diffusion equations with arbitrary polynomial growth

Nonlinear Analysis 71 (2009) 751–765 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Exponential attractors for reaction–diffusion equations with arbitrary polynomial growth Yansheng Zhong ∗ , Chengkui Zhong School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, PR China article a b s t r a c t info Article history: Received August 2008 Accepted 31 October 2008 In this paper, we study exponential attractors for an equation with arbitrary polynomial growth nonlinearity f and inhomogeneous term g First, we prove by the -trajectory method that the exponential attractor in L2 (Ω ) with g ∈ H −1 (Ω ) Second, by proving the semigroup satisfying discrete squeezing property, we obtain the exponential attractor in H01 (Ω ) with g ∈ L2 (Ω ) Because the solutions without higher regularity than L2p−2 (Ω ) for g belong only to L2 (Ω ) in the equation, the general method by proving the Lipschitz continuity between L2p−2 (Ω ) and L2 (Ω ) does not work in our case Therefore, we give a new method (presented in a theorem) to obtain an exponential attractor in a stronger topology space i.e., L2p−2 (Ω ) with g ∈ G (stated in a definition) when it is out of reach for the other known techniques © 2008 Elsevier Ltd All rights reserved Keywords: Exponential attractor Asymptotic a priori estimate Discrete squeezing property Semigroup Supercritical nonlinearity Global attractor Introduction In this paper, we consider the existence of an exponential attractor for the following reaction–diffusion equation  ∂u   − ∆u + f (u) = g ∂t  u = u(x, 0) = u0 in Ω × R+ (1.1) on ∂ Ω × R+ , in Ω , where Ω is a bounded smooth domain in Rn (n f (s) 3), f is a C function satisfying −κ (1.2) and C1 |s|p − C0 f (s)s C2 |s|p + C0 , p 2, (1.3) and inhomogeneous term g in H −1 (Ω ), L2 (Ω ), G (see Definition 2.1), respectively Exponential attractors or attractors for the reaction–diffusion equation have been extensively considered in many monographs and lectures see eg., [1,5,7–9,17,18] and the references therein Exponential attractors for unbounded domain or non-autonomous with subcritical nonlinearity have been considered in [1,5,7,8] In [9], the authors obtained finite dimensions of a global attractor in L∞ (Ω ) with nonlinearity f satisfying arbitrary growth when g ∈ L∞ (Ω ) Also the finite dimension of a global attractor in L2 (Ω ) has been proved in [18] when nonlinearity is supercritical and g ∈ L2 (Ω ) In [17], the authors obtained a global attractor in L2 (Ω ), H01 (Ω ), Lp (Ω ) with arbitrary polynomial growth nonlinearity and g ∈ H −1 (Ω ) ∗ Corresponding author E-mail address: zhongyansheng04@gmail.com (Y Zhong) 0362-546X/$ – see front matter © 2008 Elsevier Ltd All rights reserved doi:10.1016/j.na.2008.10.128 752 Y Zhong, C Zhong / Nonlinear Analysis 71 (2009) 751–765 At the same time, they also proved the existence of a global attractor in L2 (Ω ), H (Ω ), L2p−2 (Ω ) with arbitrary polynomial growth nonlinearity and g ∈ L2 (Ω ) Let us come back to our problem First, to obtain an exponential attractor in L2 (Ω ) with g ∈ H −1 (Ω ), we apply the -trajectory method as in [10] It is worth noticing that g ∈ H −1 (Ω ) is different from [10] with g ∈ L2 (Ω ) Second, to obtain an exponential attractor in H01 (Ω ) with g ∈ L2 (Ω ), we prove the semigroup satisfies a discrete squeezing property in H01 (Ω ) Obviously, it is a more regular result