Pullback Exponential Attractors for

Journal of Applied Mathematics and Physics, 2015, 3, 730-736 Published Online July 2015 in SciRes. http://www.scirp.org/journal/jamp http://dx.doi.org/10.4236/jamp.2015.37087

Pullback Exponential Attractors for Nonautonomous Reaction Diffusion Equations in H 01 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 L4loc ( R; L4 (Ω )) , and using the method developed in [3], we prove the existence of pullback exponential attractors in H 01 (Ω ) for nonlinear reaction diffusion equation with polynomial growth nonlinearity( 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 generated 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:  ∂u f (u ) g (t ), x ∈ Ω,  ∂t − ∆u + =  u |Ω = 0, u (τ ) = u τ.  

(1.1)

where Ω is a bounded smooth domain in R n , g (⋅) ∈ L2loc ( R, L2 (Ω)) , f ∈ C1 ( R, R) and there exist p ≥ 2, ci > 0, i 1, 2, 5, l > 0 such that =

c1 | u | p −c2 ≤ f (u )u ≤ c3 | u | p +c4 , f ′(u ) ≥ −l , | f ′(u ) |≤ c5 (1+ | u | p − 2 )

(1.2)

for all u ∈ R . How to cite this paper: Li, Y.J., Zhang, Y.H. and Wei, X.N. (2015) Pullback Exponential Attractors for Nonautonomous Reaction Diffusion Equations in H 01 . Journal of Applied Mathematics and Physics, 3, 730-736. http://dx.doi.org/10.4236/jamp.2015.37087

Y. J. Li et al.

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 theorem. Theorem 1.1. ([3]) Let X be an uniformly convex Banach space, B ( X ) be the set of all bounded subsets of X , {U (t ,τ ) | t ≥ τ } be a time continuous process in X . Then the process {U (t ,τ ) | 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

U (t ,τ − s ) D ⊂ B, ∀s ≤ T0

(1.3)

(2) There exist δ , θ , > 0, δ + θ < 1, T1 , l > 0 , and a finite dimension subspace X 1 ⊂ X , such that

|| U (t ,τ )u1 − U (t ,τ )u2 ||≤ l || u1 − u2 ||, ∀t ,τ ∈ [kT1 , (k + 1)T1 ], ∀k ∈ ,

(1.4)

|| ( I − Pm )(U (t , t − T1 )u1 − U (t , t − T1 )u2 ) ||≤ δ || u1 − u2 ||,

(1.5)

|| ( I − Pm )  U (t ,τ − s )u ||≤ θ , ∀t ≥ τ ,

(1.6)

s ≤ T1

for all u , u1 , u2 ∈ 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 → X 1 is a bounded projector, m is the dimension of X 1 .

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 pullback exponential attractors for the problem (1.1). For convenience, hereafter let | ⋅ | p be the norm of Lp (Ω)( p > 1), 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 H 01 (Ω) and ((u , v)) = ∫ ∇u∇vdx for all Ω

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 1 0

unique solution u (t ) ∈ C ([τ , T ]; H )  L2 ([τ , T ]; H 01 ) for any T > τ . Thanks to the existence theorem, the initial value problem is equivalent to a process {U (t ,τ ) | t ≥ τ } define by

U (t ,τ ) : H → H 01 . In addition, we assume that the function g (t ) is translation bounded in L3loc ( R; L3 (Ω)) , that is t +1

sup ∫t | g ( s ) |33ds < ∞.

(2.1)

t ∈R

By (2.1), for λ > 0 , we have

sup ∫ t ∈R

t +1 t

∞

| g ( s ) |2 ds < c, e − λ t ∫ τ eλ s | g ( s ) |2 dx ≤ ∑ ∫ t

n=0

t −n t − n −1

e − λ (t − s ) | g ( s ) |2 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:

| u (t ) |2 ≤ e − λ (t − s ) | uτ |2 +c

(2.3)

and t

∫τ e

λs

t

(|| u ( s ) ||2 +2c1 | u ( s ) | pp )ds ≤ (1 + λ (t − τ ))eλτ | uτ |2 +ceλ t + λ −1 ∫ eλ s | g ( s ) |2 ds τ

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(2.4)

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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 following inequality holds for t > τ | u (t ) |2 + || u (t ) ||2 + | u (t ) | pp ≤ c(1 +

1 1 )(1 + t − τ )e − λ (t −τ ) | uτ |2 +c(1 + ) t −τ t −τ

(2.5)

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

| U (t ,τ )uτ |2 + || U (t ,τ )uτ ||2 + | U (t ,τ )uτ | pp ≤ r for any uτ ∈ D and t − τ ≥ T .

