Pullback Exponential Attractors for Nonautonomous Reaction Diffusion Equations in H01

(1)

http://dx.doi.org/10.4236/jamp.2015.37087

Pullback Exponential Attractors for

Nonautonomous Reaction Diffusion

Equations in

1

0

H

Yongjun Li, Yanhong Zhang, Xiaona Wei

School of Mathematics, Lanzhou City University, Lanzhou, China Email: [email protected]

Received 1 March 2015; accepted 23 June 2015; published 30 June 2015

Abstract

Under the assumption that g t( ) is translation bounded in Lloc R L

4 4

( ; ( ))Ω , and using the method

developed in [3], we prove the existence of pullback exponential attractors in H01( )Ω for nonli-near reaction diffusion equation with polynomial growth nonlinonli-nearity(p≥2 is arbitrary).

Keywords

Dynamical System, Pullback Exponential Attractors, Reaction Diffusion Equation

1. Introduction

Attractor’s theory is very important to describe the long time behavior of dissipative dynamical systems gener-ated by evolution equations, and there are several kinds of attractors. In this article, we will study the existence of pullback exponential attractors (see [1]-[3]) for nonlinear reaction diffusion equation. This equation is written in the following form:

.

( ) ( ), ,

| 0,

( )

u

u f u g t x t

u uτ u_τ

Ω ∂

 _{− ∆ +} ₌ _{∈ Ω}

 ∂ 

= 

 ₌

 

(1.1)

where Ω is a bounded smooth domain in Rn, g( )⋅ ∈L_loc2 ( ,R L2( ))Ω , f∈C R R1( , ) and there exist p≥2,

0,

i

c > i=1, 2,5,l>0 such that

2 1| | 2 ( ) 3| | 4, ( ) , | ( ) | 5(1 | | )

p p p

c u − ≤c f u u≤c u +c f u′ ≥ −l f u′ ≤c + u − (1.2)

(2)

The Equation of (1.1) has been widely studied. For the autonomous case, i.e., g t( ) does not depend on the time, the asymptotic behaviors of the solution have been studied extensively in the framework of global attractor, see [4]-[6]. For the nonautonomous case, the asymptotic behaviors of the solution have been studied in the framework of pullback attractor, see [7]-[9]. Recently, the theory of pullback exponential attractor have been developed, see [1]-[3], and some methods are given to prove the existence of pullback exponential attractors.

In order to obtain the existence of pullback exponential attractors of (1.1), we will need the following theo-rem.

Theorem 1.1. ([3]) Let X be an uniformly convex Banach space, B X( ) be the set of all bounded subsets of X, { ( , ) |U tτ t≥τ} be a time continuous process in X. Then the process { ( , ) |U tτ t≥τ} exist pullback exponential attractors in X if the following conditions hold true:

(1) There exists an uniformly bounded absorbing set B⊂X , that is, for any t≥τ and D∈B X( ), there exists T₀ >0 such that

0

( , ) ,

U tτ −s D⊂B ∀ ≤s T (1.3)

(2) There exist δ θ, ,>0,δ θ+ <1, ,T l₁ >0, and a finite dimension subspace X₁⊂X, such that

1 2 1 2 1 1

||U t( , )τ u −U t( , )τ u ||≤l u|| −u ||,∀t,τ∈[kT, (k+1) ],T ∀ ∈k , (1.4)

1 1 1 2 1 2

|| (I−Pm)( ( ,U t t T u− ) −U t t T u( , − ) ) ||≤δ ||u −u ||, (1.5)

1

|| ( m) ( , ) || , ,

s T

I P U tτ s u θ t τ ≤

− _ − ≤ ∀ ≥ (1.6)

for all u u u, ,₁ ₂∈B and t∈R, where δ is independent on the choice of t, and || ||⋅ is the norm in X, I

is the identity operator, Pm:X →X₁ is a bounded projector, m is the dimension of X1.

2. Some Estimates of Equation (1.1)

In this section, we will derive some priori estimates for the solutions of (1.1) that will be used to construct pull-back exponential attractors for the problem (1.1).

