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On Boundary Value Problem for a Class of Retarded Nonlinear Partial Differential Equations
Journal of Mathematical Analysis and Applications 254, 515᎐523 Ž2001. doi:10.1006rjmaa.2000.7215, available online at http:rrwww.idealibrary.com on
On Boundary Value Problem for a Class of Retarded Nonlinear Partial Differential Equations Alexander Rezounenko Department of Mechanics and Mathematics, Kharko¨ Uni¨ ersity, Kharko¨ 310077, Ukraine Submitted by Konstantin A. Lurie Received October 2, 1998
1. INTRODUCTION This paper is devoted to a problem of nonlinear oscillations of an elastic plate in a potential supersonic gas flow. The case when the inertial forces are essentially weaker than the resistance ones is called a quasistatic case. This problem can be described by a class of retarded quasilinear partial differential equations
␥u ˙ q ⌬2 u y f
žH
⍀
ⵜu Ž x, t .
2
dx ⌬ u q
/
⭸u ⭸ x1
y q Ž ut . s d0 Ž x . , x g ⍀,
t)0
Ž 1.
with the boundary conditions u < ⭸ ⍀ s ⌬ u < ⭸ ⍀ s 0.
Ž 2.
Here ⍀ is a bounded domain in R 2 , x s Ž x 1 , x 2 ., ␥ , , are positive parameters of the system, u ˙ s ⭸⭸ut , and ⌬ is the Laplace operator. Assumptions on the scalar function f Ž s . will be given below. We rely here on the Berger approach to large deflection w1x, Žin w1x f Ž s . is a linear function.. 515 0022-247Xr01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
516
ALEXANDER REZOUNENKO
The retarded term has the form q Ž ut ; x . s
1 2 k
Hy⬁ d H0
= , x2 y
ž
a s
Ž . s
sin y 1 y sin k2
2
x1
d
x1 y k
ž
⭸ x1
q b
⭸ ⭸ x2
U
2
/
u
cos , t y Ž x 1 y .
b s
,
a
⭸
k ⭈ cos
y sin
/
Ž 3.
,
Ž y sin . ,
where ⌿U Ž x . is the extension of ⌿ Ž x . by zero outside of ⍀, and the parameter ) 1 represents the gas velocity, k s ' 2 y 1 . Formula Ž3. shows that the value of retarded term at time t uses values of uŽ s . for s g Ž t y t#, t ., where t# s l Ž y 1.y1 is a time retardation and l is the length of ⍀ along x 1 axis. That is why here and below we use the notation u t s u t Ž . s uŽ t q ., g Žyt#, 0.. The investigation of the considered problem with a Cauchy initial conditions was begun in second order in a time nonretarded setting Ž q Ž u t . ' 0.. The existence and uniqueness theorems have been obtained in w2x; the long-time behaviour for the one dimensional case was investigated by various authors Žsee, e.g., w3᎐5x and the references therein .. The analysis of influence of potential supersonic flow carried out in w6, 7x leads to the retarded equation Ž1.. The Cauchy problem for second order in the time retarded case has been investigated in w8, 9x, where the existence and properties of solutions in different spaces were studied. For a quasistatic formulations see w10, 11x. In w11x the author considered problem Ž1. ᎐ Ž3. with the following initial conditions Žcf. w12x.: u < ts0qs u o ,
u < t g Žyt# , 0. s Ž x, t . .
Ž 4.
In general we do not assume any compatibility conditions between Ž s . and u o . So even if is continuous or piecewise continuous, we do not assume that lim s ª 0 Ž s . s u o . For mechanical models Žas in our case, the model of oscillations of a plate. such initial conditions can describe a strike Žshock. at time moment t s 0 Žsee the discussion of such initial conditions in the finite dimensional case, e.g., in w13x.. It was proved w11, Theorem 2.1x that Ž1. ᎐ Ž4. have an unique solution which is continuous for all t ) 0. From the point of view of applications it is not convenient to evaluate a function in a moment of the strick Ža point of discontinuity of solution.; it
BOUNDARY PROBLEM FOR RETARDED PDEs
517
is more convenient to evaluate it in some moment t s b ) 0 when the solution is continuous. To this end, in finite dimensional problems, i.e., u ˙ s f Ž ut . ,
uŽ t . g R n ,
t G 0,
one can consider the boundary value problem with conditions Žsee, e.g., w13x. u < t g Žyt# , 0. s Ž x, t . ,
u < tsb s u b ,
b ) 0.