(2.6)

s Proof. Let F ( s ) = ∫ f (τ )dτ , then by (1.2), we get there exist ci > 0 , i = 1, 2,3, 4 , such that 0

c1 | u | p −c2 ≤ F ( s ) ≤ c3 | u | p +c4 .

(2.7)

Taking inner product of (1.1) with u in H and using (2.7), we get d | u |2 + || u ||2 +c ∫ Ω F (u )dx ≤ c(1+ | g (t ) |2 ) . dt

(2.8)

Multiply (1.1) by ut , we have | ut |2 +

since | ( g (t ), ut ) |≤

1 d (|| u ||2 +2 ∫ Ω F (u )dx) = ( g (t ), ut ), 2 dt

1 (| g (t ) |2 + | ut |2 ) , we obtain 2 d (|| u ||2 +2 ∫ F (u )dx) ≤| g (t ) |2 . Ω dt

Combining (2.7), we get d (| u |2 + || u ||2 +2 ∫ Ω F (u )dx) + || u ||2 +c ∫ Ω F (u )dx ≤ c(1+ | g (t ) |2 ) . dt

(2.9)

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

|| u ||2 +c ∫ Ω F (u )dx ≥

λ

1 | u |2 + || u ||2 +c ∫ Ω F (u )dx ≥ c(| u |2 + || u ||2 +c ∫ Ω F (u )dx) . 2 2

(2.10)

Let G (u ) = | u |2 + || u ||2 +c ∫ F (u )dx , by (2.9) and (2.10), we obtain Ω

d G (u ) + cG (u ) ≤ c(1+ | g (t ) |2 ) , dt

which imply d ((t − τ )eλ t G (u )) ≤ (1 + (λ − c)(t − τ ))G (u )eλ t + c(1+ | g (t ) |2 )(t − τ )eλ t , dt

integrating, we get t

t

(t − τ )eλ t G (u ) ≤ (1 + c(t − τ )) ∫ τ G (u )eλ s ds + c(t − τ )eλ t + c(t − τ ) ∫ τ | g ( s ) |2 eλ s ds , 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 following inequality holds for t > τ 0

| u (t ) |22 pp −− 22 ≤ c(1 +

1 )(1 + t − τ 0 )e − λ (t −τ 0 ) | uτ 0 |2 +e − λ (t −τ 0 ) | uτ 0 | pp , t −τ 0

= (t ,τ )uτ , uτ 0 U (τ 0 ,τ )uτ for any t > τ 0 > τ . Here u (t ) U=

732

(2.11)

Y. J. Li et al.

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

sup ∫ τ e | g ( s ) | λs

t >τ

3 p−4 p −1 3 p−4 p −1

ds < ∞ .

(2.12)

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

1 d ( p − 2) | u | pp +( p − 1) ∫ Ω | u |( p − 2) | ∇u |2 dx + ∫ Ω f (u ) | u | p − 2 udx = ∫ Ω g (t ) | u | udx . p dt

(2.13)

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

f (u ) | u | p − 2 u ≥ c1′ | u |2 p − 2 −c2′ , | ∫ Ω g (t ) | u | p − 2 udx |≤

c1′ 1 | u |22 pp −− 22 + | g (t ) |2 . 2 2c1′

By (2.13), we get d d λt | u | pp +c | u |22 pp −− 22 ≤ c(1+ | g (t ) |2 ), (e | u | pp ) + ceλ t | u |22 pp −− 22 ≤ ceλ t (1+ | u | pp + | g (t ) |2 ), dt dt

integrating and using (2.4), we get t

∫τ

0

t

eλ s | u |22 pp −− 22 ds ≤ c((1 + t − τ 0 )eλτ 0 | uτ 0 |2 +eλτ 0 | uτ 0 | pp +eλ t + ∫τ eλ s | g ( s ) |2 ds .

