Invariant manifolds for coupled nonlinear parabolic-hyperbolic partial differential equations
We consider an abstract system of coupled nonlinear parabolic-hyperbolic partial differential equations. This system describes, e.g., thermoelastic phenomena in various physical bodies. Several results on the existence of invariant exponentially attracting manifolds for similar problems were obtaine...
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| author | Fastovska, T. B. Фастовська, Т. В. |
| author_facet | Fastovska, T. B. Фастовська, Т. В. |
| author_sort | Fastovska, T. B. |
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| datestamp_date | 2020-03-18T20:02:57Z |
| description | We consider an abstract system of coupled nonlinear parabolic-hyperbolic partial differential equations. This system describes, e.g., thermoelastic phenomena in various physical bodies. Several results on the existence of invariant exponentially attracting manifolds for similar problems were obtained earlier. In the present paper, we prove the existence of such an invariant manifold under less restrictive conditions for a broader class of problems. |
| first_indexed | 2026-03-24T02:47:40Z |
| format | Article |
| fulltext |
UDC 517.94
T. B. Fastovska (Karazin Nat. Univ., Kharkiv)
INVARIANT MANIFOLDS
FOR COUPLED NONLINEAR PARABOLIC-HYPERBOLIC
PARTIAL DIFFERENTIAL EQUATIONS
INVARIANTNI MNOHOVYDY POV’QZANYX
NELINIJNYX PARABOLIKO-HIPERBOLIÇNYX RIVNQN|
Z ÇASTYNNYMY POXIDNYMY
We consider an abstract system of parabolic-hyperbolic coupled nonlinear partial differential equations.
This system describes, for instance, thermoelastic phenomena in various physical bodies. Several results
on the existence of invariant exponentially attracting manifolds for similar problems have been obtained
earlier. In the present paper, we prove the existence of this invariant manifold under less restrictive
conditions for a wider class of problems.
Rozhlqnuto abstraktnu systemu paraboliko-hiperboliçnyx pov’qzanyx nelinijnyx rivnqn\ z ças-
tynnymy poxidnymy. Cq systema opysu[, napryklad, termopruΩni qvywa v riznyx fizyçnyx
tilax. Deqki rezul\taty wodo isnuvannq invariantnyx mnohovydiv, wo eksponencial\no prytqhu-
gt\, dlq zadaç podibnoho typu bulo otrymano raniße. V danij roboti dovedeno isnuvannq c\oho
invariantnoho mnohovydu za menß obmeΩuval\nyx umov dlq bil\ß ßyrokoho klasu zadaç.
Introduction. We consider an abstract system of coupled parabolic-hyperbolic
differential equations
Γ w Aw F w wtt t+ = ( ), , θ , t > 0, in H, (1)
θ η θ θt t tL G w w K w w+ = ( ) + ( ), , , , t > 0, in E, (2)
where H and E are infinite-dimensional separable real Hilbert spaces and η is a
positive constant. We assume the following hypotheses to hold.
A1. Γ and A are linear positive self-adjoint operators in H with domains D( )Γ
and D A( ) , respectively, such that
D A1 2/( ) ⊂ D Γ1 2/( ) .
A2. L is a linear positive self-adjoint operator in E with discrete spectrum, i.e.,
there exists an orthonormal basis ek{ } in E such that
Le ek k k= λ , 0 < λ1 ≤ λ2 ≤ … , lim
k
k
→∞
= ∞λ . (3)
A3. F and G are nonlinear globally Lipschitz mappings
F D A D D L D: ( )/ / / *1 2 1 2 1 2( ) × ( ) × → ( )[ ]Γ Γα ,
G D A D D L E: ( )/ /1 2 1 2( ) × ( ) × →Γ α
for some 0 ≤ α < 1, i.e., there exist positive constants MF and MG such that
F w w F w w
D0 1 0 1 1 2, , ˜ , ˜ , ˜
/ *θ θ( ) − ( ) ( )[ ]Γ
≤
≤ M A w w w w LF H H E
1 2
0 0
2 1 2
1 1
2 2 1 2
/ /
/
˜ ˜ ( ˜ )−( ) + −( ) + −( )Γ α θ θ (4)
and
G w w G w w
E0 1 0 1, , ˜ , ˜ , ˜θ θ( ) − ( ) ≤
≤ M A w w w w LG H H E
1 2
0 0
2 1 2
1 1
2 2 1 2
/ /
/
˜ ˜ ( ˜ )−( ) + −( ) + −( )Γ α θ θ . (5)
© T. B. FASTOVSKA, 2005
1684 ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
INVARIANT MANIFOLDS FOR COUPLED NONLINEAR PARABOLIC-HYPERBOLIC … 1685
A4. The mapping
K D A D D L: ( )/ / *1 2 1 2( ) × ( ) × [ ]Γ β
possesses the property
L K w w K w w
E
− ( ) − ( )( )β
0 1 0 1, ˜ , ˜ ≤
≤ M A w w w wK H H
1 2
0 0
2 1 2
1 1
2 1 2
/ /
/
˜ ˜−( ) + −( )( )Γ (6)
for some 0 ≤ β ≤ 1 – α, where MK is a positive constant.
System (1), (2) is an abstract representation of certain models of thermoelasticity
(see, e.g., [1]).
The goal of this paper is to find sufficient conditions for existence of asymptotically
stable invariant manifold of the dynamical system generated by (1), (2) and relying on
this fact to formulate a reduction principle for the system considered. This principle
allows us to show that the long-time behavior of system (1), (2) is completely
determined by a nonlinear elastic system and (possibly) a finite-dimensional heat-
conduction equation. For a discussion of a general idea of reduction principles we refer
to [2].