In the present paper we formulate an analogous boundary value problem for the infinite dimensional system Ž1. ᎐ Ž3. such that Cauchy conditions Ž4. are a partial case. As an another motivation of the considered infinite dimensional boundary value problem we note that our investigation of systems with impulses has a common background with investigations of continuous dynamical systems based on the recently introduced concept of inertial manifold with delay w15x. However, our methods are different from w13, 15x. Let us first introduce the function spaces we need.
2. FORMULATION OF THE PROBLEM AND RESULT Let e k 4⬁ks1 be an orthonormal basis of L2 Ž ⍀ . consisting of the eigenfunctions of the Dirichlet problem for ⍀: ⌬ e k q k e k s 0,
e k Ž x . s 0 if x g ⭸ ⍀ ,
0 - 1 F 2 F ⭈⭈⭈ .
We use the following scale of spaces:
½
Fs s u s
⬁
⬁
ks1
ks1
Ý u k e k : 5 u 5 2s ' Ý ks u 2k - ⬁
5
,
s g ᑬ.
Ž 5.
We denote by 5 . 5 and Ž . , . . the norm and the inner product in Fo s L2 Ž ⍀ .. Let us define by PN the orthoprojection in the space Fs on the subspace spanned by e1 , . . . , e N 4 and we set Q N ' I y PN . Now we are in a position to give our initial boundary conditions u < t g Žyt# , 0. s Ž x, t . ,
Q N u < ts0qs q0 ,
PN u < tsb s p b ,
b ) 0,
Ž 6. where N is some nonnegative integer. Remark 2.1. The Cauchy conditions Ž4. are a partial case of Ž6. when N s 0.
518
ALEXANDER REZOUNENKO
DEFINITION. A strong solution of problem Ž1. ᎐ Ž3., Ž6. on an interval w0, T x is a vector-function uŽ t . g C Ž0, T ; F1 . l L2 Žyt#, T ; F2 . with derivative u ˙Ž t . g L2 Ž0, T ; Fy2 . if Eq. Ž1. is satisfied almost everywhere in t on w0, T x as an equality in Fy2 and conditions Ž6. hold. Our result is the following THEOREM. Let q0 g Q N F1 , p b g PN F0 , g L2 Žyt#, 0; F2 ., d 0 g F0 , and let f be a local Lipschitz and satisfying the condition inf lim f Ž s . G yC f , sª⬁
with some constant C f . Then there exists b 0 such that for any b F b 0 the problem Ž1. ᎐ Ž3., Ž6. has a strong solution on any inter¨ al w0, T x. This solution is unique and satisfies the property uŽ t . g L2 Ž0, T ; F3 .. Proof of Theorem. Introduce the space Y ' C Ž yt#, b; F1 . l L2 Ž yt#, b; F2 . : y < t g Žyt# , 0. ' 0 4
Ž 7.
with the norm 5 y 5 2Y ' max
2 1
yŽ s.
sg w0, b x
q
b
H0
2 2
yŽ t.
dt.
In the space Y = PN L2 Ž ⍀ . we will use the following norm:
Ž y ; p0 .
2 YN
' max
sg w0, b x
yŽ s.
2 1
q
b
H0
y Ž s.
2 2
ds q 5 p 0 5 12 .
The next assertion is of importance to us Žsee w7, 8x.: LEMMA 2.1.
If uŽ t . g L2 Žyt#, T ; F2q 2 . then
q Ž ut .
2 2
F Ct#
t
Htyt#
uŽ .
2 2q 2
d ,
0 F - 14 ,
and the map u ª q Ž u, t . is linear and continuous from L2 Žyt#, T ; F2q 2 . to L2 Ž0, T ; F2 .. Rewrite Ž1. in the form u ˙Ž t . q Au Ž t . q M Ž u t . s 0.
Ž 8.
BOUNDARY PROBLEM FOR RETARDED PDEs
519
Here A ' Žy⌬ D . 2␥y1 and M Ž u t . ' yf Ž ⵜu Ž t .
2
. ⌬ uŽ t . y
⭸ uŽ t . ⭸ x1
y q Ž u t . q d o ␥y1 .
