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.
Trang 1GLOBAL 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 Tuan 1 , Nguyen Thi Hue 1 , Nguyen Thi Hong 1 , Nguyen Xuan Tu 2
Email: nguyentuandhsd@gmail.com
1 Sao Do University
2 Hung Vuong University
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 >
t t
(1)
whereΩis a bounded domain in Nwith 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 ∈ ,
where C, 0 are two positive constants, 0 < <b l1
with l1 is the first eigenvalue of the operator −D
in Ω with the homogeneous Dirichlet boundary condition;
(H2) The external force g H∈ − 1( ).Ω
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 L2−inner product and L2−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 C I X L I X( , ), ( , )p and H m p, ( , )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 L2( )Ω by utilizing the compactness method and weak
Trang 2convergence 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 ,
0 0 (0)= ∈ ( )Ω
and
1
1 1
, ,
, , , ( ),
,
,
H H
g
ϕ
∞
−
〈 〉 + 〈∇ ∇ 〉 + 〈∇ ∇ 〉 + 〈 〉
= 〈 〉
0( ) ∞( ),
∈W H= Ω ∩L Ω
a.e t∈[0, ]T
We now prove the existence and uniqueness
result for problem (1)
Theorem 1 Assume that conditions (H1), (H2)
0∈ 0( )Ω
given, there exists a unique weak solution u to
problem (1) on the interval [0, ]T Furthermore,
1 0 ([0, ]; ( )),
1 1
0 ( )∈ ( ( ),0 Ω 0( ))Ω ∀ ∈[0, ],
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 j
w of L2( )Ω which is also orthogonal
in 1
0( )Ω
H , consisting of normalized eigenfunctions
for −D in 1
0( )Ω
Step 1 (Feado-Galerkin scheme).
Given an integer n, denote by the projection on
span( , , )w …wn ⊂H ( )Ω We look for a function u n of the form
1
( ) ( )
=
=∑n
j
satisfying
1 ( )
, , ( ), ( , ), 1 ,
| ,
w
Ω
=
〈∂ − D∂ 〉
= 〈D 〉 − 〈 〉 + ≤ ≤
=
for a.e t T≤ We get a system of ODEs in the
variables a t k( ) of the form
(1 ) , ( ),
( + k k)= − k k+ k − n k
subject to the initial condition
1
0 ( ) (0)= , Ω
According to standard existence theory for ODEs,
there exists a solution on some interval (0, )T n The
a priori estimates below imply that in fact T n = +∞
Step 2 (Energy estimates).
Multiplying the equation (2) by a k, then summing over k and adding the results, we get
(3) Using (H1) we deduce that
So
where ε>0 is small enough so that
1
1− b ε− >0
l
Integrating on (0, ),t t∈(0, )T , leads to the following estimate
In particular, we see that { }u n is bounded in
1 0 (0, ; ( ))
Using the boundedness of { }u n in 1
0 (0, ; ( ))
it is easy to check that{Du n}is bounded in
2(0, ; − 1( ))Ω
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
n n
Step 3 (Passage to limits)
From (3), we get
(4) Integrating (4) from 0 to T, we have
Hence
0 Ω ( ) ≤
∫ ∫T f u u dxdt C n n
Trang 3We now prove that { ( )}f u n is bounded in L Q1( )T
Putting h s( )= f s( )− f(0)+γs, where γ >
Note that
2
( ) ( ( ) (0))
( ) ( ) 0
γ
ξ γ γ
= − +
′
= + ≥ − ≥
for all s∈ , we have
1
2
0
{| | 1} {| | 1}
| | 1
( ) 2
( )
| | 1
| ( ) |
| ( ) | | ( ) |
( ) sup | ( ) |
||
.| | ( ) | (0) |.||
sup | ( ) | | |
T
T T
T
T
n
Q
s
C
γ
Ω
∩ > ∩ ≤
≤
≤
≤
∫ ∫
∫
∫
＼ ＼
Hence it implies that { ( )}h u n , and therefore
{ ( )}f u n is bounded in L Q1( )T
Now, we prove the boundedness of {∂t n u}
In the first equation in (2), replacingwk by ∂t n u ,
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 n u }is bounded in 2 1
0 (0, ; ( ))Ω
So, up to a subsequence,
2 1
0 weakly in (0, ; ( ))
∂t n u u t L T H Ω ,
2 1 weakly in (0, ; −( ))
D∂t n u Du t L T H Ω
0( )Ω ⊂⊂ ( )Ω ⊂ −( )Ω + ( )Ω
Aubin-Lions-Simon compactness lemma, we
obtain u n →u strongly in L2(0, ; ( ))T L2 Ω Hence
we may assume, up to a subsequence, that
→
n
u u a.e in Q T Since f is continuous, it
follows that f u( )n → f u( ) a.e inΩ×[0, ]T We
obtain that h u( )∈L Q1( )T and for all test functions
1
0 ∞([0, ]; 0( ) ∞( ))
∈C T H Ω ∩L Ω
∫Q T h u dxdt n φ ∫Q T h u dxdtφ
Hence f u( )∈L Q1( )T and
( ) ( ) ,
([0, ];φ∈C∞ T H ( )Ω ∩L∞( ))Ω
By standard arguments, we can check that u
satisfies the initial condition u(0) =u0 and this implies that u is a weak solution of problem (1)
Step 4 (Uniqueness and continuous dependence
of the solutions).