than [18] with finite dimensions of a global attractor in L2 (Ω ) when g ∈ L2 (Ω ) In particular, we construct an exponential attractor in L2p−2 (Ω ) with g ∈ G Since there is no higher regularity than L2p−2 (Ω ) for solutions of (1.1) when g ∈ G belonging only to L2 (Ω ) (see Remark 2.1.b), then general method by proving the Lipschitz continuity between L2p−2 (Ω ) and L2 (Ω ) (see [9]) does not work in our case To overcome this difficulty, we give a new method, namely asymptotic a priori estimate, which concerns the relations of radius and cardinal number of coverings for some set in two different topologies (see Lemma 2.1, Remark 2.4 and Theorem 2.1) In the concrete applications, we can first obtain an exponential attractor in a weaker topology space by a general method In some sense, it implies the change of radius and cardinal number of covering for sets {S n (B)}∞ n=1 , where B is a bounded set And then, the change of radius and cardinal number of covering for {S n (B)}∞ in the stronger topology space is easily obtained (see the proof of Theorem 2.1) n=1 Furthermore, we obtain the existence of an exponential attractor in the stronger topology space This article is organized as follows In Section 2, we first recall some basic results, and then, give our important technique tool, i.e., Theorem 2.1, which will guarantee the existence of an exponential attractor in a stronger topology space In Section 3, we prove the existence of exponential attractors in L2 (Ω ) with g ∈ H −1 (Ω ), i.e., Theorem 3.1 In Section 4, we obtain an exponential attractor in H01 (Ω ) with g ∈ L2 (Ω ), i.e., Theorem 4.1 In Section 5, we prove the existence of an exponential attractor in L2p−2 (Ω ) with g ∈ G, i.e., Theorem 5.2 Throughout this paper we use the following notation: Hilbert spaces L2 (Ω ), H01 (Ω ) equipped with the usual scalar products and norms (·, ·) and ((·, ·)) In particular, | · |X denote the Banach space X norm Ω is a bounded smooth domain in Rn C denotes any positive constant which may be different from line to line even in the same line (sometimes for special differentiation, we also denote the different positive constants by C1 , C2 , ) Preliminary results We start with the definition of the set G Definition 2.1 We define the set G by a class of functions: G = {g ∈ L2 (Ω )| g ∈ ∪2 such that, for every bounded subset B ⊂ E, there exists a constant C = C (B) such that distE (Sn B, M ) Cd−α n , where dist denotes the non-symmetric Hausdorff distance between sets Remark 2.2 We have given the definition of exponential attractors for discrete times (n ∈ Z+ ) The extension of this definition to the continuous case (t ∈ R+ ) is straightforward (see e.g [3]) Remark 2.3 We note that the existence of an exponential attractor M for the map L automatically implies the existence of the global attractor A and the embedding A ⊂ M We note however that, in contrast to the global attractor, an exponential attractor is not uniquely defined Definition 2.3 S is said to satisfy the discrete squeezing property on B if there exists an orthogonal projection PN of rank N such that for every u and v in B, PN (Su − S v) (I − PN )(Su − S v) ⇒ Su − S v u−v The following lemma concerns the covering of sets in two different topologies (see [17] Lemma 5.3) Lemma 2.1 For any > 0, the bounded subset B of Lp (Ω )(p > 0) has a finite -net in Lp (Ω ) if there exists a positive constant M = M ( ) which depends