Abstract
In this paper, we study the existence and long-time behavior of weak solutions to a nonclasscial diffusion
equation with exponential nonlinearity. We prove the existence of a global attractor of the dynamical
system associated to the equation. The main novelty of the results obtained is that no restriction on the
upper growth of the nonlinearity is imposed.
Keywords: Nonclasscial diffusion equation; global attractor; exponential nonlinearity.
Tóm tắt
Trong bài báo này, chúng tôi nghiên cứu sự tồn tại và dáng điệu tiệm cận của nghiệm yếu phương trình
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được là hàm phi tuyến không bị giới hạn về tốc độ tăng trưởng.
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NGÀNH TOÁN HỌC
Tạp chí Nghiên cứu khoa học - Đại học Sao Đỏ, ISSN 1859-4190 Số 4(59).2017 63
GLOBAL ATTRACTOR FOR NONCLASSICAL DIFFUSION EQUATION
WITH EXPONENTIAL NONLINEARITY
TẬP HÚT TOÀN CỤC CỦA PHƯƠNG TRÌNH KHUẾCH TÁN KHÔNG
CỔ ĐIỂN VỚI ĐIỀU KIỆN TĂNG TRƯỞNG KIỂU MŨ
Nguyen Viet Tuan1, Nguyen Thi Hue1, Nguyen Thi Hong1, Nguyen Xuan Tu2
Email: nguyentuandhsd@gmail.com
1Sao Do University
2Hung Vuong University
Date received: 30/10/2017
Date received after review: 20/12/2017
Date accept: 28/12/2017
Abstract
In this paper, we study the existence and long-time behavior of weak solutions to a nonclasscial diffusion
equation with exponential nonlinearity. We prove the existence of a global attractor of the dynamical
system associated to the equation. The main novelty of the results obtained is that no restriction on the
upper growth of the nonlinearity is imposed.
Keywords: Nonclasscial diffusion equation; global attractor; exponential nonlinearity.
Tóm tắt
Trong bài báo này, chúng tôi nghiên cứu sự tồn tại và dáng điệu tiệm cận của nghiệm yếu phương trình
khuếch tán không cổ điển với điều kiện hàm phi tuyến tăng trưởng và tiêu hao kiểu mũ. Chúng tôi chứng
minh sự tồn tại của tập hút toàn cục của hệ động lực sinh bởi phương trình. Tính mới lạ của kết quả thu
được là hàm phi tuyến không bị giới hạn về tốc độ tăng trưởng.
Từ khóa: Phương trình khuếch tán không cổ điển; tập hút toàn cục; tăng trưởng kiểu mũ.
1. INTRODUCTION
In this paper, we study the existence and long-time
behavior of solutions to the following nonclasscial
diffusion equation
0
( ) ( ), , 0,
( , ) 0, , 0,
( ,0) ( ), ,
− D −D + = ∈Ω >
= ∈∂Ω >
= ∈Ω
t tu u u f u g x x t
u x t x t
u x u x x
(1)
where Ω is a bounded domain in N with smooth
boundary ∂Ω . This equation arises as a model
to describe physical phenomena, such as non-
Newtonian flows, soil mechanics and heat
conduction theory (see [1]). In the past years, the
existence and long-time behavior of solutions to
nonclassical diffusion equations has been studied
extensively, for both autonomous case [5, 6] and
non-autonomous case [2, 3, 4], and even in the
case with finite delay. To study problem (1), we
assume the following assumptions:
(H1) : → f f: is a continuously differentiable
function satisfying
( ) ,′ ≥ −f u
2 0( ) , for all ,≥ − − ∈f u u u C ub
where 0, C are two positive constants, 10 < <b l
with 1l is the first eigenvalue of the operator −D
in Ω with the homogeneous Dirichlet boundary
condition;
(H2) The external force 1( ).−∈ Ωg H
Now, we introduce some notations. Unless
otherwise specified, it is understood that we
consider spaces of functions acting on the domain
Ω . Let ,⋅ ⋅ and ⋅‖‖ denote the 2 −L inner product
and 2 −L norm, respectively. We will also consider,
with standard notation, spaces of functions defined
on an interval I with values in Banach space X
such as ( , ), ( , )pC I X L I X and , ( , )m pH I X , with the
usual norms.