We assume that u1 and u2 are two solutions of (1) with initial data u10 and u20, respectively Denote
1 2
= −
( )
1 2
t u t u u f u f u u t
where f sˆ( )= f s( )+ s Here because u t( ) does not belong to 1
0 := ( )Ω ∩ ∞( )Ω
choose u t( ) as a test function
Let
if , ( ) if | | ,
if
>
= ≤
− < −
k
Consider the corresponding Nemytskii mapping
ˆ : →k
B W W defined as follows B u xˆ ( )( )k =B u x k( ( )),
for all x∈ Ω We notice that ‖B u uˆ ( )k − ‖W→0
as k→ ∞ Now multiplying (5) by B uˆ ( )k , then integrating over Ω we get
Thus
Note that f sˆ ( ) 0′ ≥ and sB s k( ) 0≥ for all s∈ ,
we get ∫Ωfˆ′( )ξ uB u dxˆk( ) ≥0
{∈Ω:| ( , )| }≤ | |∇ ≥0
∫x u x t k u dx
Since the above inequalities, we get
Integrating from 0 to t, where t∈(0, )T , then letting k→ ∞, we obtain
Trang 4By the Gronwall inequality of integral form, we get
for all [0, ].t∈ T
0 ([0, ]; ( ))
u C T H , in particular, we get the uniqueness if u( )0 0.=
3 EXISTENCE OF A GLOBAL ATTRACTOR
Theorem 1 allows us to define a continuous
0( ) 0(
( Ω → Ω
problem (1) by the formula S t u( ) : ( ),0 =u t where
(.)
u is the unique weak solution of (1) with the
initial datum 1
0∈ 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 { ( )}S t t≥0 possesses a compact
global attractor in 1
0( )Ω
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 B0 in 1
0( )Ω
H and S t( ) is asymptotically compact in 1
0( )Ω
H , that is, for any t>0, it can be decomposed in the form
1 2
( )= ( )+ ( ),
where for any bounded subset B in 1
0( )Ω
we have
i) S t1( ) is a continuous mapping from 1
0( )Ω
itself and ( ) sup 1( ) 0 as ;
∈
= → → +∞
B
y B
ii) The operators S t2( ) are uniformly compact for
t large, i.e.,
0
2( )
≥
t t
S t B is relatively compact in
1
0( )Ω
H for some t>0
It is clear that we only need to verify conditions i)
and ii) above for the absorbing set B0
Lemma 1 Assume that (H1), (H2) hold Then
there exists a bounded absorbing set in 1
0( )Ω
the semigroup S t( ).
Proof Multiplying the equation (1) by u t( ), we have
By the Cauchy inequality, we get
Thus,
1 min 1 ; 0
= − − − − >
b
γ ε l b l ε
l
According to Gronwall Lemma, we obtain
Now, we can choose T1 and ρ0 such that
2 0 ( ) ,
∇u t ≤ρ
‖ ‖ for all t T≥ 1 and for all u0∈B This completes the proof
Recall that in this paper we only assume the external force g H∈ − 1( )Ω However, we know that for any g H∈ − 1( )Ω and ε >0 given, there is a
2( )
∈ Ω
gε L , which depends on g and ε, such that
1 ( )
− Ω
− H <
‖ ‖
To make the asymptotic regular estimates, we decompose the solution S t u( ) 0 =u t( ) of problem (1) as follows
0 1 0 2 0 ( ) = ( ) + ( ) ,
where S t u1( ) 0 =u t1( ) and S t u2( ) 0 =u t2( ), that is the decomposition is of the following form
u vε wε, where v tε( ) is the unique solution of the following problem
( )
0 0
( ) ( ) , ,
( , ) | 0, ( , ) | ( ,
6 )
t t
t
g g
ε
l l
− D − D + − +
= − >
(6)
and w t( ) is the unique solution of the following problem
( )
0
( ) ( ) ,
7 ( , ) | 0, ( , ) | 0
t
− 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 d0 depending on l1,, such that for every t≥0,
0
1 2
1( ) 0 ( ) ( 0 −
Ω ≤ ∇ d t+
H
‖ ‖ ‖ ‖)
Proof Multiplying the first equation of (6) by v, then integrating over Ω we get
Trang 5Note that f s′( )≥ −and
0
,
〈 − 〉 ≤ ∇ + −
we get
Similarly to the proof of Lemma 1, we obtain
0
2
1( ) 0 ≤ ( ∇ 0 ) −d t+
‖ ‖ ‖ ‖
The proof is complete
Lemma 3 Let hypotheses (H1), (H2) hold Then,
there exists a positive constant M such that for
0∈ 0( )Ω
u H , there exists T >0 large enough,
0 ( ), ,
− Ω ∇
H
2
2
2( ) 0 H ( )Ω≤ , for all ≥
Proof Multiplying the first equation of (7) by
−Dw, we get
when t t B≥ 0( ) Notice that, similarly to the proof
of Lemma 1, we obtain a T >0 large enough such
2( ) 0 H ( )Ω≤ ,∀ ≥
The proof is complete
( )Ω ∩ ( )Ω ⊂⊂ ( )Ω
is compact, we obtain
Lemma 4 Let { ( )}S t2 t≥0 be the solution semigroup
of (7) Then for large enough, S T B2( ) is relatively compact in 1
0( )Ω
By Lemma 1, the semigroup S t( ) has a bounded absorbing set B0 in 1
0( )Ω
H Moreover, the semigroup S t( ) is asymptotically compact in
1
0( )Ω
limit set =w( )B0 is the compact global attractor for S t( ) in 1
0( )Ω
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