For convenience, hereafter let | |⋅ p be the norm of ( )( 1),

p

L Ω p> and c an arbitrary constant, which may

difference from line to line and even in the same line. We define H=L2( )Ω with scalar product ( )⋅ and norm | |⋅ ; let (( ))⋅ and || ||⋅ denote the scalar product and norm of H₀1( )Ω and (( , ))u v u vdx

Ω

= ∇ ∇

_∫

for all

1 0

, ( )

u v∈H Ω , set λ is the first eigenvalue of −∆.

For the initial value problem (1.1), we know from [4]-[6] that for any initial datum u₀∈H, there exists a

unique solution 2 1

0

( ) ([ , ]; ) ([ , ]; )

u t ∈C

τ

T H L

τ

T H for any T>τ.

Thanks to the existence theorem, the initial value problem is equivalent to a process { ( , ) |U tτ t≥τ} define by

1 0

( , ) :

U t

τ

H→H .

In addition, we assume that the function g t( ) is translation bounded in L3loc( ;R L3( ))Ω , that is

1 ₃

3

sup t | ( ) | .

t t R

g s ds

+

∈

∫

< ∞

(2.1)

By (2.1), for λ >0, we have

1 ₂ ₂ ₍ ₎ ₂

1 0

sup t | ( ) | , t t s| ( ) | t n t s | ( ) | .

t t n

t R n

g s ds c e λ _τeλ g s dx e λ g s ds c ∞

+ ₋ − ₋ ₋

− −

∈ =

< ≤

∑

≤

∫

(2.2)

Lemma 2.1. ([7]-[9]) Assume that f and g satisfy (1.2) and (2.2), u t( ) be a weak solution of (1.1), then for any t≥τ, we have the following inequality:

2 ( ) 2

| ( ) | t s | |

u t ≤e−λ − u_τ +c (2.3)

and

2 2 1 2

1

(|| ( ) || 2 | ( ) | ) (1 ( )) | | | ( ) |

t _s _p _t t _s

p

eλ u s c u s ds t eλτ u_τ ceλ eλ g s ds

τ λ τ λ τ

−

+ ≤ + − + +

(3)

Lemma 2.2. Assume that f and g satisfy (1.2) and (2.2), u t( ) be a weak solution of (1.1), then the follow-ing inequality holds for t>

τ

2 2 1 ( ) 2 1

| ( ) | || ( ) || | ( ) |p (1 )(1 ) t | | (1 )

p

u t u t u t c t e u c

t t

λ τ τ τ

τ − − τ

+ + ≤ + + − + +

− − (2.5)

Obviously, for any bounded D⊂H , there exist T r, >0, such that

2 2

|U t( , )

τ

uτ | +||U t( , )

τ

uτ || +|U t( , )

τ

uτ|pp≤r for any uτ∈D and t− ≥τ T. (2.6)

Proof. Let

0

( ) s ( )

F s =

∫

f

τ τ

d , then by (1.2), we get there exist ci>0, i=1, 2,3, 4, such that

   

1| | 2 ( ) 3| | 4

p p

c u − ≤c F s ≤c u +c . (2.7)

Taking inner product of (1.1) with u in H and using (2.7), we get

2 2 2

| | || || ( ) (1 | ( ) | )

d

u u c F u dx c g t

dt + +

∫

Ω ≤ + . (2.8)

Multiply (1.1) by u_t, we have

2 1 2

| | (|| || 2 ( ) ) ( ( ), ),

2

t t

d

u u F u dx g t u

dt Ω

+ +

_∫

=

since | ( ( ), ) | 1(| ( ) |2 | | )2 2

t t

g t u ≤ g t + u , we obtain

2 2

(|| || 2 ( ) ) | ( ) |

d

u F u dx g t

dt +

∫

Ω ≤ .