A similar problem for coupled parabolic-hyperbolic partial differential equations
was studied in [3] (but under rather restrictive hypotheses in the right-hand sides) and
in [4] under the condition Γ = I. For instance, in [4] it was proved that the
exponentially attracting invariant surface of the form
M = ( ) ∈ ( ) × ( ) ∈{ }w w w w w w D A D w w E, , ( , ) : ( , ) , ( , )/ /Φ Γ Φ1 2 1 2 (7)
exists provided that
η
λ
λ λα α β> + +[ ]+2
1
1 1M M MF G K , (8)
where λ1 is the minimal point of the spectrum of L.
Our goal is to find less restrictive condition on the constants of the problem which,
nevertheless, allows to prove reduction principle. Instead of relations between the
diffusivity parameter η and Lipschitz constants in (4) – (6), we obtain a spectral
condition. It allows to dispose of any conditions on the relation between η and MF .
In some cases (for certain parameters α and β and space E ), we also have no
conditions on MG /η and MK /η.
The paper is organized as follows. Section 1 contains the statement of our main
result on the existence of invariant manifold (Theorem 1), in Section 2 we prove some
auxiliary lemmas and the main theorem, in Section 3 we apply the results obtained to
some parabolic-hyperbolic problems.
1. Statement of main result. Rewrite system (1), (2) in the following way:
d
dt
V V V+ =� �( ) , t > 0, (9)
where V = w t w t tt
T( ), ( ), ( )θ( ) ,
� =
0 0
0 0
0 0
1
−
−
I
A
L
Γ
η
is an operator with the domain D A1 2/( ) × D A1 2/( ) × D L( ), and
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
1686 T. B. FASTOVSKA
�( )V =
0
1Γ− ( )
( ) + ( )
F w w
G w w K w w
t
t t
, ,
, , ,
θ
θ
.
We consider equation (9) with the initial condition
V V
t = =0 0 (10)
in the scale of spaces H σ = D A1 2/( ) × D Γ1 2/( ) × D L( )σ (where σ ∈ R ) equipped
with the norms
V A w w L
H H Eσ
σθ= + +( )1 2
0
2 1 2
1
2
0
2 1 2
/ /
/
Γ ,
where V = w w0 1 0, , θ( ) . The linear problem
d
dt
V V+ =� 0 (11)
generates in the spaces H σ C0 -semigroup
e t−� =
U
e
t
Lt
0
0 −
η , (12)
where Ut is a C0 -group in the space D A1 2/( ) × D Γ1 2/( ) generated by the equation
Γ w Awtt + = 0, t > 0, in H. (13)
Definition 1. A function V t( ) is said to be a mild solution to problem (9), (10)
on the interval 0, T[ ] i f V t( ) ∈ C T0 1 2, ; /[ ]( )−H α ∩ L T0, ;[ ]( )H α , V( )0 = V0
and for almost all t ∈ 0, T[ ]
V t( ) = e
t−� V0 +
0
t
te V d∫ − − ( )� �( ) ( )τ τ τ .
The contraction principle allows to establish the following result on the existence of
mild solution to (9), (10) (see [4] for the case Γ = I ):
Proposition 1. Let V0 ∈ H α −1 2/ and let one of the following assertions take
place: α + β < 1 or α + β = 1 and MK < η, where MK is the constant
from (6).
Then there exists the unique mild solution of the Cauchy problem (9), (10) and,
for any σ such that α – 1
2
≤ σ < min ,1 1
2
−
β ,
V C Tt ∈ [ ]( )0, ; H σ . (14)
It follows from Proposition 1 that for any α – 1
2
≤ σ < min ,1 1
2
−
β problem (9),
(10) generates a dynamical system H σ, St( ) with the evolution operators S Vt = V t( ),
where V t( ) = w t w t tt( ), ( ), ( )θ( ) is a mild solution to (1), (2) with the initial data V =
= w w0 1 0, , θ( ) .
Let Pn be an orthoprojector onto lin e e en1 2, , ,…{ } and Qn = I – Pn , where
n ∈ N and ei{ } is an orthonormal basis of eigenfunctions of L. We can define such
orthoprojectors in each of the spaces D L( )σ due to the fact that these spaces can be
identified with the spaces of formal rows
k k kc e=
∞∑{ 1
:
k k kc=
∞∑ 1
2 2λ σ < ∞}. We also
denote P0 = 0 and λ0 = 0.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
INVARIANT MANIFOLDS FOR COUPLED NONLINEAR PARABOLIC-HYPERBOLIC … 1687
Now we state our main result.
Theorem 1. Let in addition to hypotheses A1 – A4 the following spectral
condition hold:
A5. There exists N ∈ N ∪ {0} such that
η >
2
1
MF
N Nλ λ+ +
+
2 1
1
MG N N
N N
λ λ
λ λ
α α
+
+
+( )
−
+
2 1
1
MK N N
N N
λ λ
λ λ
α β α β
+
+ +
+
+( )
−
. (15)
We define the orthoprojector
P =
I
I
PN
0 0
0 0
0 0
in the space H 0 and the orthoprojector Q = I – P . Then, by any σ satisfying
the inequality α – 1
2
≤ σ < min ,1 1
2
−
β , there exists the function Φ: D A1 2/( ) ×
× D Γ1 2/( ) × P D LN ( )σ → Q D LN ( )σ such that
L W W C W W
E
σ
σ σ
Φ Φ( ) ( )1 2 1 2−( ) ≤ − H (16)
for any W1, W2 ∈ H σ , where Cσ is a positive constant. The surface
M = w w w w w w D A D P D LN, , ( , , ) : ( , , ) ( )/ /θ θ θ σ+( ) ∈ ( ) × ( ) ×{ Φ Γ1 2 1 2 ,
Φ( , , ) ( )w w Q D LNθ σ∈ } , (17)
is invariant with respect to the semigroup St in the space Hσ , i.e., StM ⊆ M
and exponentially attracting, i.e., for any mild solution V t( ) to problem (9) there
exists VM ∈ M such that
0
2 2
01
∞
∫ − < +( )e V t S V C Vt
t
µ
α σ( ) M (18)
and
V t S Vt( ) − M σ < Ce Vt− +( )µ
σ1 0 , t > 0, (19)
where µ = η
λ λN N+ +1
2
.