Write the variation of constants formula for the solution of Ž8.: u Ž t . s eyt A u 0 y
t yŽ ty . A
H0 e
M Ž u . d .
If we write uŽ t . s y Ž t . q ¨ Ž t ., where ¨ Ž t. '
½
eyt A u 0
if t G 0,
Ž t. ,
if t g Ž yt#, 0 . ,
Ž 9.
then y Ž t . should satisfy
¡ ¢0
t yŽ ty . A
yH e y Ž t . s~ 0
M Ž y q ¨ . d ,
if t G 0, if t g Ž yt#, 0 . .
LEMMA 2.2. Let y i g Y such that 5 y i 5 Y F CT and let p 0i g PN L2 Ž ⍀ ., i s 1, 2. Then there exists constant D T ) 0 such that for all g w0, b x one has Ay1 r4 Ž M Ž y1 q ¨1 . y M Ž y2 q ¨2 . . F D T
Ž y 1 ; p10 . y Ž y 2 ; p02 .
Here D T is some constant; ¨ i is defined by Ž9. with u 0i s p 0i q q0 . Proof of Lemma 2.2. Using Lemma 2.1 we obtain Ay1 r4 Ž M Ž y1 q ¨1 . y M Ž y2 q ¨2 . . F CT A1r4 Ž y 1 Ž . y y 2 Ž . . q CT A1r4 Ž ¨ 1 Ž . y ¨ 2 Ž . . q Ct#
ž ž
Hyt#
q Ct#
Hyt#
1r2
A1r2 Ž y 1 Ž s . y y 2 Ž s . .
2
A1r2 Ž ¨ 1 Ž s . y ¨ 2 Ž s . .
2
ds
ds
/ /
1r2
.
YN .
520
ALEXANDER REZOUNENKO
Denote ⌳ i s 2i eigenvalues of the operator A, and using estimate Žsee, e.g., w14x. 5 A␣ eyt A u 5 F
ž
␣y 
␣y
q ⌳1
t
eyt⌳ 1 5 A  u 5
/
Ž 10 .
and definition of ¨ i Ž t . by Ž9., we get
Hyt# F F
A1r2 Ž ¨ 1 Ž s . y ¨ 2 Ž s . .
H0
ds 2
A1r2 eys A Ž p10 y p 02 .
H0
2
ž
1 4s
ds
1r4
q ⌳1
F 5 p10 y p 02 5 12
eys⌳ 1 5 p10 y p 02 5 12 ds
/
H0
ž
1 4s
1r4
q ⌳1
eys⌳ 1 ds s 5 p10 y p 02 5 12 ␣ ;
/
1
ž / 4
.
Here we have denoted
␣ Ž b; k . '
b
H0
ž
k s
k
q ⌳1
eys ⌳ 1 ds.
/
Ž 11 .
Note that for k - 1, ␣ Ž b; k . ª 0 when b ª 0. So we have Ay1 r4 Ž M Ž y1 q ¨1 . y M Ž y2 q ¨2 . . F CTX max 5 y 1 y y 2 5 1 q 5 p10 y p 02 5 1 q
ž
w0, b x
ž
b
H0
y1 Ž s . y y 2 Ž s .
2 2
q5 p10 y p 02 5 1 ␣ Ž b;
1r2
ds
/
1 1r2 4
.
/
.
This completes the proof of Lemma 2.2. Let us fix g L2 Žyt#, 0; F2 ., q0 g Q N F1 , and p b g PN L2 Ž ⍀ .. Consider the map F Ž y; p 0 .: Y = PN L2 Ž ⍀ . ª Y = PN L2 Ž ⍀ . defined as follows:
F Ž y ; p0 . '
¡ ¢0
t yŽ ty . A
~yH0 e
M Ž y q ¨ . d ,
if t G 0; if t g Ž yt#, 0 .
e A b pb q
b A
H0
e PN M Ž y q ¨ . d .
0
.
BOUNDARY PROBLEM FOR RETARDED PDEs
521
Here the function ¨ Ž t . is defined by Ž9. with u 0 s p 0 q q0 g F1. Note that q0 is fixed but p 0 is a variable of F. We deduce the second coordinate of F from the equation
p b ' PN u Ž b . s eyA b p 0 y
b yŽ by . A
H0
e
PN M Ž u . d .