(2.14)

0

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

1 d 2 p−4 | u |22 pp −− 22 +(2 p − 3) ∫ Ω | u |2 p − 4 | ∇u |2 dx + ∫ Ω f (u ) | u |2 p − 4 udx = ∫ Ω g (t ) | u | udx . 2 p − 2 dt By (2.1), we get

f (u ) | u |2 p − 4 u ≥ c1′′| u |3 p − 4 −c2′′ . Using Young’s inequality 3 p−4

| ∫ Ω g (t ) | u |2 p − 4 udx |≤

c1′′ 3 p − 4 | u |3 p − 4 + c | g (t ) |3pp−−14 . 2 p −1

By the above inequality, we have 3 p−4

3 p−4

d d | u |22 pp −− 22 ≤ c(1+ | g (t ) |3pp−−14 ), ((t − τ 0 )eλ t | u |22 pp −− 22 ) ≤ (1 + λ (t − τ 0 )eλ t | u |22 pp −− 22 +c(t − τ 0 )eλ t (1+ | g (t ) |3pp−−14 ), dt dt p −1 p −1

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 uniformly pullback bounded absorbing set in H , H 01 , Lp (Ω), L2 p − 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 B ⊂ H ∩ H 01 ∩ Lp (Ω) ∩ L2 p − 2 (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 t +1

sup ∫ t | g ( s ) |44 ds < ∞, u (t ) be a weak solution of (1.1), then the process generated by the equation (1.1) have t ∈R

an uniformly pullback bounded absorbing set B ⊂ L4 p − 6 (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 .

3. Pullback Exponential Attractors In this section, we will use Theorem 1.1 to prove that the process generated by Equation (1.1) exists a pullback

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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 η > 0 such that sup ∫ t ∈R

t +η t

| g ( s ) |2 ds < ε .

(3.1)

Obviously, g (t ) is normal in L2loc ( R, H ) implying that g (t ) is translation bounded in L2loc ( R, H ) . We set A = −∆ , since A−1 is a continuous compact operator in H , by the classical spectral theorem, there exist a sequence {λ j }∞j =1 , 0 < λ1 ≤ λ2 ≤  ≤ λ j ≤  , λ j → +∞ as j → +∞, and a family of elements {e j }∞j =1 of

H 01 which are orthogonal in H such that Ae j = λ j e j , ∀j ∈  + . Let H m = span{e1 , e2 , , em } in H and P : H → H m is a orthogonal projector. For any u ∈ H , we write u = Pu + ( I − P )u  u1 + u2 .

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. By Theorem 2.4, there exists T0 > 0 , such that U (t , t − τ ) B ⊂ B for any t − τ ≥ T0 . Let B′ = ∪t∈R ∪τ ≤T0 U (t , t − τ ) B , we obtain B′ is also an uniformly pullback bounded absorbing set in

H 01 ∩ L2 p − 2 (Ω) and U (t , t − τ ) B′ ⊂ B′ for any t ≥ τ . We set u1 (t ) = U (t ,τ )u1τ , u2 (t ) = U (t ,τ )u2τ to be solutions associated with Equation (1.1) with initial data u1τ ,u2τ ∈ B′ , since B′ is the uniformly pullback bounded absorbing set in H 01 ∩ L2 p − 2 (Ω) , so there exists 2 p−2 (t ) u1 (t ) − u2 (t ) , by (1.1), we get M > 0 such that || uiτ ||≤ M ,|| ui (t ) ||≤ M ,| ui (t ) |2 p − 2 ≤ M , i = 1, 2. Let w=

wt − ∆w + f (u1 (t ) − u1 (t )) = 0.

(3.2)

Taking inner product of (3.2) with −∆w in H , we have 1 d || w ||2 + || ∆w ||2 +( f (u1 (t ) − u1 (t )), −∆w) = 0. 2 dt

(3.3)

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

| ( f (u1 ) − f (u2 ), −∆w) |≤ ∫ Ω | f (u1 ) − f (u2 ) | | ∆w | dx ≤

2 1 1 | ∆w |2 + ∫ Ω | f (u1 ) − f (u2 ) | dx , 2 2

and 2

= ∫ Ω | f ′(u1 + θ (u2 − u1 )) | u2 − u1 | dx ≤ c ∫ Ω (1+ | u1 | ∫ Ω | f (u1 ) − f (u2 ) | dx ≤ c ∫ Ω (1+ | u1 |2 p − 3 + | u2 |2 p − 3 ) | u1 − u2 | dx ≤ c(1+ | u1 |44 pp −− 66 + | u2 |44 pp −− 66 ) | w |2 , 2

2

p−2

+ | u2 | p − 2 ) 2 | u2 − u1 |2 dx

By Lemma 2.5, we get 2

∫ Ω | f (u1 ) − f (u2 ) | dx ≤ c || w || Using (3.3), we obtain

2

.