Combining (2.7), we get

2 2 2 2

(| | || || 2 ( ) ) || || ( ) (1 | ( ) | )

d

u u F u dx u c F u dx c g t

dt + +

∫

Ω + +

∫

Ω ≤ + . (2.9)

Thanks to Poincaré inequality || ||u ≥λ|u|, we have

2 2 1 2 2 2

|| || ( ) | | || || ( ) (| | || || ( ) )

2 2

u +c

∫

_ΩF u dx≥λ u + u +c

∫

_ΩF u dx≥c u + u +c

∫

_ΩF u dx . (2.10)

Let 2 2

( ) | | || || ( )

G u = u + u +c

∫

_ΩF u dx, by (2.9) and (2.10), we obtain

2

( ) ( ) (1 | ( ) | )

d

G u cG u c g t

dt + ≤ + ,

which imply

2

(( ) t ( )) (1 ( )( )) ( ) t (1 | ( ) | )( ) t

d

t e G u c t G u e c g t t e

dt

λ λ λ

τ λ τ τ

− ≤ + − − + + − ,

integrating, we get

2

(t−τ)e G uλt ( )≤ +(1 c t( −τ))

∫

_τtG u e ds c t( ) λs + ( −τ)eλt+c t( −τ)

∫

_τt| ( ) |g s e dsλs , using (2.3) and (2.4), we get the inequality (2.5).

Lemma 2.3. Assume that f and g satisfy (1.2) and (2.1), u t( ) be a weak solution of (1.1), then the fol-lowing inequality holds for t>τ₀

0 0

( ) ( )

2 2 2

2 2 0

0

1

| ( ) | p (1 )(1 ) t | | t | |p

p p

u t c t e u e u

t

λ τ λ τ

τ τ

− − − −

−

− ≤ + ₋ + − + , (2.11)

Here

0 0

( ) ( , ) , ( , )

(4)

By the assumption (2.1) and for λ >0, we get

3 4 1 3 4

1

sup | ( ) |

p

t _s _p

p

t _p

eλ g s ds

τ τ

− − −

>

∫

₋ < ∞

. (2.12)

Proof. Multiply (1.1) with | |u p−2u, we obtain

( 2) 2 2 ( 2)

1

| |pp ( 1) | |p | | ( ) | |p ( ) | |p

d

u p u u dx f u u udx g t u udx

p dt

− − −

Ω Ω Ω

+ −

_∫

∇ +

_∫

=

_∫

. (2.13)

By (1.2) and Young’s inequality, we have

2 2 2

1 2

( ) | |p | | p

f u u − u≥c u′ − −c′, 2 1 2 2 2

2 2 1 1 | ( ) | | | | | | ( ) | 2 2 p p p c

g t u udx u g t

c − − − Ω ′ ≤ + ′

∫

.

By (2.13), we get

2 2 2 2 2 2

2 2 2 2

| |p | | p (1 | ( ) | ), ( t| | )p t| | p t(1 | |p | ( ) | ),

p p p p p

d d

u c u c g t e u ce u ce u g t

dt dt

λ λ λ

− −

+ ≤ + + ≤ + +

integrating and using (2.4), we get

0 0

2 2 2 2

2 2 0

| | ((1 ) | | | | | ( ) |

t _s _p _p _t t _s

p p

eλ u ds c t eλτ u_τ eλτ u_τ eλ eλ g s ds

τ τ τ

−

− ≤ + − + + +

∫

. (2.14)

Multiply (1.1) with | |u 2p−4u, we obtain

2 2 2 4 2 2 4 2 4

2 2

1

| | (2 3) | | | | ( ) | | ( ) | |

2 2

p p p p

p d

u p u u dx f u u udx g t u udx

p dt

− − − −

− + − _Ω ∇ + _Ω = _Ω

−

∫

.

By (2.1), we get

2 4 3 4

1 2

( ) | | p | |p

f u u − u≥c u′′ − −c′′.

Using Young’s inequality

3 4

2 4 1 3 4 1

3 4 3 4

1

| ( ) | | | | | | ( ) |

2

p

p p p

p p

p

c

g t u udx u c g t

− − − − − − Ω − ′′ ≤ +

∫

.

By the above inequality, we have

3 4 3 4

2 2 1 2 2 2 2 1

2 2 3 4 0 2 2 0 2 2 0 3 4

1 1

| | (1 | ( ) | ), (( ) | | ) (1 ( ) | | ( ) (1 | ( ) | ),

p p

p p t p t p t p

p p p p p

p p

d d

u c g t t e u t e u c t e g t

dt dt

λ λ λ

τ λ τ τ

− −

− − − − −

− −

≤ + − ≤ + − + − +

integrating and using (2.12) and (2.14), we get (2.11) holds.