Remark 1. In the case of N = 0, condition (15) coincides with (8) and the
manifold M given by (17) transforms to form (7). Therefore, our Theorem 1 is a
generalization of the result from [4] for the case Γ ≠ I.
We also note that if α = β = 0 and λN +1 – λN → ∞ , there exists N0 such that
condition (15) holds for any N ≥ N0. Thus, in this case an exponentially attracting
invariant manifold exists for any choice of parameters of the problem.
2. Construction of the invariant manifold. To construct an invariant manifold,
we consider (following [5]) the integral equation
V t B V tW( ) [ ]( )= , t ≤ 0, (20)
where B VW[ ] = J VW �( )[ ]. Here, J VW[ ] is as follows:
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
1688 T. B. FASTOVSKA
J V t e W e PV d e QV dW
t
t
t
t
t[ ]( ) ( ) ( )( ) ( )= − +− − −
−∞
− −∫ ∫� � �
0
τ ττ τ τ τ (21)
and W ∈ PH α .
We seek the solution of equation (20) in the space
Y V e V Lt
α
µ
α= ∈ − ∞( ]( ){ }: , ;2 0 H ,
where ηλN < µ < ηλN +1.
To prove Theorem 1, we need some preliminaries.
Lemma 1.
U w w w wt
T
D A D
T
D A D
( , ) ( , )/ / / /0 1 0 11 2 1 2 1 2 1 2( ) × ( ) ( ) × ( )=
Γ Γ
, t ∈ R, (22)
e P e PLt
N D L N
t
N E
N− ≤η σ ηλθ λ θσ( )
, t ∈ R, σ ≥ 0, (23)
e P e e PLt
N D L N
t t
N E
N− − −≤ +[ ]η σ ηλ σ ηλθ λ λ θσ( ) 1
1 , t ∈ R, σ < 0, (24)
e QV
t
e QVt
N
tN−
+
−≤
+ ( )
+�
σ σ
σ σ ηλ
η
σ ηλ1
1 0
1 , t > 0, σ > 0, (25)
e QV e QVt
N
tN−
+
−≤ +�
σ
σ ηλλ 1 0
1 , t > 0, σ ≤ 0. (26)
The proof is standard. We refer to [5, p. 88] and [6, p. 425] for details.
Lemma 2. Let a positive self-adjoint operator L be a generator of a strongly
continuous semigroup e t−L in a Hilbert space �. Assume that λmin > 0 is the
minimal point of the spectrum of L. Then, for any 0 ≤ β ≤ 1 and µ ≥ 0, the
mapping
f t J f t e I f d
t
t( ) ( )( ) ( ) ( )( )→ = +
− ∞
− −∫L
L Lβ τ βµ τ τ
is continuous from L2( ; )R � into
L D2 1
R; L −( )( )β and the estimate
R R
∫ ∫+( ) ≤ +( ) +
L Lµ λ µ
λ
α β
α β
I J f t dt f t dt( )( ) ( )min
( )
min
� �
2 2
2
2
holds for any –β ≤ α ≤ 1 – β.
For the proof of the lemma we refer to [4].
Lemma 3. For every W ∈ PH σ a n d σ ∈ 0 1,[ ], the operator JW is
continuous from Y0 into Yσ and, for any V1, V2 ∈ Y0 , the estimates
J PV J PV PV PVW W Y
N
N
Y1 2 1 2
1
0
[ ] − [ ] ≤ +
−
−
σ µ
λ
µ ηλ
σ
(27)
and
J QV J QV QV QVW W Y
N
N
Y1 2
1
1
1 2 0
[ ] − [ ] ≤
−
−+
+σ
λ
ηλ µ
σ
(28)
hold.
Proof. First, we prove relation (27):
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
INVARIANT MANIFOLDS FOR COUPLED NONLINEAR PARABOLIC-HYPERBOLIC … 1689
J PV J PVW W Y1 2[ ] − [ ]
σ
≤
− ∞
−∫ ∫ −
0 0
1 2 0
2 1 2
t
te e PV PV d dtµ τ µτ τ( )
/
+
+ λ τσ µ ηλ τ µτ
N
t
te e PV PV d dtN
− ∞
− −∫ ∫ −
0 0
1 2 0
2 1 2
( )( )
/
.
Each of the integrals on the right-hand side has the form
R∫ ( * )( )e f dτ τ , where
e t( ) = e tδ for t ≤ 0 and e t( ) ≡ 0 for t > 0. Hence, using the Fourier transformation
and Plancherel formula, we get (27) (for a similar argument see also [4] (Lemma 2.2)).
Similarly, using Lemma 2, we obtain
J QV J QVW W Y1 2[ ] − [ ]
σ
≤
≤ 1
0
1 2 0
2 1 2
η
η τσ
σ η µ τ µτ
− ∞ − ∞
− − −∫ ∫ −
t
L tL e e QV QV d dt( ) ( )( )
/
≤
≤
λ
ηλ µ
σ
N
N
YQV QV+
+ −
−1
1
1 2 0
.
It is easy to see that relations (27) and (28) entail continuity of JW .
