to the study of
the stabilization of solutions of the first boundary value problem in a cylindrical
domain Q = (0, 1) × {t > 0} for the system of nonlinear integro-differential
parabolic equations
·
¸
·
¸
∂U
∂
∂U
∂V
∂
∂V
=
a(S)
,
=
a(S)
, (x, t) ∈ Q,
∂t
∂x
∂x
∂t
∂x
∂x
U (0, t) = V (0, t) = 0, U (1, t) = ψ1 , V (1, t) = ψ2 , t ≥ 0,
U (x, 0) = U0 (x), V (x, 0) = V0 (x), x ∈ (0, 1),
(1)
(2)
(3)
where
Zt ·µ
S(x, t) = 1 +
0
∂U
∂x
¶2
µ
∂V
+
∂x
¶2 ¸
dτ,
(4)
dx dτ.
(5)
or
Zt Z1 ·µ
S(t) = 1 +
0 0
∂U
∂x
¶2
µ
+
∂V
∂x
¶2 ¸
Here ψ1 = Const, ψ2 = Const, a(S), U0 (x) and V0 (x) are given functions.
The characteristic feature of equations (1), (4) and (1), (5) is the appearance
of nonlinear members depending on the integral of searched functions in the
coefficients of higher derivatives.
System (1), (4) arises as a model describing the penetration of the electromagnetic field into a substance [1].
1
In the literature this author is frequently referred to as T. A. Dzhangveladze.
c Heldermann Verlag www.heldermann.de
ISSN 1072-947X / $8.00 / °
58
T. JANGVELADZE AND Z. KIGURADZE
A lot of scientific works are devoted to the investigation of the problem given
in [1] and to some of its generalizations. These questions are considered in
[2]–[9] and in a number of other works as well.
The study of equations of type (1), (4) began started in [1]. In this work,
in particular, theorems of the existence of a generalized solution of the first
boundary value problem for a(S) = S and the uniqueness for more general
cases are proved. The case a(S) = S p , 0 < p ≤ 1, is studied in [2], where
a theorem of the existence and uniqueness of a solution of problem (1)–(4) is
proved. Investigations for multidimensional space cases are carried out in [3]
for the first time.
In [4], [5] an operational scheme with the so-called conditionally closed operators is proposed. This scheme is applied for the solution of problems of (1)–(4)
type [4], [5].
Note that investigations of equations of (1), (4) type are also carried out in
[6], [7].
In the work [5] some generalization of equations of type (1), (4) is proposed.
In particular, assuming the temperature of the considered body to be constant
throughout the material, i.e., depending on time, but independent of the space
coordinates, the process of penetration of the magnetic field into the material
is modelled by averaged integro-differential equations of type (1), (5).
The purpose of this note is to continue the study of the asymptotic behavior
of solutions of the equations (1), (4) which began in [8], [9]. In the present
paper estimates of stabilization rate as t → ∞ of solutions of problems (1)–
(4) and (1)–(3), (5) are obtained for the case a(S) = S p , 0 < p ≤ 1. We
will use the scheme of [10] in which the adiabatic shearing of incompressible
fluids with temperature-dependent viscosity is studied. We should note that
boundary conditions (2) are used here taking into account the physical problem
considered in [11].
We assume that (U, V ) = (U (x, t), V (x, t)) is a solution of (1)–(4) on [0, 1] ×
∂U ∂V ∂U ∂V ∂ 2 U ∂ 2 V
[0, ∞) such that U, V,
,
,
,
,
,
are all in C 0 ([0, ∞); L2 (0, 1)),
2
2
∂x ∂x ∂t ∂t ∂x ∂x
∂ 2U ∂ 2V
∂2U ∂ 2V
while
,
are in C 0 ((0, ∞); L2 (0, 1)) and 2 , 2 are in L2,loc ((0, ∞);
∂t ∂x ∂t ∂x
∂t ∂t
L2 (0, 1)) (see [1], [2], [4], [5], [10]).
The main purpose of this work is to prove the following statement.
Theorem. Assume
a(S) = S p , 0 < p ≤ 1,
U0 (0) = V0 (0) = 0, U0 (1) = ψ1 , V0 (1) = ψ2 ,
ψ12 + ψ22 6= 0, U0 , V0 ∈ W22 (0, 1).
Then for the solution of problem (1)–(4) the following estimates are true as
t → ∞:
∂V (x, t)
∂U (x, t)
= ψ1 + O(t−1−p ),
= ψ2 + O(t−1−p ),
(6)
∂x
∂x
STABILIZATION RATE OF SOLUTION OF A NONLINEAR EQUATION
∂U (x, t)
= O(t−1 ),
∂t
59
∂V (x, t)
= O(t−1 ),
∂t
(7)
uniformly in x on [0, 1].