Hence p 0 s e A b p b q H0b e A PN M Ž u . d . A fixed point of operator F gives unknown coordinates PN uŽ0. of initial data uŽ0. and the solution on the interval w0, b x. Having and full uŽ0. we arrive at initial conditions Ž4. and hence one can use all results on the continuation of the solution and its properties obtained in w11x. Let us show that F is contaction in Y = PN L2 Ž ⍀ .. We will denote two coordinates of F as Ž F1; F2 .T . Let us estimate F1 , using Lemma 2.2, Ž10., and Ž11.: A1r4 Ž F1 Ž y 1 ; p10 . Ž t . y F1 Ž y 2 ; p 02 . Ž t . . t
F
H0
F
t
H0
A1r4 eyŽ ty . A Ž M Ž y1 q ¨1 . y M Ž y2 q ¨2 . . d
ž
1 2Ž t y .
1r2
q ⌳1
/
eyŽ ty .⌳ 1 Ay1r4 Ž M Ž y1 q ¨1 . yM Ž y2 q ¨2 . . d
F ␣ b,
1
ž / 2
D T Ž y 1 ; p10 . y Ž y 2 ; p 02 .
YN .
In the same way, we deduce A1r2 Ž F1 Ž y 1 ; p10 . Ž t . y F1 Ž y 2 ; p 02 . Ž t . . F ␣ b,
3
ž / 4
D T Ž y 1 ; p10 . y Ž y 2 ; p 02 .
YN .
Remark 2.2. We need an a priori estimate 5 y i 5 Y F CT to use Lemma 2.2. It can be obtained as in w11, Theorem 2.1x.
522
ALEXANDER REZOUNENKO
Now let us estimate the second coordinate F2 : A1r4 Ž F2 Ž y 1 ; p10 . Ž t . y F2 Ž y 2 ; p 02 . Ž t . . F
b
A1r4 e A PN Ž M Ž y1 q ¨1 . y M Ž y2 q ¨2 . . d
H0
F ⌳1r2 N F DT
b ⌳ N
H0
Ay1r4 Ž M Ž y1 q ¨1 . y M Ž y2 q ¨2 . . d
e
e⌳ N b y 1
Ž y 1 ; p10 . y Ž y 2 ; p02 .
⌳1r2 N
YN .
Combine the last three estimates to get F Ž y 1 ; p10 . y F Ž y 2 ; p 02 . F ␣ Ž b,
1 2 2
.
2 YN
q ␣ Ž b,
3 2 4
.
⌳N b q ⌳y1 y 1. N Že
= D T2 Ž y 1 ; p10 . y Ž y 2 ; p 02 .
2
2
YN .
Choosing b small enough, we obtain a contraction of the map F. The proof of the theorem is complete. COROLLARY. We consider here the same class of strong solutions as in w11x and under the same conditions as nonlinear function f. So all results on long-time asymptotic beha¨ iour of solutions to Ž1. ᎐ Ž3., Ž6., including the existence of a finite dimensional attractor, are ¨ alid.
REFERENCES 1. M. Berger, A new approach to the large deflection of a plate, J. Appl. Mech. 22 Ž1955., 465᎐472. 2. Zh. N. Dmitrieva, On the theory of nonlinear oscillations of thin rectangular plate, J. So¨ iet Math. 22 Ž1978., 48᎐51. 3. I. D. Chueshov, ‘‘The Mathematical Foundations of Nonregular Oscillation Theory of Infinite Dimensional Systems ŽLecture Notes.,’’ Kharkov Univ. Press, Kharkov, 1991. 4. J. K. Hale, ‘‘Asymptotic Behavior of Dissipative Systems,’’ Amer. Math. Soc., Providence, 1988. 5. D. Sevcovic, Existence and limiting behaviour for damped nonlinear evolution equations with nonlocal terms, Comment. Math. Uni¨ . Carolinae 31, No. 2 Ž1990., 283᎐293. 6. E. A. Krasil’shchikova, ‘‘The Thin Wing in a Compressible Flow,’’ Nauka, Moscow, 1978. 7. I. D. Chueshov, On a certain system of equations with delay, occurring in aeroelasticity, J. So¨ iet Math. 58 Ž1992., 385᎐390.