Lemma 4. For every W ∈ PH α , the operator BW is continuous from Yα
into itself and
B V B V W W V VW W Y N Y1 21 2 1 1 2 1 21[ ] [ ] ( )− ≤ + +( ) − + −− −
α α
λ λ ρ µα σ α σ
σ
for every W1, W2 ∈ PH α and V1, V2 ∈ Yα , where
ρ µ
µ
λ
µ ηλ
λ
µ ηλ
λ
ηλ µ
λ
ηλ µ
α α β α α β
( ) = +
−
+
−
+
−
+
−
+
+
+
+
+
+
M M M M M
F N G
N
N K
N
N G
N
N K
N
1
1
1
1
.
Proof. We can rewrite BW in the following way:
B VW[ ] = J F V P G V L L P K w wW N N0 11, [ ], [ ] ( ) ,Γ− −+ [ ][ ]
η
ηβ
β β +
+
1 0 0
ηβ
µ β µ β, , e J e L Q Kt t
N
− −( )( )L +
0 0 0, , [ ]e J e Q G Vt t
N
− ( )( )µ µ
L ,
where
L = −η µL I .
Consequently,
B V B VW W Y1 21 2[ ] − [ ]
α
≤
≤
− ∞
− − − −∫ +[ ] −
0
1
2
1 2
2
1 2
1λ λ θ θα σ µ ηλ α σ µ ηλ
σe e P Pt
N
t
N N
N( ) ( )
/
+
+
− ∞
∫ ( ) − ( )
0
2
1 1 2 2
2
1 2
e w w w wtµ , ,
/
+
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
1690 T. B. FASTOVSKA
+ J F V P G V L L P K w wW N N0 11
1 1 1 1, [ ], [ ] ( ) ,Γ− −+ [ ][ ]
η
ηβ
β β –
– J F V P G V L L P K w wW N N
Y
0 11
2 2 2 2, [ ], [ ] ( ) ,Γ− −+ [ ][ ]
η
ηβ
β β
α
+
+ 1
0
1 1 2 2
2
1 2
ηβ
β µ β
− ∞
−∫ [ ] − [ ]( )( )
J e L Q K w w K w wt
N , ,
/
+
+
− ∞
∫ −( )( )
0
0
1 2
2
1 2
J e Q G V G Vt
N
µ [ ] [ ]
/
.
It follows from Lemma 2 and (6) that
1
0
1 1 2 2
2
1 2
ηβ
β µ β
− ∞
−∫ [ ] − [ ]( )( )
J e L Q K w w K w wt
N , ,
/
≤
M
V V
K N
N
Y
λ
ηλ µ
α β
α
+
+
+ −
−1
1
1 2 .
(29)
Similarly, we get from (5) that
− ∞
∫ −( )( )
0
0
1 2
2
1 2
J e Q G V G Vt
N
µ [ ] [ ]
/
≤
M
V VG N
N
Y
λ
ηλ µ
α
α
+
+ −
−1
1
1 2 . (30)
The statement of the lemma can be easily deduced from (27), (29), and (30).
Lemma 5. Let µ =
η λ λN N+( )+1
2
and hypotheses A1 – A5 hold. Then equa-
tion (20) has the unique solution V t W( ; ) in the space Yα . For any σ such that
σ < min ,1 1
2
−
β , this solution possesses the properties
V t C( ) , ,∈ − ∞( ]( )0 H σ , (31)
and
sup
t
te V V C W W
≤
−{ } ≤ −
0
1 2 1 2
µ
σ σ σ (32)
for any W1, W2 ∈ PH α , where Cσ is a positive constant. Moreover, for every
s ∈ − ∞( ), 0 and for almost t ∈ s, 0[ ], the function V t( ) satisfies
V t( ) = e V st s− −�( ) ( ) +
s
t
t se V d∫ − − ( )� �( ) ( )τ τ. (33)
Proof. By Lemma 3,
B V B V V VW W Y
N N
Y1 2
1
1 22
[ ] − [ ] ≤
+( )
−+
α α
ρ
η λ λ
for every V1, V2 ∈ Yα . Since hypothesis A5 holds and ρ
η λ λN N+( )
+1
2
< 1, by
contraction principle we have that equation (20) has the unique solution in Yα .
Relation (33) can be obtained by direct calculation and (31) follows from this
representation, therefore, we prove here only (32). As
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
INVARIANT MANIFOLDS FOR COUPLED NONLINEAR PARABOLIC-HYPERBOLIC … 1691
V1 – V2 = B V t B V tW W1 21 2[ ] − [ ]( ) ( ) = e W Wt− −� ( )1 2 + B V t B V t0 1 0 2[ ] − [ ][ ]( ) ( ) ,
we have
e PV PVtµ
σ1 2− ≤ e w w w wt
D A D
µ ( , ) ( , ) / /1 1 2 2 1 2 1 2− ( ) × ( )Γ +
+ e N t
D L
( )
( )
µ ηλ θ θ σ
− −1 2 + M e e V V dF
t
t
0
1 2∫ − −µ τ µ τ
α τ( ) +
+ M M e e V V dG N K N
t
tNλ λ τσ σ β µ ηλ τ µ τ
α+( ) −+ − −∫
0
1 2
( )( ) +
+ M M e e V V dG K
t
tλ λ τσ σ β µ ηλ τ µ τ
α1 1
0
1 2
1+( ) −+ − −∫ ( )( ) .