on , such that (i) B has a finite (3M )(q−p)/q ( )p/q -net in Lq (Ω ) for some q, q > 0; (ii) Ω (|u| M ) where Ω (|u| |u|p 1/p < 2−(2p+2)/p for any u ∈ B, M ) = {x ∈ Ω ||u(x)| M } Remark 2.4 From the proof of this lemma (see [17] Lemma 5.3), we not only obtain a finite -set in Lp (Ω ) for B but also obtain that both covering sets in Lp (Ω ) and Lq (Ω ) have the same cardinal number Furthermore, we can choose the same central points of both covering sets Now, we give our main theorem which describes our new technique to construct an exponential attractor in a stronger topology space Theorem 2.1 Assume that p q > and Ω ⊂ Rn is bounded Let {S n }∞ n=1 be a discrete Lipschitz continuous semigroup p q on L (Ω ) and L (Ω ) and B0 is a positively invariant bounded absorbing set in Lp (Ω ), respectively, and satisfying the following conditions: q (i) {S n }∞ n=1 has an exponential attractor in L (Ω ) with the radius of covering sets decreasing as S (B0 ) ⊂ i M ∪j=i1 BLq (Ω ) (aij , center aij of the radius (ii) For any R 2i ) ∩ S (B0 ) , i = 1, 2, , where R = supx∈B0 x R 2i q i in L (Ω ) and the cardinal number Mi = > 0, there exist positive constants M = M ( ) = C · 1/p |u|p Ω (|u| M ) < 2−(2p+2)/p N0 K0i γ q−γ · Lp (Ω ) R 2i , i = 1, 2, , that is and BLq (Ω ) (aij , R 2i ) denote a ball with , i = 1, 2, , where N0 , K0 are positive constants (γ > q), such that for any u ∈ B0 p Then, {S n }∞ n=1 has an exponential attractor in L (Ω ) γ Remark 2.5 In the concrete application of Theorem 2.1, M = M ( ) = C · q−γ , where γ depending on inhomogeneous term g, for example see our Theorem 5.1, and general q = By general method see [6,3,11], it is easy to construct an exponential attractor in L2 (Ω ) such that the radius of the covering set decreases as 2Ri , i = 1, 2, or θ i R, i = 1, 2, , θ < and the cardinal number increases as Mi = N0 · K0i , i = 1, 2, Here in our Theorem 2.1, we only prove the case similar cases are also true by the same argument in the following proof of Theorem 2.1 R , 2i the other 754 Y Zhong, C Zhong / Nonlinear Analysis 71 (2009) 751–765 Now, we prove Theorem 2.1 Proof of Theorem 2.1 We will proceed by induction Step We consider -net in Lp (Ω ) for the set S m (B0 ) (m will be fixed later) From Lemma 2.1 and the assumption (ii), it follows that S m (B0 ) has a finite -net in Lp (Ω ) if it has a finite (3M1 )(q−p)/q ( )p/q -net in Lq (Ω ) with M1 = C · γ /(q−γ ) From our assumption (i), we know that there exist i0 ∈ N such that R p/q (3M1 )(q−p)/q 2i0 R 2i0 −1 , for given small (3M1 )(q−p)/q ( )p/q − net (i), therefore, for the set S i0 (B0 ), we have Letting m = i0 and N p/q (3M1 )(q−p)/q N ( − net )Lp (Ω ) = N R ≡N Lq (Ω ) (2.1) = N ( 2Ri0 )Lq (Ω ) , and combining with Remark 2.4 and the assumption − net Lq (Ω ) i 2i0 Lq (Ω ) = N0 · K00 (2.2) Step Let be replaced by Similarly, we consider -net in Lp (Ω ) for the set S i0 +m1 (B0 ) (m1 will be fixed later) Using Lemma 2.1 and the assumption (ii) again, we know that S i0 +m1 (B0 ) has a finite -net in Lp (Ω ) if it has a finite (3M2 )(q−p)/q ( 22 )p/q -net in Lq (Ω ) with M2 = C · ( )γ /(q−γ ) From the assumption (i), we know that there exist k1 ∈ N such that R p/q (3M2 )(q−p)/q 2i0 +k1 R 2i0 +k1 −1 , for given (3M2 )(q−p)/q ( 22 )p/q − net Letting m1 = k1 and N , (2.3) = N ( 2i0R+k1 )Lq (Ω ) and combining with Remark 2.4 and the Lq (Ω ) assumption (i), therefore, for the set S i0 +k1 (B0 ), we have N − net =N Lp (Ω ) (3M2 )(q−p)/q R ≡N Step Let 2i−1 2i − net Lq (Ω ) i +k1 2i0 +k1 be replaced by p/q Lq (Ω ) = N0 · K00 , similarly, we consider (2.4) 2i -net in Lp (Ω ) for the set S i0 +k1 +k2 +···+ki−1 +mi (B0 ) (mi will be fixed later) From Lemma 2.1 and the assumption (ii), it follows that S i0 +k1 +k2 +···+ki−1 +mi (B0 ) has a finite 2i -net in Lp (Ω ) if it has a finite (3Mi )(q−p)/q ( 22 )p/q -net in Lq (Ω ) with Mi = C · ( 2i )γ /(q−γ ) From the assumption (i), there exist ki ∈ N such that i R 2i0 +k1 +···ki (3Mi )(q−p)/q p/q 2i R 2i0 +k1 +···+ki −1 (3Mi )q−p/q ( 22i )p/q − net Letting mi = ki and N Lq (Ω ) , for given 2i , (2.5) = N ( 2i0 +kR1 +···ki )Lq (Ω ) and combining with Remark 2.4 and the assumption (i), therefore for the set S i0 +k1 +···+ki (B0 ), we have N 2i − net Lp (Ω ) =N ≡N (3Mi )q−p/q p/q 2i − net Lq (Ω ) R 2i0 +k1 +···ki i +k1 +···ki Lq (Ω ) = N0 · K00 (2.6) Now, (3Mi−1 )(q−p)/q ( 2 )p/q on (3Mi )(q−p)/q ( 22 )p/q together with Mi = C · ( 2i )γ /(q−γ ) by above procedure, we have i−1 (3Mi−1 )(q−p)/q (3Mi )(q−p)/q i p/q 2i−1 p/q 2i = Mi−1 Mi (q−p)/q · 2p/q Y Zhong, C Zhong / Nonlinear Analysis 71 (2009) 751–765 γ /(q−γ ) = γ /(q−γ ) 2i−1 (q−p)/q 2i = 2γ /(q−γ ) (q−p)/q 755 · 2p/q · 2p/q = (2)γ (q−p)/(q−γ )q · 2p/q > (2.7) From (2.7), noticing that the ratio of (3Mi−1 )(q−p)/q ( 2 )p/q and (3Mi )(q−p)/q ( 22 )p/q is a constant for given γ , p, q, independent of , i, and combining with inequalities (2.1), (2.3) and (2.5) and the assumption (i), therefore, we can choose k1 = k2 = · · · = ki = · · · ≡ [(2)γ (q−p)/(q−γ )q · 2p/q ] + 1, where [a] is the integer part of a and in what follows m0 [(2)γ (q−p)/(q−γ )q · 2p/q ] + 1, such that i−1 i S i0 +k1 +···+ki (B0 ) = S i0 +i·m0 (B0 ) Ni BLp (Ω ) aij , = S i0 [S m0 ]i (B0 ) ⊂ j =1 where aij ∈ S i0 [S m0 ]i (B0 ) and the balls (BLp (Ω ) (aij , (2.2), (2.4) and (2.6), it follows that i +i·m0 Ni = N0 · K00 i m = N0 K00 · [K0 ]i , 2i 2i ∩ S i0 [S m0 ]i (B0 ) , i = 1, 2, (2.8) )) in Lp (Ω ) Again from the assumption (i) and combining with estimates i = 1, 2, (2.9) Combining (2.8) and (2.9), it is easy to deduce that the existence of an exponential attractor M d for the discrete semigroup (S m0 )n in Lp (Ω ) and from the Lipschitz property of discrete semigroup {S n }∞ n=1 , we can set m0 S i (M d ) M1d = i =1 Using the same argument as in [3], it is easy to deduce that M1d is an exponential attractor in Lp (Ω ) for discrete semigroup {S n }∞ n =1 Finally, we formulate the celebrated so-called Aubin–Lions lemma Lemma 2.2 Let p1 ∈ (1, ∞], p2 ∈ [1, ∞) Let X be a Banach space and Y , Z be separable and reflexive Banach spaces such that Y → → X → Z Then for any τ ∈ (0, ∞), {u ∈ Lp1 (0, τ ; Y ); u ∈ Lp2 (0, τ ; Z )} → → Lp1 (0, τ ; X ) (2.10) Exponential attractor in L (Ω ) with g ∈ H −1 (Ω ) In this section, we will prove an exponential attractor in L2 (Ω ) for (1.1) with g ∈ H −1 (Ω ) We start with the following general existence and uniqueness of solutions which can be obtained by the normal Faedo–Galerkin methods, see [14,16] for details Lemma 3.1 Let the assumptions (1.2) and (1.3) hold and g ∈ H −1 (Ω ) Then for any initial date u0 ∈ L2 (Ω ) and any T > 0, there exists a unique solution u for (1.1) which satisfies u ∈ L2 (0, T ; H01 (Ω )) ∩ Lp (0, T ; Lp (Ω )), ∀T > 0, u ∈ C (R ; L (Ω )), + and the mapping u0 → u(t ) is continuous in L2 (Ω ) If, furthermore, g ∈ L2 (Ω ) and u0 ∈ H01 (Ω ), then u ∈ C ([0, T ); H01 (Ω )) ∩ L2 (0, T ; H (Ω )), ∀T > By Lemma 3.1, we can define the operator semigroup {S (t )}t S (t )u0 : L2 (Ω ) × R+ → L2 (Ω ), in L2 (Ω ) for both g ∈ H −1 (Ω ) and g ∈ L2 (Ω ) as follows: (3.1) which is continuous in L2 (Ω ) Remark 3.1 From [17], we note that the semigroup S (t ) has a global attractor in H01 (Ω ) for (1.1) with g ∈ H −1 (Ω ) It implies that the semigroup S (t ) possesses a bounded and positively invariant absorbing set B0 in H01 (Ω ) More precisely, we will take B0 = ∪t t0 S (t )B0 , where B0 is a bounded absorbing set in H01 (Ω ) and t0 is a time such that S (t )B0 ⊂ B0 for t t0 and where the closure is taken in the weak topology of H01 (Ω ) For the set B0 , the following facts are true 756 Y Zhong, C Zhong / Nonlinear Analysis 71 (2009) 751–765 Lemma 3.2 Let the assumptions (1.2) and (1.3) hold and g ∈ H −1 (Ω ) and B0 be defined as above Then, for any u0 ∈ B0 , the solution u(t ) = S (t )u0 ∈ B0 , for any t > Furthermore, if u0 is the limit, for the weak topology of H01 (Ω ), of a sequence {u0n }, where u0n ∈ ∪t S (t )B0 , for every n, then there exists at least a subsequence of solutions, {un (t ) = S (t )u0n } converges to the solution u(t ) for the weak topology of H01 (Ω ), for any t Proof For convenience, we denote g by Di f i + h(x)(= Σin=0 Di f i + h(x)), where f i , h ∈ L2 (Ω )(i = 1, n) Now multiplying (1.1) by u, after the standard integration by parts and using the assumption (1.3), we have d dt |u|22 + |∇ u|22 + Ω f (u)u = Di f i , u + h, u =− Ω f˜ · ∇ u + h, u , (3.2) which implies that d dt |u|22 + |∇ u|22 + C (|f˜ |22 , λ1 , |h|22 , |Ω |), |u|p Ω where f˜ = (f , , f n ), |f˜ |22 = Σin=1 |f i |22 and |u(t )|22 · (3.3) denotes the L2 -inner product From (3.3), it follows that e−α t |u(0)|22 + C (|f˜ |22 , λ1 , |h|22 , |Ω |) (3.4) where α is a positive constant s Meanwhile, let F (s) = f (τ )dτ ; then by (1.3) again, it follows that C˜1 |s|p − k F (s) C˜2 |s|p + k (3.5) Therefore, C˜1 Ω |u|p − k|Ω | Ω F (u) C˜2 Ω |u|p + k|Ω | (3.6) Also noticing that |∇ u + f˜ |22 2|∇ u|22 + 2|f˜ |22 , (3.7) by (3.5)–(3.7), we infer from (3.3) that d dt |u|22 + C |∇ u + f˜ |22 + Ω F (u) C (|f˜ |22 , λ1 , |h|22 , |Ω |) (3.8) Integrating the inequality above from t to t + 1, we have t +1 |∇ u + f˜ |22 + Ω t F ( u) C (|f˜ |22 , λ1 , |h|22 , |Ω |) + |u(0)|22 , (3.9) On the other hand, multiplying (1.1) by ut , we obtain |ut |22 + d dt |∇ u|22 + d dt Ω F (u) = Di f i , ut + h, ut =− d dt f˜ , ∇ u + h, ut (3.10) By the Hölder inequality and the Cauchy inequality, it follows from (3.7) and (3.10) that d dt |∇ u + f˜ |22 + Ω F ( u) |h|22 (3.11) Combining with (3.9) and (3.11), by the uniform Gronwall inequality, we obtain |∇ u(t ) + f˜ |22 + Ω F (u(t )) |h|22 + C (|f˜ |22 , λ1 , |h|22 , |Ω |) + |u0 |22 , Combining with the definition of B0 , therefore S (t )u0 ∈ B0 , ∀t ∀t > 0 for u0 ∈ ∪t t0 B0 (3.12) Y Zhong, C Zhong / Nonlinear Analysis 71 (2009) 751–765 757 Now we assume that u0 is the weak limit of H01 (Ω ), of a sequence {u0n }, where u0n ∈ ∪t t0 S (t )B0 Now we set (t ) = un (t ) − u(t ) (where un (t ), u(t ) are the solutions, to the initial value u0n , u0 , respectively), which satisfies the following equation: d (t ) − ∆vn (t ) + (t )vn (t ) = 0, (0) = u0n − u0 , (3.13) dt where l(t ) := f (sun (t ) + (1 − s)u(t ))ds Since f (u) −κ , then, obviously, l(t ) −κ also Since {u0n } is bounded sequence in H01 (Ω ) and H01 (Ω ) → → L2 (Ω ), It implies