Lemma 2.1, lemma 2.2 and lemma 2.3 show that the process generated by the equation (1.1) have an un-iformly pullback bounded absorbing set in H H L, ₀1, p( ),Ω L2p−2( )Ω , that is

Theorem 2.4. Assume that f and g satisfy (1.2) and (2.1), u t be a weak solution of( ) (1.1), then the process generated by the equation (1.1) have an uniformly pullback bounded absorbing set

1 2 2

0 ( ) ( )

p p

B⊂H∩H ∩L Ω ∩L − Ω , that is, for any bounded set D⊂H , there exists T₀ >0, such that

( , )

U t t−τ D⊂B for any t− ≥τ T₀.

In fact, using the same proof as in Lemma 2.3, we can get the following result.

Lemma 2.5. Assume that f satisfies (1.2), g t is translation bounded in ( ) L4loc( ;R L4( ))Ω , that is

1 ₄

4

sup t | ( ) | ,

t t R

g s ds

+

∈

∫

< ∞ u t( )

be a weak solution of (1.1), then the process generated by the equation (1.1) have

an uniformly pullback bounded absorbing set B⊂L4p−6( )Ω , that is, for any bounded set D⊂H , there exists

0 0

T > , such that U t t( , −τ)D⊂B for any t− ≥τ T₀.

3. Pullback Exponential Attractors

(5)

exponential attractor.

First we assume that the function g t( ) is normal ([10]) in L2loc( ,R H), that is, for any ε >0, there exists

0

η> such that

2

sup t | ( ) |

t t R

g s ds

η

ε +

∈

∫

< . (3.1)

Obviously, g t( ) is normal in L2loc( ,R H) implying that g t( ) is translation bounded in

2

( , )

loc

L R H .

We set A= −∆, since 1

A− is a continuous compact operator in H, by the classical spectral theorem, there exist a sequence { }

λ

_j ∞_j₌₁, 0<λ₁≤λ₂ ≤≤λj ≤,λj → +∞as j→ +∞, and a family of elements { }ej j 1

∞ = of 1

0

H which are orthogonal in H such that Ae_j =λ_je_j, ∀ ∈j +. Let Hm =span e e{ ,₁ ₂,,em} in H and

: m

P H →H is a orthogonal projector. For any u∈H, we write

1 2

( )

u=Pu+ −I P uu +u .

Theorem 2.4. Assume that f satisfies (1.2), g t is translation bounded in ( ) L4loc( ;R L4( ))Ω and (3.1) holds, then the process generated by the equation (1.1) have a pullback exponential attractor.

Next, we will verify that the process generated by (1.1) satisfy all the conditions of Theorem 1.1.

Proof. ByTheorem 2.4, there exists T₀>0, such that U t t( , −τ)B⊂B for any t− ≥τ T₀. Let

0 ( , ) t R T

B′ = ∪ ∪∈ _τ≤ U t t−

τ

B , we obtain B′ is also an uniformly pullback bounded absorbing set in

1 2 2

0 ( )

p

H ∩L − Ω and U t t( , −τ ′)B ⊂B′ for any t≥τ.

We set u t₁( )=U t( , )τ u₁_τ, u t₂( )=U t( , )τ u₂_τ to be solutions associated with Equation (1.1) with initial data

1 , 2

u uτ τ∈B′, since B′ is the uniformly pullback bounded absorbing set in 10 2 2( )

p

H ∩L − Ω , so there exists

0

M > such that ||uiτ|| M,||u ti( ) || M u t,| i( ) |2₂pp 2₂ M

− −

≤ ≤ ≤ , i=1, 2. Let w t( )=u t₁( )−u t₂( ), by (1.1), we get

1 1

( ( ) ( )) 0.

t

w − ∆ +w f u t −u t = (3.2)

Taking inner product of (3.2) with −∆w in H, we have

2 2

1 1

1

|| || || || ( ( ( ) ( )), ) 0.