The proof of the theorem is based on a priori estimates which are obtained
with the help of a number of identities derived below.
2. Proof of the theorem. Now let us proved to obtaining a priori estimates.
Lemma 1. For solving problem (1) − (4) the following estimations are true:
Zt Z1 µ
0 0
∂U
∂τ
Zt Z1 µ
¶2
dx dτ ≤ C,
0 0
∂V
∂τ
¶2
dx dτ ≤ C, t ≥ 0.
(8)
Proof. Let us differentiate the first equation of system (1) with respect to t:
"
2
∂ 2U
∂
p ∂ U
=
S
+ pS p−1
∂t2
∂x
∂t ∂x
µ·
¸3
∂U
∂x
·
∂U ∂V
+
∂x ∂x
¸2 ¶#
.
(9)
Multiplying equation (9) by ∂U/∂t and integrating with respect to x on the
interval (0, 1), we have
1
1d Z
2 dt
µ
0
∂U
∂t
¶2
µ
Z1
dx +
S
p
0
¶2
S p−1
0
µ
Z1
S p−1
dx + p
0
µ
Z1
+p
∂ 2U
∂t ∂x
∂U ∂V
∂x ∂x
¶2
¶3
∂U
∂x
∂ 2U
dx +
∂t ∂x
∂ 2U
dx = 0.
∂t ∂x
(10)
Integration from 0 to t gives the identity
1
1Z
2
0
µ
∂U
∂t
¶2
dx +
p
+
2
µ
S
0 0
p−1
S
p
0 0
p(p − 1)
−
4
Zt Z1
µ
Zt Z1
∂V
∂x
¶2
µ
Zt Z1
S
p−2
0 0
µ
∂ ∂U
∂τ ∂x
¶2
∂ 2U
∂τ ∂x
∂U
∂x
¶2
µ
1
p Z p−1 ∂U
dx dτ +
S
4
∂x
¶4
dx −
0
¶4 ·µ
∂U
∂x
1
1Z
dx dτ =
2
¶2
µ
0
µ
∂V
+
∂x
¶2 ¸
dx dτ +
∂U (x, 0)
∂t
¶2
µ
¶2
1
pZ
dx +
4
µ
0
∂U0
∂x
¶4
dx.
It follows that
Z1
0
µ
∂U
∂t
¶2
Zt Z1 µ
dx + 2
0 0
∂ 2U
∂τ ∂x
¶2
Zt Z1
S p−1
dx dτ + p
0 0
∂V
∂x
µ
∂ ∂U
∂τ ∂x
¶2
dx dτ ≤ C.
Here and below c, C and Ci denote the positive constants dependent only on
ψi = Const, i = 1, 2, U0 (x), V0 (x) and consequently independent of t.
60
T. JANGVELADZE AND Z. KIGURADZE
Similarly, using the second equation of system (1), we get
Z1 µ
0
∂V
∂t
¶2
Zt Z1 µ
dx + 2
0 0
∂ 2V
∂τ ∂x
¶2
µ
Zt Z1
S p−1
dx dτ + p
0 0
∂U
∂x
¶2
µ
¶2
∂ ∂V
∂τ ∂x
dx dτ ≤ C
and, therefore
Z1 ·µ
0
∂U
∂t
¶2
µ
∂V
+
∂t
¶2 ¸
Zt Z1 ·µ
dx + 2
0 0
Zt Z1
S p−1
+p
0 0
∂
∂τ
·µ
∂U
∂x
¶2 µ
∂ 2U
∂τ ∂x
¶2
µ
∂ 2V
+
∂τ ∂x
¶2 ¸
dx dτ
¶2 ¸
∂V
∂x
dx dτ ≤ C.
Note that
Zt Z1
p
S
p−1
0 0
∂
∂τ
Z1
µ
−p
0
·µ
∂U
∂x
∂U0
∂x
¶2 µ
¶2 µ
∂V
∂x
∂V0
∂x
µ
∂V
+
∂x
¶2 ¸
µ
Z1
S p−1
dx dτ = p
0
¶2
∂U
∂x
·µ
Zt Z1
dx − p(p − 1)
¶2 ¸µ
∂U
∂x
¶2 µ
¶2 µ
S
p−2
0 0
∂V
∂x
∂U
∂x
∂V
∂x
¶2
dx
¶2
¶2
dx dτ.