BOUNDARY PROBLEM FOR RETARDED PDEs
523
8. I. D. Chueshov and A. V. Rezounenko, Global attractors for a class of retarded quasilinear partial differential equations, C. R. Acad. Sci. Paris Ser. I 321 Ž1995., 607᎐612 Ždetailed version: Mat. Fiz. Anal. Geom. 2, No. 3 Ž1995., 363᎐383.. 9. L. Boutet de Monvel, I. D. Chueshov, and A. V. Rezounenko, Long-time behaviour of strong solutions for a class of retarded nonlinear P.D.E.s, Comm. Partial Differential Equations 22 Ž1997., 1453᎐1474. 10. I. D. Chueshov, A problem on non-linear oscillations of shallow shell in a quasistatic formulation, Math. Notes 47 Ž1990., 401᎐407. 11. A. Rezounenko, On singular limit dynamics for a class of retarded nonlinear partial differential equations, Mat. Fiz. Anal. Geom. 4, No. 1r2 Ž1997., 193᎐211. 12. J. K. Hale, ‘‘Theory of Functional Differential Equations,’’ Springer-Verlag, Berlinr HeidelbergrNew York, 1977. 13. N. B. Azbelev, V. P. Maksimov, and L. F. Rakhmatulina, ‘‘Introduction to the Theory of Functional Differential Equations,’’ Nauka, Moscow, 1991. 14. R. Temam, ‘‘Infinite Dimensional Dynamical Systems in Mechanics and Physics,’’ Springer-Verlag, BerlinrHeidelbergrNew York, 1988. 15. A. Debussche and R. Temam, Some new generalizations of inertial manifolds, Discrete Continuous Dynam. Systems 2 Ž1996., 543᎐558.
On Boundary Value Problem for a Class of Retarded Nonlinear Partial Differential Equations Alexander Rezounenko Department of Mechanics and Mathematics, Kharko¨ Uni¨ ersity, Kharko¨ 310077, Ukraine Submitted by Konstantin A. Lurie Received October 2, 1998
1. INTRODUCTION This paper is devoted to a problem of nonlinear oscillations of an elastic plate in a potential supersonic gas flow. The case when the inertial forces are essentially weaker than the resistance ones is called a quasistatic case. This problem can be described by a class of retarded quasilinear partial differential equations
␥u ˙ q ⌬2 u y f
žH
⍀
ⵜu Ž x, t .
2
dx ⌬ u q
/
⭸u ⭸ x1
y q Ž ut . s d0 Ž x . , x g ⍀,
t)0
Ž 1.
with the boundary conditions u < ⭸ ⍀ s ⌬ u < ⭸ ⍀ s 0.
Ž 2.
Here ⍀ is a bounded domain in R 2 , x s Ž x 1 , x 2 ., ␥ , , are positive parameters of the system, u ˙ s ⭸⭸ut , and ⌬ is the Laplace operator. Assumptions on the scalar function f Ž s . will be given below. We rely here on the Berger approach to large deflection w1x, Žin w1x f Ž s . is a linear function.. 515 0022-247Xr01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
516
ALEXANDER REZOUNENKO
The retarded term has the form q Ž ut ; x . s
1 2 k
Hy⬁ d H0
= , x2 y
ž
a s
Ž . s
sin y 1 y sin k2
2
x1
d
x1 y k
ž
⭸ x1
q b
⭸ ⭸ x2
U
2
/
u
cos , t y Ž x 1 y .
b s
,
a
⭸
k ⭈ cos
y sin
/
Ž 3.
,
Ž y sin . ,
where ⌿U Ž x . is the extension of ⌿ Ž x . by zero outside of ⍀, and the parameter ) 1 represents the gas velocity, k s ' 2 y 1 . Formula Ž3. shows that the value of retarded term at time t uses values of uŽ s . for s g Ž t y t#, t ., where t# s l Ž y 1.y1 is a time retardation and l is the length of ⍀ along x 1 axis. That is why here and below we use the notation u t s u t Ž . s uŽ t q ., g Žyt#, 0.. The investigation of the considered problem with a Cauchy initial conditions was begun in second order in a time nonretarded setting Ž q Ž u t . ' 0.. The existence and uniqueness theorems have been obtained in w2x; the long-time behaviour for the one dimensional case was investigated by various authors Žsee, e.g., w3᎐5x and the references therein .. The analysis of influence of potential supersonic flow carried out in w6, 7x leads to the retarded equation Ž1.. The Cauchy problem for second order in the time retarded case has been investigated in w8, 9x, where the existence and properties of solutions in different spaces were studied. For a quasistatic formulations see w10, 11x. In w11x the author considered problem Ž1. ᎐ Ž3. with the following initial conditions Žcf. w12x.: u < ts0qs u o ,
u < t g Žyt# , 0. s Ž x, t . .