that there exists a subsequence of u0n converging in L2 (Ω ) And since u0n weak converge to u0 in H01 (Ω ), it follows that u0n converge to u0 in L2 (Ω ) Now multiplying (3.13) by (t ) and integrating over Ω , we have d dt |vn |22 + 2|∇vn |22 2κ|vn |22 (3.14) Using the Gronwall inequality, we obtain that e2κ t |u0n − u0 |22 , |un (t ) − u(t )|22 (3.15) Taking the limit on both sides of (3.15), we have lim un (t ) = u(t ), n→∞ in L2 (Ω ) (3.16) Again using the the fact that un (t ) ∈ B0 is bounded in H01 (Ω ), it implies that un (t ) has a weak converging subsequence {unj (t )}, ∀t > Combining with (3.16), it is easy to deduce that the subsequence {unj (t )} is weakly converging to u(t ) in H01 (Ω ) for any t > Therefore, u(t ) ∈ B0 and Lemma 3.2 is proved Now we consider difference of two solutions of (1.1) starting from B0 Proposition 3.1 Let the assumptions of Lemma 3.2 hold and let u1 and u2 be two solutions of (1.1) starting from B0 Then, there exists a constant > such that |∇(u1 − u2 )(s)|22 ds |u1 (s) − u2 (s)|22 ds C (3.17) where the positive constant C is independent of u1 (0) and u2 (0) Proof We set v(t ) = u1 (t ) − u2 (t ) This function satisfies the equation dv − ∆v + l(t )v = 0, dt v|∂ Ω = 0, (3.18) where l(t ) := f (su1 (t ) + (1 − s)u2 (t ))ds Since f (u) −κ , then, obviously, l(t ) Multiplying Eq (3.18) by v(t ) and integrating over Ω , we obtain the inequality d dt |v|22 + 2|∇v|22 −κ also 2κ|v|22 (3.19) Using the Gronwall inequality, we have |v(τ )|22 e2κ(τ −s) |v(s)|22 , (3.20) and taking s ∈ (0, ) and integrating (3.19) over τ ∈ (s, ) Thus, we obtain |v(2 )|22 + 2 |∇v(τ )|22 dτ s |v(τ )|22 dτ + |v(s)|22 (3.21) s Inserting estimate (3.20) into the right-hand side of estimate (3.21), we obtain the following inequality: |∇v(τ )|22 dτ C |v(s)|22 (3.22) where the positive constant depends on Integrating (3.22) over s ∈ (0, ) then yields the inequality (3.17) and completes the proof of Proposition 3.1 758 Y Zhong, C Zhong / Nonlinear Analysis 71 (2009) 751–765 We now have Proposition 3.2 Let the assumptions of Proposition 3.1 hold and ∂ (u1 − u2 ) ∂t C u1 − u2 L2 ( ,2 ;H −1 (Ω )) L2 (0, ;L2 (Ω )) be defined as above Then, (3.23) Proof We have, again setting v = u1 − u2 , ∂v ∂t = sup L2 ( ϕ ,2 ;H −1 (Ω )) where ϕ ∈ L2 ( , ; H01 (Ω )), ϕ then have, noting that ∂v = v − l(t )v, ∂t ∂v ∂ t L2 ( ,2 ;H −1 (Ω )) ∂v , ϕ dt , ∂t L2 ( ,2 ;H01 (Ω )) = 1, and ·, · denotes the duality product between H01 (Ω ) and H −1 (Ω ) We 2 |∇v ∇ϕ|dt + sup ϕ (3.24) |l(t )v ϕ|dt (3.25) Furthermore, |∇v||∇ϕ|dt v |l(t )v||ϕ|dt κ v L2 ( ,2 ;H01 (Ω )) , (3.26) C v L2 ( ,2 ;L2 (Ω )) L2 ( ,2 ;H01 (Ω )) (3.27) We thus deduce from (3.26) and (3.27) that ∂v ∂t L2 ( ,2 ;H −1 (Ω )) C v L2 ( ,2 ;H01 (Ω )) , which yields, owing to (3.17), estimate (3.23), i.e., ∂v ∂t L2 ( ,2 ;H −1 (Ω )) C v L2 ( ,2 ;L2 (Ω )) and completes the proof of Proposition 3.2 Thanks to Propositions 3.1 and 3.2 and Lemma 2.2, we now prove that the existence of exponential attractor in L2 (Ω ) for Eq (1.1) with g ∈ H −1 (Ω ) To so, we use the method of -trajectories (see [11] for more details) We introduce the space of trajectories X = {w : (0, ) → L2 (Ω ), w is a solution of (1.1) on (0, )}, where is as above, which we endow with the topology of L2 ( , ; L2 (Ω )) We then set B = {w ∈ X , w(0) ∈ B0 } For the set B , we have the following result Proposition 3.3 Let the assumptions of Proposition 3.1 hold and B be defined