(3.4)

d || w ||2 ≤ c || w ||2 , hence dt

|| w(t ) ||2 ≤|| w(τ ) ||2 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 1 d || w2 ||2 + | ∆w2 |2 +( f (u1 ) − f (u2 ), −∆w2 ) = 0 . 2 dt

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(3.6)

Y. J. Li et al. 2 1 1 | ∆w2 |2 + ∫ Ω | f (u1 ) − f (u2 ) | dx . 2 2

| ( f (u1 ) − f (u2 ), −∆w2 ) |≤ ∫ Ω | f (u1 ) − f (u2 ) | | ∆w2 | dx ≤ Taking into (3.4) account, we obtain

d || w2 ||2 + | ∆w2 |2 ≤ c || w ||2 , dt

Using the Poincaré inequality λn || w2 ||2 ≤| ∆w2 |2 , we get

d || w2 ||2 +λn || w2 ||2 ≤ c || w ||2 , by Gronwall’s dt

t

Lemma, we have || w2 (t ) ||2 ≤ e − λn (t −τ ) || w(τ ) ||2 +ce − λn t ∫ e − λn s || w( s ) ||2 ds . Using (3.5), we get τ

|| w2 (t ) ||2 ≤ (e − λn (t −τ ) + c

ec (t −τ )

λn

) || w(τ ) ||2 .

(3.7)

Let u= (t ) u1 (t ) + u2 (t ) , u1 (t ) be the project in PH . Taking inner product of (1.1) with −∆u2 , we get 1 d = u2 ) ( g (t ), −∆u2 ). || u2 ||2 + || ∆u2 ||2 +( f (u (t )), −∆ 2 dt

1 1 Since | ( g (t ), −∆u2 ) |≤| g (t ) |2 + | ∆u2 |2 , | ( f (u (t )), −∆u2 ) |≤ ∫ | f (u ) |2 dx + | ∆u2 |2 , and by Poincaré inΩ 4 4 2 2 equality λn || u2 || ≤| ∆u2 | , we have d || u2 ||2 +λn || u2 ||2 ≤ c(1+ | g (t ) |2 ). dt

By Gronwall’s lemma, we get t

| u2 (t ) |2 ≤ e − λn (t −τ ) | uτ |2 +ce − λn t ∫ τ eλn s (1+ | g ( s ) |2 )ds . By (3.1), we obtain that there exists c > 0 , such that there exists η > 0 , such that

t

t +1

∫t

| g ( s ) |2 ds < c for any t ∈ R , and for any ε > 0 ,

ε

∫ t −η | g ( s) | ds < 3 , so we get

t

t

τ

τ

2

e − λn t ∫ e − λn s | g ( s ) |2 ds = ∫ e − λn (t − s ) | g ( s ) |2 ds t −η

t

≤ ∫ t −η e − λn (t − s ) | g ( s ) |2 ds + ∫ t −η −1 e − λn (t − s ) | g ( s ) |2 ds t −η −1 − λ ( t − s ) n

+ ∫ t −η − 2 e ≤

ε 3

t −η − k

| g ( s ) | ds +  + ∫ t 2

−η − k −1

e

− λn ( t − s )

+ ce − λnη (1 + e − λn + e −2 λn +  + e − k λn + ) ≤

t

1

τ

λn

and e − λn t ∫ eλn s ds ≤

| g ( s ) | ds +  2

, we have

e − λnη 3 1 − e − λn

ε

+

| u2 (t ) |2 ≤ e − λn (t −τ ) | uτ |2 +c(

1

λn

+

e − λnη ). 3 1 − e − λn

ε

+

(3.8)

Let T1 = t − τ =1 , by (3.5), we get

|| U (t ,τ )u1τ − U (t ,τ )u2τ ||≤ ec || u1τ − u2τ || .

(3.9)

Since λn → +∞ , for 0 < ε < 1 , from (3.7) and (3.8), there exist m ∈  + , δ , θ , > 0, δ + θ < 1 such that

|| ( I − Pm )(U (t , t − T1 )u1 − U (t , t − T1 )u2 ) ||≤ δ || u1 − u2 ||,

(3.10)

|| ( I − Pm )  U (t ,τ − s )u ||≤ θ , ∀t ≥ τ .

(3.11)

s ≤ T1

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By Theorem 2.4 and (3.9)-(3.11), we know that the process {U (t ,τ ) | 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).

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