Hence, using the Hölder inequality, we get
sup
t
te PV PV
≤
−{ }
0
1 2
µ
σ ≤ W W1 2− σ + sup ( )
/
t
F
t
tM e d
≤
−∫
0
0
2
1 2
µ τ τ +
+ M M e d V VG N K N
t
t
Y
Nλ λ τσ σ β µ ηλ τ
α
+( )
−+ − −∫
0
2
1 2
1 2
( )( )
/
+
+ M M e d V VG K
t
t
Yλ λ τσ σ β µ ηλ τ
α1 1
0
2
1 2
1 2
1+( )
−+ − −∫ ( )( )
/
.
This estimate implies that
sup
t
te PV PV
≤
−{ }
0
1 2
µ
σ ≤
≤ W W1 2− σ +
M M M M M
V VF G N K N
N
G N K N
N
Y2 2 2 1 2µ
λ λ
µ ηλ
λ λ
µ ηλ
σ σ β σ σ β
α
+ +
−
+ +
−
−
+ +
( ) ( )
.
(34)
Similarly, we obtain
e QV QVtµ
σ1 2− ≤ M L e Q e V V dG
t
L t
− ∞
− − −∫ −σ η µ τ µ τ
α τ( )( )
1 2 +
+ M L e Q e PV PV dK
t
L t
− ∞
+ − − −∫ −σ β η µ τ µ τ
α τ( )( )
1 2 .
Therefore,
sup
t
te QV QV
≤
−{ }
0
1 2
µ
σ ≤ a t V V Y1 1 2( ) −
α
+ a t e PV PV
t
t
2
0
1 2( ) sup
≤
−{ }µ
σ , (35)
where
a t M L e Q dG
t
L t
1
2
1 2
( ) ( )( )
/
=
− ∞
− − −∫ σ η µ τ τ ,
a t M L e Q dK
t
L t
2( ) ( )( )=
− ∞
+ − − −∫ σ β η µ τ τ .
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
1692 T. B. FASTOVSKA
Using Lemma 1, we have
a t1( ) ≤
≤
M
e d
t
e dG
t
N
t
t
tN N
η
ηλ τ σ
τ
τσ
σ ηλ µ τ
σ
ηλ µ τ
− ∞
+
− −
− ∞
− −∫ ∫( ) +
−
+ +2 21
2 2
2
2
1 2
1 1( )( ) ( )( )
/
≤
≤
MG N
N Nη
ηλ
ηλ µ
κ
ηλ µσ
σ
σ
σ
+
+ +
−
( )
−
+
−
1
2
1
2
1
1 2
1 2/
≡ a1 (36)
for any 0 < σ < 1, where κs = s e ds s
0
∞ − −∫ τ ττ for 0 < s < 1 and κ0 = 0. Since
κs > 0, we can also estimate a t1( ) ≤ a1 for σ ≤ 0.
Similarly,
a t2( ) ≤
M
t
e dK
t
N
tN
η
ηλ σ β
τ
τσ β
σ β
σ β
ηλ µ τ
+
− ∞
+
+
+
− −∫ ( ) + +
−
+
1
1( )( )
≤
≤
MK
N
N
Nη
κ
ηλ µ
ηλ
ηλ µσ β
σ β
σ β
σ β
+
+
+
− +
+
+
+−( )
+
( )
−
1
1
1
1
( ) ≡ a2. (37)
From (34) and (35) we obtain
sup
t
te V V
≤
−{ }
0
1 2
µ
σ ≤ ( )1 2 1 2+ −a W W σ +
+ a a
M M M M M
V VF G N K N
N
G K
1 2
1 1
1
1 21
2 2 2
+ + + +
−
+ +
−
−
+ +
( )
( ) ( )µ
λ λ
µ ηλ
λ λ
µ ηλ
σ σ β σ σ β
α . (38)
It follows from Lemma 3 that
V V W WY N1 2
1
1 1 21 1− ≤ − + +( ) −− − −
α
ρ λ λα σ α σ
σ( ) .
Hence, (38) implies (32).
Proof of Theorem 1. We define the mapping
Φ( ) ( ) ( , ) ( )W e Q G V K w w d V W= ( ) +( ) = −
− ∞
−∫
0
0η τ τ τ� ,
where V t( ) = ( , , )w w θ is the solution to (20). Relation (32) and standard arguments
(see, e.g., [5]) imply that the manifold M generated by Φ is forward invariant and
possesses the property (16). To prove the tracking properties, we rely on the idea
applied by Miclavcic∨ ∨ [7] in the theory of inertial manifolds. We extend the mild
solution V t( ) to (9), (10) on the semiaxis − ∞( ], 0 by the formula V t( ) =
= w w t A0 1
1
01, , +( )( )− θ .
Consider the function
Z t
V t B V t t
e V B V t
PV
t
PV
0
0
0
0
0 0 0
( )
( ) [ ]( ), ,
( ) [ ]( ) , ,
( )
( )
=
− + ≤
− +[ ] >
−�
in the space
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
INVARIANT MANIFOLDS FOR COUPLED NONLINEAR PARABOLIC-HYPERBOLIC … 1693
Z Z= = < ∞
− ∞
∞
∫Z t Z e Z t dtt( ): ( )2 2 2µ
α .
It is easy to see that
V C VYα σ
2
0
2≤ , V t C( ) , ,∈ − ∞( ]( )0 H σ
and
Z C V0 01Z ≤ +( )σ , sup
t
te Z C V
∈
{ } ≤ +( )
R
µ
σ σ0 01
for any σ such that α – 1
2
≤ σ < min ,1 1
2
−
β .
Define the integral operator � : Z → Z by the formula
�[ ]( )Z t = Z t0( ) –
t
te P V Z V d
∞
− −∫ +( ) − ( )[ ]� � �( ) ( ) ( ) ( )τ τ τ τ τ +
+
− ∞
− −∫ +( ) − ( )[ ]
t
te Q V Z V d� � �( ) ( ) ( ) ( )τ τ τ τ τ .