The paper is organized as follows. In Section 2,
we prove the existence and uniqueness of weak
solutions to problem (1) in the space 2 ( )ΩL by
utilizing the compactness method and weak
64
NGHIÊN CỨU KHOA HỌC
Tạp chí Nghiên cứu khoa học - Đại học Sao Đỏ, ISSN 1859-4190 Số 4(59).2017
convergence techniques in Orlicz spaces. The
existence of a global attractor for the continuous
semigroup associated to the problem is studied in
the last section.
2. EXISTENCE AND UNIQUENESS OF WEAK
SOLUTIONS
Definition 1. A function ( )u t is called a weak
solution of problem (1) on the interval [0, ]T ,
where 0>T , with initial datum 10 0(0) ( )∈= Ωu u H if
1 1 2 1
0 0([0, ]; ( )), ( ) ( ), (0, ; ( )),∈ Ω ∈ ∈ ΩT tu C T H f u L Q u L T H
and
1
1 1
0
,
,
, , , ( ),
, ,
t t L L
H H
u u u f u
g
ϕ ϕ ϕ ϕ
ϕ
∞
−
〈 〉 + 〈∇ ∇ 〉 + 〈∇ ∇ 〉 + 〈 〉
= 〈 〉
for all test functions 10 ( ) ( ),
∞∈ = Ω ∩ ΩW H Lϕ and
a.e. [0, ]∈t T .
We now prove the existence and uniqueness
result for problem (1).
Theorem 1. Assume that conditions (H1), (H2)
hold. Then for any 10 0 ( )Ω∈u H and any 0>T
given, there exists a unique weak solution u to
problem (1) on the interval [0, ]T . Furthermore,
1
0([0, ]; ( )),∈ Ωu C T H and the mapping
1 1
0 0 0( ) ( ( ), ( )) [0, ],∈ Ω Ω ∀ ∈u u t C H H t T that is,
the weak solutions depend continuously on the
initial data.
Proof. We use the Feado-Galerkin method. We
recall that there exists a smooth orthonormal
basis 1{ }
∞
=j jw of
2 ( )ΩL which is also orthogonal
in 10 ( )ΩH , consisting of normalized eigenfunctions
for −D in 10 ( )ΩH .
Step 1 (Feado-Galerkin scheme).
Given an integer n , denote by the projection on
the subspace 11 0span( , , ) ( ) ⊂ Ωn Hw w . We look
for a function nu of the form
1
( ) ( )
=
= ∑
n
n j j
j
u t a t w
satisfying
1
0 ( )
0 0
,
, ( ), ( , ), 1 ,
|
,
t n t n k H
n k n k k
n t n
u u
u f u g k n
u P u
w
w w w
Ω
=
〈∂ − D∂ 〉
= 〈D 〉 − 〈 〉 + ≤ ≤
=
for a.e. ≤t T . We get a system of ODEs in the
variables ( )ka t of the form
(1 ) , ( ),( )+ = − + −k k k k k n kd a a g f udt ν ν w w
(2)
subject to the initial condition
1
00 ( )
(0) , .
Ω
=k k Ha u w
According to standard existence theory for ODEs,
there exists a solution on some interval (0, )nT . The
a priori estimates below imply that in fact = +∞nT .
Step 2 (Energy estimates).
Multiplying the equation (2) by ka , then summing
over k and adding the results, we get
(3)
Using (H1) we deduce that
So
where 0>ε is small enough so that
1
1 0.− − >b ε
l
Integrating on (0, ), (0, )∈t t T , leads to the following
estimate
In particular, we see that { }nu is bounded in
1
0(0, ; ( ))
∞ ΩL T H .