2

d

w w f u t u t w

dt + ∆ + − −∆ = (3.3)

Taking into account (1.2) and Holder inequality, it is immediate to see that

2 2

1 2 1 2 1 2

1 1

| ( ( ) ( ), ) | | ( ) ( ) | | | | | | ( ) ( ) |

2 2

f u −f u −∆w ≤

∫

_Ω f u − f u ∆w dx≤ ∆w +

∫

_Ω f u −f u dx,

and

2 ₂ ₂ ₂ _{2 2} ₂

1 2 1 2 1 2 1 1 2 2 1

2 3 2 3 4 6 4 6 2

1 2 1 2 1 4 6 2 4 6

| ( ) ( ) | | ( ( )) | | (1 | | | | ) | |

(1 | | | | ) | | (1 | | | | ) | | ,

p p

p p p p

p p

f u f u dx f u u u u u dx c u u u u dx

c u u u u dx c u u w

θ − −

Ω Ω Ω

− − − −

− −

Ω

′

− = + − − ≤ + + −

≤ + + − ≤ + +

∫

By Lemma 2.5, we get

2 ₂

1 2

| f u( ) f u( ) | dx c w|| || .

Ω − ≤

∫

(3.4)

Using (3.3), we obtain 2 2

|| || || ||

d

w c w

dt ≤ , hence

2 2 ( )

|| ( ) ||w t ≤|| ( ) ||w

τ

ec t−τ . (3.5) Let w=w₁+w₂, w₁ be the project in PH. Taking inner product of (3.2) with −∆w₂ in H, we have

2 2

2 2 1 2 2

1

|| || | | ( ( ) ( ), ) 0

2

d

w w f u f u w

(6)

2 2

1 2 2 1 2 2 2 1 2

1 1

| ( ( ) ( ), ) | | ( ) ( ) | | | | | | ( ) ( ) |

2 2

f u − f u −∆w ≤

∫

_Ω f u − f u ∆w dx≤ ∆w +

∫

_Ω f u − f u dx.

Taking into (3.4) account, we obtain

2 2 2

2 2

|| || | | || ||

d

w w c w

dt + ∆ ≤ ,

Using the Poincaré inequality

λ

n||w₂||2≤ ∆| w₂|2, we get

2 2 2

2 2

|| || _n|| || || ||

d

w w c w

dt +λ ≤ , by Gronwall’s

Lemma, we have ||_{w t}₂( ) ||2 _e λn(tτ)|| ( ) ||_w 2 _ce λnt t_eλns|| ( ) ||_{w s} 2 _ds

τ τ

− − − −

≤ +

_∫

. Using (3.5), we get

( ) ( )

2 2

2

|| ( ) || ( n ) || ( ) ||

c t t

n

e

w t e c w

τ

λ τ _τ

λ − − −

≤ + . (3.7)

Let u t( )=u t₁( )+u t₂( ), u t₁( ) be the project in PH. Taking inner product of (1.1) with −∆u₂, we get

2 2

2 2 2 2

1

|| || || || ( ( ( )), ) ( ( ), ).

2

d

u u f u t u g t u

dt + ∆ + −∆ = −∆

Since | ( ( ), ₂) | | ( ) |2 1| ₂|2 4

g t −∆u ≤ g t + ∆u , | ( ( ( )), ₂) | | ( ) |2 1| ₂|2 4

f u t −∆u ≤

∫

_Ω f u dx+ ∆u , and by Poincaré

in-equality

λ

n||u₂|| |2≤ ∆u₂|2, we have

2 2 2

2 2

|| || _n|| || (1 | ( ) | ).

d

u u c g t

dt +λ ≤ +

By Gronwall’s lemma, we get

( )

2 2 2

2

|_{u t}( ) | _eλnt τ |_u | _ce λnt t_eλns(1 | ( ) | )_{g s} _ds

τ _τ

− − −

≤ +

_∫

+ .