The reasons analogous to given in Lemma 4 for the operator B VW[ ] lead to the
estimate
� �Z Z q Z Z1 2 1 2[ ] − [ ] ≤ −Z Z
for every Z1, Z2 ∈ Z, i.e., � is a contraction in Z. This implies that the equation
Z = �[ ]Z has the unique solution Z ∈ Z and the estimate
Z q Z C VZ Z≤ − ≤ +( )−( )1 11
0 0 σ (39)
holds.
Using the same arguments as in Lemma 5, we obtain
sup
t
te Z C V
∈
{ } ≤ +( )
R
µ
σ σ1 0 . (40)
It follows from (39) and (40) that V t( ) + Z t( ) is the desired trajectory emanating from
V* = V( )0 + Z( )0 [4].
In the case of α < min ,1 1
2
−
β , we can apply the results obtained to reduce
system (1), (2), to the following system:
Γ Φw Aw F w w w wtt t t+ = + ( )( ), , , ,v v , t > 0, in H, (41)
v v v vt N t t N tL P G w w w w P K w w+ = +( ) +η , , ( , , ) ( , )Φ , t > 0, in P EN (42)
with initial conditions
w wt = =0 0 , w wt t = =0 1, v vt = =0 0. (43)
Denote
�
Φ w w tt( ), ( ), ( )τ τ v( ) =
=
0
F w w w w t
P G w w w w P K w w
t t
N t t N t
( ), ( ), ( ), ( ), ( )
( ), ( ), ( ), ( ), ( ) ( ), ( )
τ τ τ τ
τ τ τ τ τ τ τ
v v
v v
+ ( )( )
+ ( )( ) + ( )
Φ
Φ
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
1694 T. B. FASTOVSKA
A mild solution to this system on the interval 0, T[ ] is a function
W t w t w t C T D A D P D Lt N( ) ( ), ( ), , , ( )/ /= ( ) ∈ [ ] ( ) × ( ) ×( )v 0 1 2 1 2Γ α (44)
such that
0
2
T
t E
L w t w t dt∫ ( ) < ∞α Φ ( ), ( ), v (45)
and
W t S
w
w S B w w dt
t
t t( ) ( ), ( ), ( )=
+ ( )∫ −
0
1
0
0v
vτ τ τ τ τΦ
for almost all t ∈ 0, T[ ].
Proposition 2. Let w w0 1 0, , v( ) ∈ D A1 2/( ) × D Γ1 2/( ) × P D LN
α −( )1 2/ and let
conditions of Theorem 1 hold. Then problem (41) – (43) has a mild solution. If
α < min ,1 1
2
−
β , then the solution is unique and any mild solution ˜ ( )W t =
= ˜ , ˜ ,w wt v( ) to this problem generates the mild solution to problem (1), (2) by the
formula
w t w t t w t w t w t w tt t t( ), ( ), ( ) ˜ ( ), ˜ ( ), ˜ ( ), ˜ ( ),θ( ) = ( )( )v + vΦ . (46)
Proof. Let V t( ) = w t w t tt( ), ( ), ( )θ( ) be a mild solution to (9) with initial
conditions V0 = ( w0 , w1, v0 + Φ ( w0 , w1, v0)) . The property (44) holds by
Proposition 1. We get (45) from the fact that PV t( ) = W t( ), QV t( ) = Φ W t( )( ) and
0
2
0
2
T T
QV t dt V t dt∫ ∫≤ < ∞( ) ( )α α ,
consequently, W t( ) is a mild solution to (41) – (43).
Let W1 = w w1 1 1, , v( ) , W2 = w w2 2 2, , v( ) ∈ D A1 2/( ) × D Γ1 2/( ) × P D LN ( )α . If
α < min ,1 1
2
−
β , we use the Lipschitz properties (16) and (4) – (6) to get
� �Φ Φ( ) ( )W W1 2
2
−
H α
≤
≤ M M L P C M L PF G N K N+( ) +( ) +[ ]+α
α
α β1 2 ×
×
w w w wD A D D L1 2
2
1 2
2
1 2
2
1 2
1 2 1 2− + − + −( )( ) ( )/ / ( )
/
Γ v v α ≤
≤ M M M C W WF N G N K+ +( ) +( ) −+λ λα β
α α
1 2
1 2 H .
The uniqueness of the solution to reduced system and relation (46) follow immediately
from the estimate obtained.
Theorem 1 and Proposition 2 allow us to obtain a reduction principle for problem
(1), (2). The point is that by Theorem 1 for any mild solution V t( ) = w t w t tt( ), ( ), ( )θ( )
to problem (1), (2) with initial data V0 ∈ H σ , where α – 1
2
≤ σ < min ,1 1
2
−
β ,
there exists a mild solution W t( ) =
˜ ( ), ˜ ( ), ( )w t w t tt v( ) to reduced system such that
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
INVARIANT MANIFOLDS FOR COUPLED NONLINEAR PARABOLIC-HYPERBOLIC … 1695
A w t w t
H
1 2 2/ ( ) ˜ ( )−( ) + Γ1 2 2/ ( ) ˜ ( )w t w tt t H
−( ) +
+ θ( ) ( ) ˜ ( ), ˜ ( ), ( )t t w t w t tt E− − ( )v vΦ 2 ≤ Ce t−µ
for any t ≥ 0, where C, µ > 0. Therefore, under conditions of Theorem 1, the long-
time behavior of solutions to (1), (2) can be described by solutions to the reduced
problem. Moreover, if α < min ,1 1
2
−
β , then by Proposition 2 every limiting
regime of the reduced system appears in system (1), (2).