We have
Z1 ·µ
0
∂U
∂t
¶2
µ
∂V
+
∂t
¶2 ¸
Zt Z1 ·µ
dx + 2
0 0
∂ 2U
∂τ ∂x
¶2
µ
∂ 2V
+
∂τ ∂x
¶2 ¸
dx dτ ≤ C. (11)
From this, taking into consideration the relations
Zt Z1 µ
0 0
Zt Z1
0 0
µ
∂U
∂τ
¶2
Zt Z1 µ
∂V
∂τ
¶2
dx dτ ≤
0 0
Zt Z1
µ
dx dτ ≤
0 0
∂ 2U
∂τ ∂x
¶2
∂ 2V
∂τ ∂x
¶2
dx dτ,
dx dτ,
we get a priori estimates (8).
Lemma 2. The following estimations are true:
1
1
cϕ 1+2p (t) ≤ S(x, t) ≤ Cϕ 1+2p (t), 0 ≤ x ≤ 1, t ≥ 0,
where
·µ
Zt Z1
S 2p
ϕ(t) = 1 +
0 0
∂U
∂x
¶2
µ
+
∂V
∂x
¶2 ¸
dx dτ.
(12)
STABILIZATION RATE OF SOLUTION OF A NONLINEAR EQUATION
61
Proof. From (4) it follows that
µ
∂S
∂U
=
∂t
∂x
¶2
µ
+
∂V
∂x
¶2
, S(x, 0) = 1.
Let us multiply this equation by S 2p :
µ
1 ∂S 1+2p
∂U
=
1 + 2p ∂t
∂x
¶2
µ
S
2p
∂V
+
∂x
¶2
S 2p .
Now let us introduce the notations:
∂V
∂U
, σ2 = S p
,
σ1 = S p
∂x
∂x
then (1) can be rewritten as
∂U
∂σ1
=
,
∂t
∂x
∂V
∂σ2
=
.
∂t
∂x
We have
1 ∂S 1+2p
= σ12 + σ22 ,
1 + 2p ∂t
Z1
σ12 (x, t)
Z1 Zx
σ12 (y, t)dy
=
+2
Z1
σ22 (x, t)
∂U (ξ, t)
dξ dy,
∂t
σ2 (ξ, t)
∂V (ξ, t)
dξ dy,
∂t
Z1 Zx
σ22 (y, t)dy
=
σ1 (ξ, t)
0 y
0
+2
0 y
0
Zt
(13)
Z1
(σ12 + σ22 )dx dτ.
ϕ(t) = 1 +
0 0
From (8) and (13) we get
t
Z
1
1
S 1+2p = (σ12 + σ22 )dτ +
1 + 2p
1 + 2p
0
=
Zt Z1 ³
σ12 (y, τ )
+
´
σ22 (y, τ )
Zt Z1 Zx
dy dτ + 2
σ1 (ξ, τ )
0 0 y
0 0
Zt Z1 Zx
+2
σ2 (ξ, τ )
0 0 y
Zt
Z1
³
1
∂V (ξ, τ )
dξ dy dτ +
∂τ
1 + 2p
´
σ12 (y, τ ) + σ22 (y, τ ) dy dτ + C +
≤2
∂U (ξ, τ )
dξ dy dτ
∂τ
0 0
1
≤ (2 + C1 )ϕ(t),
1 + 2p
i.e.,
1
S(x, t) ≤ Cϕ 1+2p (t).
(14)
62
T. JANGVELADZE AND Z. KIGURADZE
Analogously,
Zt Z1 ³
´
1
1+2p
S
=
σ12 (y, τ ) + σ22 (y, τ ) (y, τ )dy dτ
1 + 2p
0 0
Zt Z1 Zx
+2
0 0 y
Zt Z1 Zx
∂U (ξ, τ )
∂V (ξ, τ )
σ1 (ξ, τ )
dξ dy dτ + 2
σ2 (ξ, τ )
dξ dy dτ
∂τ
∂τ
y
0 0
t 1
´
1
1
1Z Z ³ 2
+
≥
σ1 (y, τ ) + σ22 (y, τ ) dy dτ − C2 ≥ ϕ(t) − C3 .
1 + 2p
2
2
(15)
0 0
From (4) it follows that S(x, t) ≥ 1. So
C3 S 1+2p ≥ C3 .
(16)
Taking into account (15) and (16) we easily get
µ
¶
1
1
+ C3 S 1+2p ≥ ϕ(t),
1 + 2p
2
or
1
S(x, t) ≥ cϕ 1+2p (t).
(17)
Finally, from (14) and (17) we obtain (12).