Using the boundedness of { }nu in
1
0(0, ; ( ))
∞ ΩL T H ,
it is easy to check that { }D nu is bounded in
2 1(0, ; ( ))− ΩL T H .
Therefore, up to passing to a subsequence, there
exists a function u such that
1
0
2 1
weakly-star in (0, ; ( )),
weakly in (0, ; ( )).
∞
−
Ω
D D Ω
n
n
u u L T H
u u L T H
Step 3 (Passage to limits).
From (3), we get
(4)
Integrating (4) from 0 to T , we have
Hence
0
( ) .
Ω
≤∫ ∫
T
n nf u u dxdt C
NGÀNH TOÁN HỌC
Tạp chí Nghiên cứu khoa học - Đại học Sao Đỏ, ISSN 1859-4190 Số 4(59).2017 65
We now prove that { ( )}nf u is bounded in
1( )TL Q .
Putting ( ) ( ) (0)= − +h s f s f sγ , where > γ .
Note that
2
2 2 2
( ) ( ( ) (0))
( ) ( ) 0
h s s f s f s s
f s s s
γ
ξ γ γ
= − +
′= + ≥ − ≥
for all ∈s , we have
1
2
0
{| | 1} {| | 1}
| | 1
( )
2
( )
| | 1
| ( ) |
| ( ) | | ( ) |
( ) sup | ( ) |
||
. | |
( ) | (0) |. ||
sup | ( ) | . | |
.
T n T n
T
TT
T
T
n
n n nQ u Q u
n n TQ s
n n n L QQ
n TL Q
s
h u dxdt
h u u dxdt h u dxdt
h u u dxdt h s Q
f u u dxdt f u
u h s Q
C
γ
Ω
∩ > ∩ ≤
≤
≤
≤ +
≤ +
≤ +
+ +
≤
∫ ∫
∫ ∫
∫
∫
\ \
Hence it implies that { ( )}nh u , and therefore
{ ( )}nf u is bounded in
1( )TL Q .
Now, we prove the boundedness of { }∂t nu .
In the first equation in (2), replacing kw by ∂t nu ,
and then using the Cauchy inequality, we get
Hence, by choosing ε small enough, we arrive at
Integrating from 0 to t , we can deduce that
{ }∂t nu is bounded in
2 1
0(0, ; ( ))ΩL T H .
So, up to a subsequence,
2 1
0 weakly in (0, ; ( ))∂ Ωt n tu u L T H ,
2 1 weakly in (0, ; ( )).−D∂ D Ωt n tu u L T H
Because 1 2 1 10 ( ) ( ) ( ) ( )
−Ω ⊂⊂ Ω ⊂ Ω + ΩH L H L , by the
Aubin-Lions-Simon compactness lemma, we
obtain →nu u strongly in
2 2(0, ; ( ))ΩL T L . Hence
we may assume, up to a subsequence, that
→nu u a.e. in TQ . Since f is continuous, it
follows that ( ) ( )→nf u f u a.e. in [0, ]Ω× T . We
obtain that 1( ) ( )∈ Th u L Q and for all test functions
1
0 0([0, ]; ( ) ( ))
∞ ∞∈ Ω ∩ ΩC T H Lφ ,
( ) ( ) .→∫ ∫
T T
nQ Q
h u dxdt h u dxdtφ φ
Hence 1( ) ( )∈ Tf u L Q and
( ) ( ) ,
T T
nQ Q
f u dxdt f u dxdtφ φ→∫ ∫
for all 10 0 ([0, ]; ( ) ( ))
∞ ∞∈ Ω ∩ ΩC T H Lφ .
By standard arguments, we can check that u
satisfies the initial condition 0(0) =u u and this
implies that u is a weak solution of problem (1).
Step 4 (Uniqueness and continuous dependence
of the solutions).