as above Then, B is closed in the topology of L2 (0, ; L2 (Ω )) Proof Let {wn } be a sequence of trajectories belonging to B which converges to some w From Lemma 3.2, we note that wn (t ) ∈ B0 , for every t ∈ [0, ] and for every n Using the same argument as in Lemma 3.2, we can pass to the limit in the equation to prove that, at least for a subsequence, wn converges to a solution w of the equation on [0, ] which is weakly continuous from [0, ] onto H01 (Ω ) and that w(0) ∈ B0 Therefore, this yields that B is a complete metric space (we note that this is not necessarily the case for X , see [11,13]) and completes the proof of Proposition 3.3 Y Zhong, C Zhong / Nonlinear Analysis 71 (2009) 751–765 759 We then define the operators L(t ) : X → X , t 0, by (L(t )w)(s) = u(t + s), s ∈ [0, ], where u is the unique solution of Eq (1.1) such that u|[0, ] = w We finally set L = L( ) Then, it follows from estimate (3.17) that, if w1 , w2 ∈ B , Lw1 − Lw2 L2 (0, ;H01 (Ω )) C w1 − w2 (3.28) X and it follows from estimate (3.23) that ∂ (Lw1 − Lw2 ) ∂t L2 (0, ;H −1 (Ω )) C w1 − w2 X (3.29) Furthermore, we have the following result: Proposition 3.4 Let the assumptions of Proposition 3.1 hold and let the operator L(t ) be defined as above Then, L(t ) is Lipschitz from B onto X , ∀t > 0, and that the mapping t → L(t )w is Lipschitz, ∀w ∈ X Proof Let the space X = L2 (Ω ) in Lemma 2.1 of [11] and combining with estimate (3.20), it is easy to deduce L(t ) is Lipschitz from B onto X , ∀t > Now we prove the mapping t → L(t )w is Lipschitz, ∀w ∈ X Similarly, Let X = L2 (Ω ) in Lemma 2.2 of [11], it is equivalent to prove that {χ ; χ ∈ C } be bounded in space Lq (0, ; L2 (Ω )) with some q ∈ (1, ∞], this is easily deduced from the following lemma p < ∞ and any bounded subset B ⊂ L2 (Ω ), the Lemma 3.3 Let the assumptions of Proposition 3.1 hold Then, for any following estimate is valid: Ω |ut (s)|p C (e−α t |u0 |22 + 1) for any u0 ∈ B, (3.30) where α is a positive constant and the constant C depends on p, , n (space dimensional), λ1 , g , |Ω | and ut (s) = d dt (S (t )u0 )|t =s Proof Using a similar argument as Lemma 5.20 of [17], we can prove Lemma 3.3 From estimate (3.30), we complete the proof of Proposition 3.4 Remark 3.2 Thanks to estimates (3.28) and (3.29), it follows (see [11] for the details of the proof) that the semigroup L(t ) possesses an exponential attractor M on B , that is, M is compact for the topology of X , is positively invariant (i.e., L(t )M ⊂ M , ∀t 0), has a finite fractal dimension and attracts exponentially fast (B ) (again, for the topology of X ) We now introduce the mapping e : X → L2 (Ω ) defined by e(w) = w( ) (i.e., e maps an -trajectory onto its endpoint) For the mapping e, we have Proposition 3.5 Let the assumptions of Proposition 3.1 hold and let the mapping e be defined as above Then, e is Lipschitz Proof Let the space X = L2 (Ω ) in Lemma 2.1 of [11] and combining with estimate (3.20), it is easy to deduce Proposition 3.5 Now, we have Theorem 3.1 Let the assumptions (1.2) and (1.3) hold and g ∈ H −1 (Ω ) Then, for the semigroup S (t ) generated by Eq (1.1) possesses an exponential attractor M in L2 (Ω ) Moreover, these exponential attractors can be chosen such that dimF M , L2 (Ω ) C (3.31) and there exists a constant α > such that for every bounded subset B of L (Ω ), there exists a constant C = C (B) such that distL2 (Ω ) (S (t )B, M ) C e−α t (3.32) Based