Lemma 3. The following esimates are true:
cϕ
2p
1+2p
(t) ≤
Z1 ³
´
2p
σ12 (x, t) + σ22 (x, t) dx ≤ Cϕ 1+2p (t), t ≥ 0.
(18)
0
Proof. Taking into account (12), we get
Z1
+
σ22 )dx
=
0
µ
+
∂V
∂x
·µ
Z1
(σ12
S
2p
0
¶2 ¸
(" Z1
2p
dx ≥ cϕ 1+2p (t)
0
∂U
∂x
¶2
µ
∂V
+
∂x
∂U
dx
∂x
#2
¶2 ¸
dx ≥ cϕ
2p
1+2p
Z1 ·µ
(t)
0
" Z1
+
0
∂V
dx
∂x
#2 )
∂U
∂x
¶2
2p
= (ψ12 + ψ22 )cϕ 1+2p (t),
or
Z1 ³
´
2p
σ12 (x, t) + σ22 (x, t) dx ≥ cϕ 1+2p (t).
(19)
0
At the same time, from (11) we derive
Z1 µ
0
∂U
∂t
¶2
Z1 µ
dx ≤ C,
0
∂V
∂t
¶2
dx ≤ C, t ≥ 0.
(20)
STABILIZATION RATE OF SOLUTION OF A NONLINEAR EQUATION
63
Let us multiply the first and second equations of system (1) by U (x, t) and
V (x, t), respectively. Using the boundary conditions (2), we have
Z1
µ
1
¶2
Z
∂U
∂U
U
dx + S p
∂t
∂x
0
dx = ψ1 σ1 (1, t),
0
Z1
µ
1
Z
∂V
∂V
V
dx + S p
∂t
∂x
0
¶2
dx = ψ2 σ2 (1, t).
0
Using these equalities, (12), (19), (20) and the maximum principle
|U (x, t)| ≤ max |U0 (y)|, |V (x, t)| ≤ max |V0 (y)|, 0 ≤ x ≤ 1, t ≥ 0,
0≤y≤1
0≤y≤1
we have
( Z1
³
σ12 (x, t)
+
σ22 (x, t)
)2
´
( Z1
dx
)2
≤2
0
≤ 2C1 ϕ
2p
1+2p
(t)
µ
S
p
0
2p
1+2p
≤ 4C1 ϕ
∂U
∂x
dx
+
V
à Z1
0
+C2
≤ 8C1 ϕ
·
2p
n³
max |U0 (y)|
´2
0≤y≤1
(t)
+
³
∂σ2
∂x
+
ψ22 )
+ (ψ2 σ2 (1, t))2
´
+
0
∂σ1
∂x
"
2p
1+2p
´2 o¸
(t)
¶2
Z1
σ22 dx
dx +
0
#
Z1
(ψ12
+
ψ22 )
(σ12
+
σ22 )dx
+ C4
0
"
≤ 8C1 ϕ
dx
!2
max |V0 (y)|
Z1 µ
σ12 dx
#
dx + C3 ≤ 8C1 ϕ
2p
1+2p
)2 #
0≤y≤1
( Z1
(ψ12
)
¶2
∂V
∂x
³
0
Z1 µ
p
¶2
≤ 4C1 ϕ 1+2p (t) (ψ12 + ψ22 ) σ12 (1, t) + σ22 (1, t)
"
2p
1+2p
S
∂U
dx
U
∂t
(t) (ψ1 σ1 (1, t)) +
∂V
dx
∂t
µ
0
2
!2 #
( Z1
+
0
à Z1
0
+2
0
)2
¶2
"
+
σ22 (x, t)dx
0
"( Z1
)2
( Z1
σ12 (x, t)dx
Z1
(ψ12
(t)
+
(σ12
ψ22 )
+
σ22 )dx
0
2p
≤ Cϕ 1+2p (t)
Z1 ³
+
(σ12
2p
cϕ 1+2p (t)
0
´
σ12 (x, t) + σ22 (x, t) dx,
0
i.e.,
Z1 ³
´
#
Z1
C4
2p
σ12 (x, t) + σ22 (x, t) dx ≤ Cϕ 1+2p (t).
0
Finally, using this estimate and (19) we obtain (18).
+
σ22 )dx
64
T. JANGVELADZE AND Z. KIGURADZE
Lemma 4. The following inequalities take place:
2p
ct
≤
Z1 ³
´
σ12 (x, t) + σ22 (x, t) dx ≤ Ct2p , t ≥ 1,
(21)
0
ct ≤ S(x, t) ≤ Ct, 0 ≤ x ≤ 1, t ≥ 1.