We assume that 1u and 2u are two solutions of (1)
with initial data 10u and 20u , respectively. Denote
1 2= −u u u , then it satisfies
( )1 2ˆ ˆ( ( ) ( )) 0, 0, 5t tu u u f u f u u t∂ − D∂ − D + − − = ∀ >
where ˆ ( ) ( ) .= + f s f s s Here because ( )u t does
not belong to 10: ( ) ( )
∞= Ω ∩ ΩW H L , we cannot
choose ( )u t as a test function.
Let
if ,
( ) if | | ,
if .
>
= ≤
− < −
k
k s k
B s s s k
k s k
Consider the corresponding Nemytskii mapping
ˆ : →kB W W defined as follows ˆ ( )( ) ( ( )),=k kB u x B u x
for all ∈Ωx . We notice that ˆ ( ) 0− →k WB u u‖ ‖
as →∞k . Now multiplying (5) by ˆ ( )kB u , then
integrating over Ω we get
Thus
Note that ˆ ( ) 0′ ≥f s and ( ) 0≥ksB s for all ∈s ,
we get ˆ ˆ( ) ( ) 0.
Ω
′ ≥∫ kf uB u dxξ
Therefore 2
{ :| ( , )| }
| | 0.
∈Ω ≤
∇ ≥∫ x u x t k u dx
Since the above inequalities, we get
Integrating from 0 to t , where (0, )∈t T , then
letting →∞k , we obtain
66
NGHIÊN CỨU KHOA HỌC
Tạp chí Nghiên cứu khoa học - Đại học Sao Đỏ, ISSN 1859-4190 Số 4(59).2017
By the Gronwall inequality of integral form, we get
for all [0, ].t T∈
Hence 10([0, ] (; ))∈ Ωu C T H , in particular, we get
the uniqueness if ( )0 0.=u
3. EXISTENCE OF A GLOBAL ATTRACTOR
Theorem 1 allows us to define a continuous
semigroup 1 10 0( ) () : )( Ω Ω→S t H H associated to
problem (1) by the formula 0( ) : ( ),=S t u u t where
(.)u is the unique weak solution of (1) with the
initial datum 10 0 ( )Ω∈u H . The aim of this section
is to prove the following result.
Theorem 2. Assume that (H1), (H2) hold. Then
the semigroup 0{ ( )} ≥tS t possesses a compact
global attractor in 10 ( )ΩH .
To prove this theorem, by the classical abstract
results on existence of global, we need to show
that the semigroup ( )S t has a bounded absorbing
set 0B in
1
0 ( )ΩH and ( )S t is asymptotically
compact in 10 ( )ΩH , that is, for any 0>t , it can be
decomposed in the form
1 2( ) ( ) ( ),= +S t S t S t
where for any bounded subset B in 10 ( )ΩH ,
we have
i) 1( )S t is a continuous mapping from
1
0 ( )ΩH into
itself and 1( ) sup ( ) 0 as ;
∈
= → → +∞B
y B
r t S t y t‖ ‖
ii) The operators 2 ( )S t are uniformly compact for
t large, i.e.,
0
2 ( )
≥
t t
S t B is relatively compact in
1
0 ( )ΩH for some 0>t .
It is clear that we only need to verify conditions i)
and ii) above for the absorbing set 0B .
Lemma 1. Assume that (H1), (H2) hold. Then
there exists a bounded absorbing set in 10 ( )ΩH for
the semigroup ( )S t .
Proof. Multiplying the equation (1) by ( )u t , we have
By the Cauchy inequality, we get
Thus,
where 1 1
1
min 1 ; 0.
= − − − − >
b
γ ε l b l ε
l
According to Gronwall Lemma, we obtain
Now, we can choose 1T and 0ρ such that
2
0( ) ,∇ ≤u t ρ‖ ‖ for all 1≥t T and for all 0 ∈u B . This
completes the proof.