By (3.1), we obtain that there exists c>0, such that t1| ( ) |2

t g s ds c

+

<

∫

for any t∈R, and for any ε >0,

there exists η>0, such that _{| ( ) |}2

3 t

tη g s ds

ε

− <

∫

, so we get

( )

2 2

( ) 2 ( ) 2

1

1 ₍ ₎ ₂ ₍ ₎ ₂

2 1 2 | ( ) | | ( ) | | ( ) | | ( ) | | ( ) | | ( ) | (1 ) 3 3

n n n

n n

n n n n

t t

t s t s

t _{t s} t _{t s}

t t

t _{t s} t k _{t s}

t t k

k

e e g s ds e g s ds

e g s ds e g s ds

ce e e e

λ λ λ

τ τ

η

λ λ

η η

η _λ η _λ

η η

λ η λ λ λ

ε ε − − − − − − − − − − − − − − ₋ ₋ − − ₋ ₋ − − − − − − − − − = ≤ + + + + + ≤ + + + + + + ≤ +

∫



∫

   1 n n e e λ η λ − − −

and nt t ns 1

n e λ _τeλ ds

λ

− _≤

∫

, we have

( ) 2 2 2 1 | ( ) | | | ( ). 3 1 n n n t n e

u t e u c

e λ η λ τ τ λ ε λ − − − − ≤ + + +

− (3.8)

Let T₁= − =t τ 1, by (3.5), we get

1 2 1 2

||U t( , )

τ

u_τ−U t( , )

τ

u_τ||≤ec||u_τ −u_τ|| . (3.9) Since λ → +∞_n , for 0< <ε 1, from (3.7) and (3.8), there exist m∈+, δ θ, ,>0,δ θ+ <1 such that

1 1 1 2 1 2

|| (I−Pm)( ( ,U t t T u− ) −U t t T u( , − ) ) ||≤δ ||u −u ||, (3.10)

1

|| ( m) ( , ) || , .

s T

I P U tτ s u θ t τ ≤

(7)

By Theorem 2.4 and (3.9)-(3.11), we know that the process { ( , ) |U tτ t≥τ} generated by (1.1) satisfy all the conditions of Theorem 1.1.

Funds

This work was supported by the National Nature Science Foundation of China (11261027) and Longyuan youth innovative talents support programs of 2014, and the innovation Funds of principal (LZCU-XZ2014-05).

References

[1] Langa, J., Miranville, A. and Real, J. (2010) Pullback Exponential Attractors. Discrete and Continuous Dynamical Systems—Series A, 26, 1329-1357.

[2] Czaja, R. and Efendiev, M. (2011) Pullback Exponential Attractors for Non-Autonomous Equations, Part I: Semilinear parabolic Problems. Journal of Mathematical Analysis and Applications, 381, 748-765.

http://dx.doi.org/10.1016/j.jmaa.2011.03.053

[3] Li, Y., Wang, S. and Zhao, T. (2015) The Existence of Pullback Exponential Attractors for Non-Autonomous Dynam-ical System and Application to Non-Autonomous Reaction Diffusion Equations. J. Appl. Anal. Comp (in press).

[4] Chepyzhov, V. and Vishik, M. (2002) Attractors for Equations of Mathematics Physics. 49, American Mathematical Society Colloquium Publications, AMS.

[5] Temam, R. (1997) Infinite-Dimensional Dynamical Systems in Mechanics and Physics. 68, Springer, New York.

http://dx.doi.org/10.1007/978-1-4612-0645-3

[6] Ladyzhenskaya, O. (1991)Attractors for Semigroups and Evolution Equations. Cambridge University Press, Cam-bridge, UK. http://dx.doi.org/10.1017/CBO9780511569418

[7] Song, H. and Wu, H. (2007) Pullback Attractor for Nonlinear Autonomous Reaction Diffusion Equations. Journal of Mathematical Analysis and Applications, 325, 1200-1215. http://dx.doi.org/10.1016/j.jmaa.2006.02.041

[8] Song, H. (2010) Pullback Attractors of Nonautonomous Reaction Diffusion Equation in 1 0

H . Journal of Differential Equations, 249, 2357-2376. http://dx.doi.org/10.1016/j.jde.2010.07.034

[9] Li, Y. and Zhong, C. (2007) Pullback Attractors for the Norm-to-Weak Continuous Process and Application to the Non-Autonomous Reaction Diffusion Equations. Appl. Math. Comp., 190, 1020-1029.

http://dx.doi.org/10.1016/j.amc.2006.11.187