(22)
Proof. From (18) taking into account the relation
1
´
dϕ(t) Z ³ 2
=
σ1 (x, t) + σ12 (x, t) dx
dt
0
we get
2p
dϕ(t)
≤ Cϕ 1+2p (t).
dt
1+2p
From this we have ct
≤ ϕ(t) ≤ Ct1+2p , t ≥ 1. Now taking into account
(12) and (18) from the last inequality we obtain (21) and (22).
2p
cϕ 1+2p (t) ≤
Lemma 5. ∂U/∂t and ∂V /∂t statisfy the inequality
Z1 ·µ
0
∂U
∂t
¶2
µ
+
∂V
∂t
¶2 ¸
dx ≤ Ct−2 , t ≥ 1.
(23)
Proof. By Schwarz’s inequality, (10) yields
µ
1
d Z
dt
0
∂U
∂t
¶2
µ
Z1
dx +
S
p
0
∂ 2U
∂t ∂x
µ
Z1
+2p2
S p−2
0
¶2
dx ≤ 2p
0
∂σ1
∂x
¶2
Z1
dx = −
0
S
p−2
0
∂U
∂x
¶2 µ
Now using (21), (22), the relations σ1 = S p
Z1 µ
µ
Z1
2
∂V
∂x
∂U
∂x
¶6
dx
¶4
dx.
(24)
∂U
∂V
, σ2 = S p
and
∂x
∂x
¶
Z1 µ
Z1
∂σ2 2
∂ 2 σ1
∂ 2 σ2
σ1 2 dx,
dx = − σ2 2 dx,
∂x
∂x
∂x
0
0
from (24) we get
1
d Z
dt
0
µ
∂U
∂t
Z1 µ
¶2
p
dx + ct
0
Z1
≤ C1 t−5p−2
σ12 (x, t)dx
∂ 2U
∂t ∂x
³h
0
≤ C2 t
0
i2
0≤x≤1
σ12 dx
+2
0
h
i2 ´
+ max σ22 (x, t)
0≤x≤1
# 1 " Z1 µ
" Z1
σ12 dx
0
1
tp−2 Z 6
dx ≤ C1 6p (σ1 + σ12 σ24 )dx
t
max σ12 (x, t)
Ã( Z1
−3p−2
¶2
2
0
∂σ1
∂x
# 1 )2
¶2
2
dx
STABILIZATION RATE OF SOLUTION OF A NONLINEAR EQUATION
( Z1
# 1 " Z1 µ
" Z1
σ22 dx
+
+2
0
0
0
" Z1
Ã( Z1
≤ C2 t
σ12 dx
−3p−2
+2
0
0
Ã" Z1 µ
≤ C3 t
+ C4 t
−3p−2 3p
t
0
≤ C3 t
p−2
4
σ22 dx
σ22 dx + 2
0
p−2
0
# 3 " Z1 µ
+ C5 t
−p−4
ctp
+
4
2
∂ U
∂t ∂x
à Z1 µ
¶2
0
∂ 2 σ1
∂x2
¶2
#1
" Z1 µ
dx
#1 ¾
¶2
4
2
dx
# 1 )2 !
∂ 2 σ2
∂x2
4
dx
2
∂ 2U
∂t ∂x
2
dx
4
0
" Z1
# 1 )2 !
¶2
# 3 " Z1 µ
σ12 dx
0
( Z1
+
∂σ2
∂x
2
σ22 dx
65
+
0
Z1
¶2
µ
dx +
0
∂ 2V
∂t ∂x
∂ 2V
∂t ∂x
#1 !
¶2
¶2
2
dx
!
dx .
Hence we have
1
d Z
dt
µ
0
∂U
∂t
¶2
1
· µ
¶2
c pZ
∂ 2U
dx + t
3
4
∂t ∂x
0
µ
∂ 2V
−
∂t ∂x
¶2 ¸
dx ≤ Ctp−2 , t ≥ 1.
Alalogously, we can show that
1
d Z
dt
µ
0
∂V
∂t
¶2
1
· µ
∂ 2V
c pZ
dx + t
3
4
∂t ∂x
¶2
0
µ
∂ 2U
−
∂t ∂x
¶2 ¸
dx ≤ Ctp−2 , t ≥ 1.