Recall that in this paper we only assume the
external force 1( )−∈ Ωg H . However, we know
that for any 1( )−∈ Ωg H and 0>ε given, there is a
2 ( )∈ Ωg Lε , which depends on g and ε, such that
1 ( )
.− Ω− <Hg g
ε ε‖ ‖
To make the asymptotic regular estimates, we
decompose the solution 0( ) ( )=S t u u t of problem
(1) as follows
0 1 0 2 0( ) ( ) ( ) ,= +S t u S t u S t u
where 1 0 1( ) ( )=S t u u t and 02 2( ) ( )=S t u u t , that is
the decomposition is of the following form
= +u v wε ε ,
where ( )v tε is the unique solution of the following
problem
( )
0 0
( ) ( )
, ,
( , ) | 0, ( , ) | ( ,
6
)
t t
t
v v v f u f w v
g g
v x t v x t u x
ε ε ε ε ε
ε
ε ε
l
l
∂Ω =
− D − D + − +
= − >
= =
(6)
and ( )w t is the unique solution of the following
problem
( )
0
( ) ( ) ,
7
( , ) | 0, ( , ) | 0.
t t
t
w w w f w u w g
w x t w x t
ε ε ε ε ε ε
ε ε
l l
∂Ω =
− D − D + − − = >
= =
As in the proof of Theorem 1, one can prove the
existence and uniqueness of solutions to (6)
and (7).
Lemma 2. Let hypotheses (H1), (H2) hold. Then
the solutions of (6) satisfy the following estimates:
there is a constant 0d depending on 1,l , such
that for every 0≥t ,
0
1
0
2
1 0 0( )
( ) ( .−
Ω
≤ ∇ +d t
H
S t u Q u e ε‖ ‖ ‖ ‖)
Proof. Multiplying the first equation of (6) by v,
then integrating over Ω we get
NGÀNH TOÁN HỌC
Tạp chí Nghiên cứu khoa học - Đại học Sao Đỏ, ISSN 1859-4190 Số 4(59).2017 67
Note that ( )′ ≥ −f s and
1 1 1
0
2 2
, ( )
1 1,
2 2− − Ω
〈 − 〉 ≤ ∇ + −
H H H
g g v v g gε ε ε ε‖ ‖ ‖ ‖ ,
we get
Similarly to the proof of Lemma 1, we obtain
02
1 0 0( ) ( ) .
−≤ ∇ +d tS t u Q u e ε‖ ‖ ‖ ‖
The proof is complete.
Lemma 3. Let hypotheses (H1), (H2) hold. Then,
there exists a positive constant M such that for
any 10 0 ( )∈ Ωu H , there exists 0>T large enough,
which depends on 12 0( ) , ,− Ω ∇Hg uε‖ ‖ ‖ ‖, such that
2
2
2 0 ( )
( ) , for all .
Ω
≤ ≥
H
S t u M t T‖ ‖
Proof. Multiplying the first equation of (7) by
−Dw , we get
when 0 ( )≥t t B . Notice that, similarly to the proof
of Lemma 1, we obtain a 0>T large enough such
that 222 0 ( )( ) , .Ω ≤ ∀ ≥HS t u M t T‖ ‖
The proof is complete.
Since the embedding 2 1 10 0( ) ( ) ( )Ω ∩ Ω ⊂⊂ ΩH H H
is compact, we obtain
Lemma 4. Let 2 0{ ( )} ≥tS t be the solution semigroup
of (7). Then for large enough, ( )2 S T B is
relatively compact in 10 ( )ΩH .
By Lemma 1, the semigroup ( )S t has a bounded
absorbing set 0B in
1
0 ( )ΩH . Moreover, the
semigroup ( )S t is asymptotically compact in
1
0 ( )ΩH due to Lemmas 2 and 4. Therefore, the w -
limit set 0( )= Bw is the compact global attractor
for ( )S t in 10 ( )ΩH .
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