3. Application to the thermoelastic models. In this section, we give examples of
several thermoelastic models with various α and β to illustrate the results obtained.
Consider a one-dimensional Mindlin – Timoshenko system
ρ β α µ µ δ θ θ θ( ) , , , , , , ,x u T u u utt t xx x x x x t t xv v v v v v v+ − + + + = ( )0 0 0 0 0 ,
ρ β µ µ θ θ( ) , , , , , , ,x u u u D u u utt t xx x x x t t x+ − − = ( )1 0 0v v v v , t > 0, x ∈ ( 0, l ),
θ ηθ δ θ θt xx tx x x t t xG u u u− + = ( )v v v v, , , , , , , ,
with Dirichlet boundary conditions
v v( , ) ( , ) ( , ) ( , ) ( , ) ( , )0 0 0 0t l t u t u l t t l t= = = = = =θ θ .
This system describes the dynamics of heat conductive elastic beam. Here, v( , )x t and
u x t( , ) are, respectively, the angle of slope of the transverse section and deflection
averaged with respect to the thickness of the beam, θ( , )x t is the temperature
variation, ρ( )x is a strictly positive continuous function. For details concerning
Mindlin – Timoshenko hypotheses see [8]. We assume that T, D , G : R8 → R are
globally Lipshitz functions. Denote w = v( , ), ( , )x t u x t T( ) and rewrite the system in
the following way:
ρ α θ θ( ) , , , ,x w w F w w wtt xx x t x− = ( )0 , t > 0, x l∈( )0, ,
(47)
θ ηθ θ θt xx x t x tG w w w K w w− = ( ) −, , , , ( , ).
Note that system (47) satisfies conditions A3, A4 with α = β = 1
2
, H = L l2 2
0( , )[ ] ,
E = L l2 0( , ), the operators Γ, L and A are given by Γ w = ρ( )x w , A = − ∂α0
2
x and
L = − ∂x
2 .
To use Theorem 1, we should check the spectral condition (15). The spectrum of
the operator − ∂x
2 with the Dirichlet boundary conditions has the form
πn
l
2
:
n ∈
R , therefore, (15) looks like
2
2 2 1
2 2 2 2 1
2 1
1
2
2 2
2
l M
N N
lM M N N
N
F G K
ηπ ηπ η+ +( ) + +
+ +( )
+
<
( )
.
This condition holds, for instance, if each item is less then 1
3
. The first one tends to
zero if N → ∞, thus, we can find N0 ∈ N ∪ {0} such that condition
2
2 2 1
1
3
2
2 2
l M
N N
F
ηπ + +( ) <
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
1696 T. B. FASTOVSKA
holds for any N > N0, namely, N0 = [r] + 1, where r = – 1
2
+
3 1
4
2
2
M lF
ηπ
−
provided r > 0 ( [r] denotes the integer part of r), otherwise, N0 = 0. Therefore, we
exclude MF from the condition on the constants of the system given in [4]. Thus, we
have obtained the following result.
Proposition 3. Assume that conditions on the parameters of the problem (47)
l
MG
< π η
6
and
M N
N N
K
η
< +
+ +
2 1
2 2 12
hold for some N > N0. Then, for any σ such that 0 ≤ σ < 1
2
, there exists the
mapping Φ : L l2 2
0( , )[ ] × L l2 0( , ) × P H lN 0
2 0σ( , ) → Q H lN 0
2 0σ( , ) holds and the
surface (17) is forward invariant in H σ = L l2 2
0( , )[ ] × L l2 0( , ) × H l0
2 0σ( , ) and
exponentially attracting.
The case of α = 0 and β = 0 in conditions A3, A4 is exemplified in a system
appearing in classical linear two-dimensional thermoelasticity (for the statement of the
problem see, e.g., [1]). We use the well-known decomposition of the displacement
vector into potential and solenoidal part v = ∇w + rot u, where w , u are scalar
functions and the rotation of u is defined by rot u = ∂ − ∂( )2 1u u, . The function u
satisfies a linear wave equation and the problem is reduced to the following coupled
system:
w wtt − + =α δ θ0 0 0∆ , t > 0, x ∈ ⊂Ω R
2,
θ η θ δt tw− + =∆ ∆ 0,
with Dirichlet boundary conditions. For this system, condition (15) has the form
2 2
10
1
1
1
δ
η λ λ
δ λ λ
η λ λN N
N N
N N+
+
++( ) +
+( )
−( ) < . (48)
Due to the fact that lim
N
N
→∞
λ = ∞, the first item tends to zero as N → ∞ and there
exists N0 ∈ N such that the first item in (48) is less then 1
2
. Hence, we obtain the
existence of the mapping Φ : L2 2
( )Ω[ ] × L2( )Ω × P HN
2σ( )Ω → Q HN
2σ( )Ω for any
– 1
2
≤ σ < 0 such that surface (17) is forward invariant and exponentially attracting
provided that
2 1
2
0 0
0 0
1
1
δ λ λ
η λ λ
N N
N N
+
+
+( )
−( ) < .
In two cases considered, Theorem 1 gives us a good sufficient condition for
establishing the existence of invariant manifold if MF is large and MK is small.
Then condition (8) may not be satisfied, while condition (15) holds.