As a consequence we get the following estimate:
·µ
1
d Z
dt
0
∂U
∂t
¶2
µ
∂V
+
∂t
¶2 ¸
1
c Z
dx+ tp
2
·µ
0
∂ 2U
∂t ∂x
¶2
µ
+
∂ 2V
∂t ∂x
¶2 ¸
dx ≤ Ctp−2 . (25)
By the Poincarè inequality
°
°
° 2
° ∂U °
° ∂ U
°
°≤°
°
°
°
∂t
∂t ∂x
°
°
°,
°
°
° ∂V
°
°
∂t
°
° 2
°
°∂ V
°≤°
°
°
∂t ∂x
°
°
°,
°
(25) gives
1
d Z
dt
0
·µ
∂U
∂t
¶2
µ
∂V
+
∂t
¶2 ¸
1 ·µ
µ
¶
¶ ¸
ctp Z
∂U 2
∂V 2
dx +
+
dx ≤ Ctp−2 .
2
∂t
∂t
0
From this we obtain (23).
Let us now estimate ∂S/∂x in L1 (0, 1).
Lemma 6. For ∂S/∂x the following estimate is true:
¯
Z1 ¯
¯ ∂S ¯
¯
¯dx ≤ Ct−p , t ≥ 1.
¯
¯
0
∂x
(26)
66
T. JANGVELADZE AND Z. KIGURADZE
Proof. We have
·
¸
∂ 2p ∂S
∂σ2
∂U
∂V
∂σ1
S
= 2σ1
+ 2σ2
= 2σ1
+ 2σ2
.
∂t
∂x
∂x
∂x
∂t
∂t
(27)
From (21) and (23) it follows that
¯ Z1
¯
¯
∂U ¯¯
¯
¯ σ1
dx¯ ≤ C1 tp t−1 = Ctp−1 ,
¯
∂t ¯
0
¯ Z1
¯
¯
∂V ¯¯
¯
¯ σ2
dx¯ ≤ C1 tp t−1 = Ctp−1
¯
∂t ¯
(28)
0
and, applying (22), (27) and (28), we get
S
2p ∂S
∂x
¶
Zt µ
∂V
∂U
2σ1
+ 2σ2
dτ,
∂τ
∂τ
=
0
¯
Z1 ¯
Zt
¯ ∂S ¯
1
¯
¯dx ≤ t−2p C1 τ p−1 dτ = Ct−p .
¯
¯
∂x
0
c
0
Now we are ready to prove the theorem. Let us estimate ∂ 2 U/∂x2 in L1 (0, 1).
We have
∂U
∂S
∂σ1
∂U ∂ 2 U
∂U −p
= σ1 S −p ,
=
,
=
S − pσ1 S −p−1 ,
2
∂x
∂x
∂t
∂x
∂t
∂x
Z1
σ12 (x, t)
Z1
σ12 (y, t)dy
≤
+2
0
¯
¯
¯ ∂U (y, t) ¯
¯
¯dy ≤ C1 t2p + C2 t−2 .
|σ1 (y, t)|¯
¯
∂t
0
From the latter we get
σ1 (x, t) ≤ Ctp ,
t ≥ 1.
(29)
Applying now (22), (23), (26) and (29), we derive
¯
¯
¯
Z1 ¯ 2
Z1 ¯
Z1 ¯
¯
¯ ∂ U (x, t) ¯
¯ ∂U −p ¯
¯
−p−1 ∂S ¯
¯
¯dx ≤
¯
¯
¯
S
dx
+
p
σ
S
1
¯
¯
¯
¯
¯
¯dx
2
∂x
0
" Z1
S
≤
−2p
0
∂x
0
# 1 " Z1 ¯
#1
¯
¯
Z1 ¯
2
2
¯ ∂U ¯2
¯
∂S ¯¯
−1−p
¯
¯σ1 S −p−1
¯ dx
dx
+
p
.
¯
¯
¯
¯dx ≤ Ct
0
Hence we have
∂t
∂t
0
0
∂x
¯
Z1 ¯ 2
¯ ∂ U (x, t) ¯
¯
¯dx ≤ Ct−1−p , t ≥ 1.
¯
¯
2
∂x
0
From this estimate, taking into account the relation
1
Z1 Zx 2
∂ U (ξ, t)
∂U (x, t) Z ∂U (y, t)
=
dy +
dξ dy,
2
∂x
∂y
∂ξ
y
0
0
STABILIZATION RATE OF SOLUTION OF A NONLINEAR EQUATION
67
it follows that
¯
Z1 Zx 2
Z1 ¯ 2
¯ ∂ U (y, t) ¯
∂U (x, t)
∂ U (ξ, t)
¯
¯dy ≤ Ct−1−p .
− ψ1 =
dξ dy ≤ ¯
¯
2
2
∂x
∂ξ
∂y
y
0
0
Thus the following asymptotic formula takes place:
∂U (x, t)
= ψ1 + O(t−1−p ).
∂x
The same estimate is valid for ∂V /∂x:
∂V (x, t)
= ψ2 + O(t−1−p ).