Ž 4.
In general we do not assume any compatibility conditions between Ž s . and u o . So even if is continuous or piecewise continuous, we do not assume that lim s ª 0 Ž s . s u o . For mechanical models Žas in our case, the model of oscillations of a plate. such initial conditions can describe a strike Žshock. at time moment t s 0 Žsee the discussion of such initial conditions in the finite dimensional case, e.g., in w13x.. It was proved w11, Theorem 2.1x that Ž1. ᎐ Ž4. have an unique solution which is continuous for all t ) 0. From the point of view of applications it is not convenient to evaluate a function in a moment of the strick Ža point of discontinuity of solution.; it
BOUNDARY PROBLEM FOR RETARDED PDEs
517
is more convenient to evaluate it in some moment t s b ) 0 when the solution is continuous. To this end, in finite dimensional problems, i.e., u ˙ s f Ž ut . ,
uŽ t . g R n ,
t G 0,
one can consider the boundary value problem with conditions Žsee, e.g., w13x. u < t g Žyt# , 0. s Ž x, t . ,
u < tsb s u b ,
b ) 0.
In the present paper we formulate an analogous boundary value problem for the infinite dimensional system Ž1. ᎐ Ž3. such that Cauchy conditions Ž4. are a partial case. As an another motivation of the considered infinite dimensional boundary value problem we note that our investigation of systems with impulses has a common background with investigations of continuous dynamical systems based on the recently introduced concept of inertial manifold with delay w15x. However, our methods are different from w13, 15x. Let us first introduce the function spaces we need.
2. FORMULATION OF THE PROBLEM AND RESULT Let e k 4⬁ks1 be an orthonormal basis of L2 Ž ⍀ . consisting of the eigenfunctions of the Dirichlet problem for ⍀: ⌬ e k q k e k s 0,
e k Ž x . s 0 if x g ⭸ ⍀ ,
0 - 1 F 2 F ⭈⭈⭈ .
We use the following scale of spaces:
½
Fs s u s
⬁
⬁
ks1
ks1
Ý u k e k : 5 u 5 2s ' Ý ks u 2k - ⬁
5
,
s g ᑬ.
Ž 5.
We denote by 5 . 5 and Ž . , . . the norm and the inner product in Fo s L2 Ž ⍀ .. Let us define by PN the orthoprojection in the space Fs on the subspace spanned by e1 , . . . , e N 4 and we set Q N ' I y PN . Now we are in a position to give our initial boundary conditions u < t g Žyt# , 0. s Ž x, t . ,
Q N u < ts0qs q0 ,
PN u < tsb s p b ,
b ) 0,
Ž 6. where N is some nonnegative integer. Remark 2.1. The Cauchy conditions Ž4. are a partial case of Ž6. when N s 0.
518
ALEXANDER REZOUNENKO
DEFINITION. A strong solution of problem Ž1. ᎐ Ž3., Ž6. on an interval w0, T x is a vector-function uŽ t . g C Ž0, T ; F1 . l L2 Žyt#, T ; F2 . with derivative u ˙Ž t . g L2 Ž0, T ; Fy2 . if Eq. Ž1. is satisfied almost everywhere in t on w0, T x as an equality in Fy2 and conditions Ž6. hold. Our result is the following THEOREM. Let q0 g Q N F1 , p b g PN F0 , g L2 Žyt#, 0; F2 ., d 0 g F0 , and let f be a local Lipschitz and satisfying the condition inf lim f Ž s . G yC f , sª⬁
with some constant C f . Then there exists b 0 such that for any b F b 0 the problem Ž1. ᎐ Ž3., Ž6. has a strong solution on any inter¨ al w0, T x. This solution is unique and satisfies the property uŽ t . g L2 Ž0, T ; F3 .. Proof of Theorem. Introduce the space Y ' C Ž yt#, b; F1 . l L2 Ž yt#, b; F2 . : y < t g Žyt# , 0. ' 0 4
Ž 7.
with the norm 5 y 5 2Y ' max
2 1
yŽ s.
sg w0, b x
q
b
H0
2 2
yŽ t.
dt.