It appears that, in the case of α + β = 0 and a rectangular domain Ω in R
2 ,
Theorem 1 holds for any choice of the parameters of problem (1), (2). Consider a two-
dimensional problem
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
INVARIANT MANIFOLDS FOR COUPLED NONLINEAR PARABOLIC-HYPERBOLIC … 1697
ρ α θ( ) , , ,x w w F w w wtt t− = ∇( )0 ∆ , t > 0, x ∈ ⊂Ω R
2,
θ η θ θt tG w w w− = ∇( )∆ , , , ,
with globally Lipschitz functions F and G. Here, H = E = L2( )Ω , the operators Γ,
L and A are given by Γ w = ρ( )x w , A = −α0 ∆ and L = – ∆, and the spectral
condition turns into
2 2 2
1
1 1 1
M M MF
N N
G
N N
K
N Nη λ λ η λ λ η λ λ+ + ++( ) +
−( ) +
−( ) < (49)
As it have been already note, the item including MF tends to zero when N goes to
infinity. We assume that Ω = ( , )0 1l × ( , )0 2l is a rectangle with
l
l
1
2
rational. For
such form of the domain, there is a spectral gap limit result, i.e., λN k( )+1 – λN k( ) → ∞
when k → ∞ for some subsequence N k( ) (see [9]). Therefore, by the choice of N,
the expression on the left-hand side of (49) can be made arbitrary small and we
establish the existence of invariant exponentially attracting manifold in the space H σ
for – 1
2
≤ σ < 1
2
.
1. Jiang S., Racke R. Evolution equations in thermoelasticity // Monogr. Surv. Pure Appl. Math. –
2000. – 112. – 453 p.
2. L¥kova O. B., Mytropol\skyj G. A. Yntehral\n¥e mnohoobrazyq v nelynejnoj mexanyke. –
M.: Nauka, 1973. – 512 s.
3. Leung A. W. Asymptotically stable invariant manifold for coupled nonlinear parabolic-hyperbolic
partial differential equations // J. Different. Equat. – 2003. – 187. – P. 184 – 200.
4. Chueshov I. D. A reduction principle for coupled nonlinear parabolic-hyperbolic partial differential
equations // J. Evol. Equat. (to appear).
5. Chueshov I. D. Introduction to the theory of infinite-dimensional dissipative systems (in Russian).
– Kharkov: Acta, 1999. – 433 p.
6. Temam R. Infinite-dimensional dynamic systems in mechanics and physics. – New York: Springer,
1988. – 500 p.
7. Miclavcic
∨ ∨
M. A sharp condition for existence of an inertial manifold // J. Dynam. Different. Equat.
– 1991. – 3. – P. 437 – 456.
8. Lagnese J. Boundary stabilization of thing plates. – Philadelphia: SIAM, 1989. – 176 p.
9. Mallet-Paret J., Sell G. Inertial manifolds for reaction-diffusion equations in higher dimension // J.
Amer. Math. Soc. – 1988. – 1. – P. 805 – 866.
Received 22.11.2004
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
|
| id | umjimathkievua-article-3719 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:47:40Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/64/4cbd9b5177d60d866f16fead1e1ba764.pdf |
| spelling | umjimathkievua-article-37192020-03-18T20:02:57Z Invariant manifolds for coupled nonlinear parabolic-hyperbolic partial differential equations Інваріантні многовиди пов'язаних нелінійних параболіко-гіперболічних рівнянь з частинними похідними Fastovska, T. B. Фастовська, Т. В. We consider an abstract system of coupled nonlinear parabolic-hyperbolic partial differential equations. This system describes, e.g., thermoelastic phenomena in various physical bodies. Several results on the existence of invariant exponentially attracting manifolds for similar problems were obtained earlier. In the present paper, we prove the existence of such an invariant manifold under less restrictive conditions for a broader class of problems. Розглянуто абстрактну систему параболіко-гіперболічних пов'язаних нелінійних рівнянь з частинними похідними. Ця система описує, наприклад, термопружні явища в різних фізичних тілах. Деякі результати щодо існування інваріантних многовидів, що експоненціально притягують, для задач подібного типу було отримано раніше. В даній роботі доведено існування цього інваріантного многовиду за менш обмежувальних умов для більш широкого класу задач. Institute of Mathematics, NAS of Ukraine 2005-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3719 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 12 (2005); 1684–1697 Український математичний журнал; Том 57 № 12 (2005); 1684–1697 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3719/4163 https://umj.imath.kiev.ua/index.php/umj/article/view/3719/4164 Copyright (c) 2005 Fastovska T. B. |
| spellingShingle | Fastovska, T. B. Фастовська, Т. В. Invariant manifolds for coupled nonlinear parabolic-hyperbolic partial differential equations |
| title | Invariant manifolds for coupled nonlinear parabolic-hyperbolic partial differential equations |
| title_alt | Інваріантні многовиди пов'язаних нелінійних параболіко-гіперболічних рівнянь з частинними похідними |
| title_full | Invariant manifolds for coupled nonlinear parabolic-hyperbolic partial differential equations |
| title_fullStr | Invariant manifolds for coupled nonlinear parabolic-hyperbolic partial differential equations |
| title_full_unstemmed | Invariant manifolds for coupled nonlinear parabolic-hyperbolic partial differential equations |
| title_short | Invariant manifolds for coupled nonlinear parabolic-hyperbolic partial differential equations |
| title_sort | invariant manifolds for coupled nonlinear parabolic-hyperbolic partial differential equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3719 |
| work_keys_str_mv | AT fastovskatb invariantmanifoldsforcouplednonlinearparabolichyperbolicpartialdifferentialequations AT fastovsʹkatv invariantmanifoldsforcouplednonlinearparabolichyperbolicpartialdifferentialequations AT fastovskatb ínvaríantnímnogovidipov039âzanihnelíníjnihparabolíkogíperbolíčnihrívnânʹzčastinnimipohídnimi AT fastovsʹkatv ínvaríantnímnogovidipov039âzanihnelíníjnihparabolíkogíperbolíčnihrívnânʹzčastinnimipohídnimi |