∂x
Let us now establish the asymptotic behaviour of the derivatives ∂U/∂t and
∂V /∂t. For this multiply (10) by t2 . By integrating on (0, t) we have
1µ
µ 2 ¶2
¶
¶
Zt Z1 µ
Zt Z1
t2 Z ∂U 2
∂U 2
∂ U
2 p
dx −
τ
dx dτ +
τ S
dx dτ
2
∂t
∂τ
∂τ ∂x
0 0
0
µ
Zt Z1
2
+p
τ S
p−1
0 0
∂U
∂x
¶3
0 0
∂ 2U
dx dτ + p
∂τ ∂x
Zt Z1
2
τ S
µ
∂V
∂x ∂x
p−1 ∂U
0 0
¶2
∂ 2U
dx dτ = 0
∂τ ∂x
and, using Schwarz’s inequality, we conclude that
t 1
µ
¶
¶
Zt Z1 µ
∂U 2
1 Z Z 2 p ∂ 2U 2
dx dτ ≤
τ
dx dτ
τ S
2
∂τ ∂x
∂τ
0 0
0 0
µ
Zt Z1
+p
2
2
τ S
p−2
0 0
∂U
∂x
¶6
µ
Zt Z1
2
2
dx dτ + p
τ S
0 0
p−2
∂U
∂x
¶2 µ
∂V
∂x
¶4
dx dτ.
From this using (6), (22) and (23) we get
µ
Zt Z1
2
τ S
p
0 0
∂ 2U
∂τ ∂x
¶2
∂ 2U
∂τ ∂x
¶2
∂2V
∂τ ∂x
¶2
∂ 2V
∂τ ∂x
¶2
dx dτ ≤ Ctp+1 .
(30)
dx dτ ≤ Ctp+1 .
(31)
dx dτ ≤ Ctp+1 ,
(32)
Hence
Z1 µ
Zt
τ
p+2
0
0
Analogously,
µ
Zt Z1
2
τ S
0 0
Zt
τ
0
p
Z1 µ
p+2
0
dx dτ ≤ Ctp+1 .
(33)
68
T. JANGVELADZE AND Z. KIGURADZE
Multiplying (9) by t3 ∂ 2 U/∂t2 , applying the formula of integrating by parts
and a priori estimates (6), (22), (30), (31), (32) and (33), we get
Zt Z1
3
τ
µ 2 ¶2
∂ U
∂τ 2
0 0
t 1
·
¸
1 Z Z 3 p ∂ ∂ 2U 2
τ S
dx dτ +
dx dτ
2
∂τ ∂τ ∂x
0 0
µ
Z1
Zt
τ 3 S p−1
+p
0 0
Zt Z1
3
+p
τ S
τ
3
µ 2 ¶2
∂ U
∂τ 2
0 0
Zt
·µ
Z1
µ
∂U
∂x
Zt
Z1
µ
3
p−1
τ S
0 0
Zt Z1
2
τ S
∂V
∂x
¶2 ¸
¶2 µ
µ
∂V
∂x ∂x
µ
∂U ∂V
∂x ∂x
¶2
∂U
∂x
∂ 2U
∂ τ ∂x
¶2
µ
t
S
0
∂2U
∂t ∂x
τ 3 S p−1
3 Z1
+
t
2
0
¶2
µ
Sp
2
dx ≤ 3
τ S
p
0 0
¶2
¶2 ¸µ
∂2U
∂τ ∂x
¶2
dx dτ
¶5
µ
+
∂U
∂x
¶2
¶3 µ
Z1
dx dτ −pt
3
¶2 ¸
∂V
∂x
S
∂ 2U
dx dτ
∂τ ∂x
µ
∂V
∂x ∂x
p−1 ∂U
0
¶2
∂ 2U
dx
∂t ∂x
Zt
µ
Z1
τ 3 S p−1
0 0
∂V
∂x
¶2 µ
∂ 2U
∂τ ∂x
¶2
dx dτ
∂U ∂V ∂ 2 U ∂ 2 V
dx dτ,
∂x ∂x ∂τ ∂x ∂τ ∂x
µ
Zt Z1
∂ 2U
∂t ∂x
∂V
+
∂x
0 0
0 0
p
µ
∂ 2U
dx dτ +p
∂x ∂τ
+2p
Z1
0 0
·µ
¶
Zt Z1
∂ 2U
∂U 2
3 p−2
dx dτ + p(p − 1)
τ S
∂τ ∂x
∂x
Zt Z1
3
t 1
µ
¶
3 Z Z 2 p ∂ 2U 2
dx =
dx dτ
τ S
2
∂τ ∂x
0 0
·µ
∂U
∂x
p−1 ∂U
0 0
µ
¶2
µ
¶3 2
Zt Z1
∂ 2U
∂ U
2 p−1 ∂U
dx + 3p
τ S
dx dτ
∂t ∂x
∂x ∂τ ∂x
0 0
Zt Z1
¶2
¸
¶3
τ 3 S p−2
+p(p − 1)
+
·
∂ ∂ 2U
dx dτ = 0,
∂τ ∂τ ∂x
p−1
τ S
p−1
∂U
∂x
¸
3
0 0
0
+3p
·
∂ ∂2U
dx dτ
∂τ ∂τ ∂x
0
S
+3p
¶2
µ
1
Z1
−pt
¶3
1 Z 3 p ∂ 2U
dx dτ +
tS
2
∂t ∂x
p
+
2
3
µ
∂V
∂x ∂x
p−1 ∂U
0 0
Zt Z1
∂U
∂x
∂ 2U