In the space Y = PN L2 Ž ⍀ . we will use the following norm:
Ž y ; p0 .
2 YN
' max
sg w0, b x
yŽ s.
2 1
q
b
H0
y Ž s.
2 2
ds q 5 p 0 5 12 .
The next assertion is of importance to us Žsee w7, 8x.: LEMMA 2.1.
If uŽ t . g L2 Žyt#, T ; F2q 2 . then
q Ž ut .
2 2
F Ct#
t
Htyt#
uŽ .
2 2q 2
d ,
0 F - 14 ,
and the map u ª q Ž u, t . is linear and continuous from L2 Žyt#, T ; F2q 2 . to L2 Ž0, T ; F2 .. Rewrite Ž1. in the form u ˙Ž t . q Au Ž t . q M Ž u t . s 0.
Ž 8.
BOUNDARY PROBLEM FOR RETARDED PDEs
519
Here A ' Žy⌬ D . 2␥y1 and M Ž u t . ' yf Ž ⵜu Ž t .
2
. ⌬ uŽ t . y
⭸ uŽ t . ⭸ x1
y q Ž u t . q d o ␥y1 .
Write the variation of constants formula for the solution of Ž8.: u Ž t . s eyt A u 0 y
t yŽ ty . A
H0 e
M Ž u . d .
If we write uŽ t . s y Ž t . q ¨ Ž t ., where ¨ Ž t. '
½
eyt A u 0
if t G 0,
Ž t. ,
if t g Ž yt#, 0 . ,
Ž 9.
then y Ž t . should satisfy
¡ ¢0
t yŽ ty . A
yH e y Ž t . s~ 0
M Ž y q ¨ . d ,
if t G 0, if t g Ž yt#, 0 . .
LEMMA 2.2. Let y i g Y such that 5 y i 5 Y F CT and let p 0i g PN L2 Ž ⍀ ., i s 1, 2. Then there exists constant D T ) 0 such that for all g w0, b x one has Ay1 r4 Ž M Ž y1 q ¨1 . y M Ž y2 q ¨2 . . F D T
Ž y 1 ; p10 . y Ž y 2 ; p02 .
Here D T is some constant; ¨ i is defined by Ž9. with u 0i s p 0i q q0 . Proof of Lemma 2.2. Using Lemma 2.1 we obtain Ay1 r4 Ž M Ž y1 q ¨1 . y M Ž y2 q ¨2 . . F CT A1r4 Ž y 1 Ž . y y 2 Ž . . q CT A1r4 Ž ¨ 1 Ž . y ¨ 2 Ž . . q Ct#
ž ž
Hyt#
q Ct#
Hyt#
1r2
A1r2 Ž y 1 Ž s . y y 2 Ž s . .
2
A1r2 Ž ¨ 1 Ž s . y ¨ 2 Ž s . .
2
ds
ds
/ /
1r2
.
YN .
520
ALEXANDER REZOUNENKO
Denote ⌳ i s 2i eigenvalues of the operator A, and using estimate Žsee, e.g., w14x. 5 A␣ eyt A u 5 F
ž
␣y 
␣y
q ⌳1
t
eyt⌳ 1 5 A  u 5
/
Ž 10 .
and definition of ¨ i Ž t . by Ž9., we get
Hyt# F F
A1r2 Ž ¨ 1 Ž s . y ¨ 2 Ž s . .
H0
ds 2
A1r2 eys A Ž p10 y p 02 .
H0
2
ž
1 4s
ds
1r4
q ⌳1
F 5 p10 y p 02 5 12
eys⌳ 1 5 p10 y p 02 5 12 ds
/
H0
ž
1 4s
1r4
q ⌳1
eys⌳ 1 ds s 5 p10 y p 02 5 12 ␣ ;
/
1
ž / 4
.
Here we have denoted
␣ Ž b; k . '
b
H0
ž
k s
k
q ⌳1
eys ⌳ 1 ds.
/
Ž 11 .
Note that for k - 1, ␣ Ž b; k . ª 0 when b ª 0. So we have Ay1 r4 Ž M Ž y1 q ¨1 . y M Ž y2 q ¨2 . . F CTX max 5 y 1 y y 2 5 1 q 5 p10 y p 02 5 1 q
ž
w0, b x
ž
b
H0
y1 Ž s . y y 2 Ž s .