∂τ ∂x
¶2
p+2
0
0
Zt
µ
Z1
τ 2S p
S p−2 dx + C3
0
τ
dx dτ +C1
Z1
dx + C2 t3
Z1 µ
Zt
0 0
∂2U
∂τ ∂x
∂ 2U
∂τ ∂x
¶2
dx dτ
¶2
dx dτ
STABILIZATION RATE OF SOLUTION OF A NONLINEAR EQUATION
2
τ S
+C4
µ
Zt Z1
Zt Z1
p−2
2
dx dτ +
τ S
0 0
0 0
Zt Z1
τ S
p−4
dx dτ + C7 t
p−1
+ C6
0 0
Z1
τ
µ
τ p+2
0
0
∂ 2U
∂τ ∂x
¶2
Zt
τ S
p−4
0 0
dx dτ + C11
0 0
Z1 µ
τ
p+2
0
Z1 µ
Zt
+C12
τ
p+2
0
0
Z1 µ
ct
µ
τ 2S p
Zt
4
+C10
0
Z1
dx dτ + C9
Zt Z1
p+3
0
∂ 2U
∂τ ∂x
∂ 2U
∂t ∂x
¶2
¶2
¶2
Z1 µ
p+2
0
Zt
+C8
∂ 2U
∂τ ∂x
Zt
4
+C5
p
0
69
dx dτ
∂ 2U
∂τ ∂x
∂ 2U
∂τ ∂x
∂2U
∂τ ∂x
¶2
dx dτ
¶2
dx dτ
¶2
dx dτ
dx dτ ≤ C13 tp+1 ,
dx ≤ Ctp+1 .
From this we have
Z1 µ
0
∂ 2U
∂t ∂x
¶2
dx ≤ Ct−2 .
(34)
Taking into account the relation
1
Z1 Zx 2
∂U (x, t) Z ∂U (y, t)
∂ U (ξ, t)
=
dy +
dξ dy ≤ Ct−1
∂t
∂t
∂t
∂ξ
y
0
Z1 Zx
+
0 y
0
" Z1 µ
#1
¶
2
∂ 2 U (ξ, t)
∂ 2 U (y, t) 2
−1
dξ dy ≤ Ct +
dy ,
∂t ∂ξ
∂t ∂y
0
from (34) we get
∂U (x, t)
= O(t−1 ).
∂t
Analogously, we can show that
∂V (x, t)
= O(t−1 ).
∂t
So the proof of the theorem is over.
Finally we note that estimates similar to (6) and (7) are true for the averaged
integro-differential problem (1)–(3), (5).
70
T. JANGVELADZE AND Z. KIGURADZE
References
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uniqueness of the solution of a class of nonlinear parabolic problems. (Russian) Differentsial’nye Uravneniya 19(1983), No. 7, 1197–1207.
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one nonlinear integro-differential parabolic equation. Reports of Enlarged Session of the
Seminar of I. Vekua Institute of Applied Mathematics 10(1995), No. 1, 36–38.
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Vekua Inst. Appl. Math., Rep. 23(1997), 51–87.
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(Received 4.06.2001; revised 8.10.2001)
Author’s addresses:
T. Jangveladze
I. Vekua Institute of Applied Mathematics
I. Javakhishvili Tbilisi State University
2, University St., Tbilisi 380043
Georgia
E-mail: djante.math@viam net.edu.ge
Z. Kiguradze
Department of Mechanics and Mathematics
I. Javakhishvili bilisi State University
1, University St., Tbilisi 380043
Georgia
E-mail: zura2806@yahoo.com