2 2
q5 p10 y p 02 5 1 ␣ Ž b;
1r2
ds
/
1 1r2 4
.
/
.
This completes the proof of Lemma 2.2. Let us fix g L2 Žyt#, 0; F2 ., q0 g Q N F1 , and p b g PN L2 Ž ⍀ .. Consider the map F Ž y; p 0 .: Y = PN L2 Ž ⍀ . ª Y = PN L2 Ž ⍀ . defined as follows:
F Ž y ; p0 . '
¡ ¢0
t yŽ ty . A
~yH0 e
M Ž y q ¨ . d ,
if t G 0; if t g Ž yt#, 0 .
e A b pb q
b A
H0
e PN M Ž y q ¨ . d .
0
.
BOUNDARY PROBLEM FOR RETARDED PDEs
521
Here the function ¨ Ž t . is defined by Ž9. with u 0 s p 0 q q0 g F1. Note that q0 is fixed but p 0 is a variable of F. We deduce the second coordinate of F from the equation
p b ' PN u Ž b . s eyA b p 0 y
b yŽ by . A
H0
e
PN M Ž u . d .
Hence p 0 s e A b p b q H0b e A PN M Ž u . d . A fixed point of operator F gives unknown coordinates PN uŽ0. of initial data uŽ0. and the solution on the interval w0, b x. Having and full uŽ0. we arrive at initial conditions Ž4. and hence one can use all results on the continuation of the solution and its properties obtained in w11x. Let us show that F is contaction in Y = PN L2 Ž ⍀ .. We will denote two coordinates of F as Ž F1; F2 .T . Let us estimate F1 , using Lemma 2.2, Ž10., and Ž11.: A1r4 Ž F1 Ž y 1 ; p10 . Ž t . y F1 Ž y 2 ; p 02 . Ž t . . t
F
H0
F
t
H0
A1r4 eyŽ ty . A Ž M Ž y1 q ¨1 . y M Ž y2 q ¨2 . . d
ž
1 2Ž t y .
1r2
q ⌳1
/
eyŽ ty .⌳ 1 Ay1r4 Ž M Ž y1 q ¨1 . yM Ž y2 q ¨2 . . d
F ␣ b,
1
ž / 2
D T Ž y 1 ; p10 . y Ž y 2 ; p 02 .
YN .
In the same way, we deduce A1r2 Ž F1 Ž y 1 ; p10 . Ž t . y F1 Ž y 2 ; p 02 . Ž t . . F ␣ b,
3
ž / 4
D T Ž y 1 ; p10 . y Ž y 2 ; p 02 .
YN .
Remark 2.2. We need an a priori estimate 5 y i 5 Y F CT to use Lemma 2.2. It can be obtained as in w11, Theorem 2.1x.
522
ALEXANDER REZOUNENKO
Now let us estimate the second coordinate F2 : A1r4 Ž F2 Ž y 1 ; p10 . Ž t . y F2 Ž y 2 ; p 02 . Ž t . . F
b
A1r4 e A PN Ž M Ž y1 q ¨1 . y M Ž y2 q ¨2 . . d
H0
F ⌳1r2 N F DT
b ⌳ N
H0
Ay1r4 Ž M Ž y1 q ¨1 . y M Ž y2 q ¨2 . . d
e
e⌳ N b y 1
Ž y 1 ; p10 . y Ž y 2 ; p02 .
⌳1r2 N
YN .
Combine the last three estimates to get F Ž y 1 ; p10 . y F Ž y 2 ; p 02 . F ␣ Ž b,
1 2 2
.
2 YN
q ␣ Ž b,
3 2 4
.
⌳N b q ⌳y1 y 1. N Že
= D T2 Ž y 1 ; p10 . y Ž y 2 ; p 02 .
2
2
YN .
Choosing b small enough, we obtain a contraction of the map F. The proof of the theorem is complete. COROLLARY. We consider here the same class of strong solutions as in w11x and under the same conditions as nonlinear function f. So all results on long-time asymptotic beha¨ iour of solutions to Ž1. ᎐ Ž3., Ž6., including the existence of a finite dimensional attractor, are ¨ alid.
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BOUNDARY PROBLEM FOR RETARDED PDEs
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