on the above results and applying the method as in [10,11], we can deduce Theorem 3.1 Exponential attractor in H01 (Ω ) with g ∈ L (Ω ) In this section, we show that an exponential attractor in H01 (Ω ) for Eq (1.1) with g ∈ L2 (Ω ) We begin with the following fact Since semigroup S (t ) has a global attractor in H01 (Ω ) for Eq (1.1) with g ∈ L2 (Ω ), see [17], it implies that the semigroup S (t ) is asymptotically compact in H01 (Ω ) From [4], to obtain our result, it remains to prove the semigroup S (t ) satisfies the 760 Y Zhong, C Zhong / Nonlinear Analysis 71 (2009) 751–765 discrete squeezing property in H01 (Ω ), that is the following lemma: Lemma 4.1 Let the assumptions of Theorem 4.1 hold Then, there exist T ∗ and a small constant δ > 0, such that for any u0 , v0 ∈ S (δ)B, if PN0 (S (T ∗ )u0 − S (T ∗ )v0 ) (I − PN0 )(S (T ∗ )u0 − S (T ∗ )v0 ) H01 (Ω ) H01 (Ω ) , then (I − PN0 )(S (T ∗ )u0 − S (T ∗ )v0 ) H01 (Ω ) u0 − v (4.1) H01 (Ω ) where B is an arbitrary bounded set in L2 (Ω ) Proof Let QN0 = (I − PN0 ) and w(t ) = u(t ) − v(t ), w2 = QN0 (w) Obviously, the function w(t ) satisfies the equation dw dt − ∆w + l(t )w = 0, w|∂ Ω = 0, (4.2) where l(t ) := f (su1 (t ) + (1 − s)u2 (t ))ds Since f (u) Now, we multiply (4.2) by −∆w2 , we have d 2dt ∇w2 + (∆w, ∆w2 ) = (−l(t )w, ∆w2 ) −κ , then, obviously, l(t ) −κ also κ(w, ∆w2 ) By orthogonal property, we have (∆w, ∆w2 ) λN0 (∆w, w2 ) = λN0 (∆w2 , w2 ) = λN0 (∇w2 , ∇w2 ) κ(w, ∆w2 ) = κ(w2 , ∆w2 ) = κ(∇w2 , ∇w2 ) So, we have d 2dt ∇w2 + λN0 (∇w2 , ∇w2 ) κ(∇w2 , ∇w2 ) Since λN0 → +∞, obviously, we can choose N0 such that λN0 > κ Therefore d dt ∇w2 + 2(λN0 − κ)(∇w2 , ∇w2 ) By the Gronwall inequality, we have w2 (t ) H01 (Ω ) e −2(λN0 −κ)t w20 H01 (Ω ) e −2(λN0 −κ)t w0 H01 (Ω ) Therefore, there exists a T ∗ such that (I − PN0 )(S (T ∗ )u0 − S (T ∗ )v0 ) v0 H01 (Ω ) , and if PN0 (S (T )u0 − S (T )v0 ) ∗ ∗ S (T )u0 − S (T )v0 ∗ ∗ H01 (Ω ) = w2 (T ∗ ) H (Ω ) (I − PN0 )(S (T )u0 − S (T )v0 ) H (Ω ) , then ∗ H01 (Ω ) (I − PN0 + PN0 )S (T )u0 − S (T )v0 ∗ (I − PN0 )(S (T )u0 − S (T )v0 ) ∗ u0 − v w0 H01 (Ω ) = 16 u0 − ∗ H01 (Ω ) 16 ∗ H01 (Ω ) ∗ H01 (Ω ) H01 (Ω ) It implies that S (t ) satisfies the discrete squeezing property and the proof of Lemma 4.1 is finished Corollary 4.1 Let the assumptions of Theorem 4.1 hold Then, the semigroup S (t ) is uniformly Hölder continuous on [0, T ∗ ]× B1 in the topology of H01 (Ω ), i.e., |S (t1 )u10 − S (t2 )u20 |H (Ω ) C (|u10 − u20 |H (Ω ) + |t1 − t2 | ) (4.3) for ui0 ∈ B1 and t1 , t2 T ∗ where B1 = ∪t t0 S (t )B0 , B0 is a bounded absorbing set in H01 (Ω ) and t0 is a time such that S (t )B0 ⊂ B0 for t t0 and where the closure is taken in the strong topology of H01 (Ω ) and the constant C depends on B1 , T ∗ Proof First, it is easy to obtain the Lipschitz continuity with respect to the initial condition from Lemma 4.1 It remains to prove the Lipschitz continuity with respect to the time t From Lemma 3.1, we know that the solution u(t ) ∈ C ([0, T ], H01 (Ω )) for the initial value belongs to B1 , therefore u(t1 ) − u(t2 ) = t2 t1 ∂t u(s)ds (4.4) Y Zhong, C Zhong / Nonlinear Analysis 71 (2009) 751–765 761 which implies that t2 |u(t1 ) − u(t2 )|H (Ω ) = ∂t u(s)ds H01 (Ω ) t1 t2 t1 |∂t u(s)|H (Ω ) ds |t1 − t2 | t2 |∂t u(s)| t1 ds H01 (Ω ) (4.5) To finish estimate (4.5), differentiating the Eq (1.1) by t and denoting θ (t ) = ∂t u(t ), we will derive ∂t θ (t ) − ∆θ (t ) + f (u(t ))θ (t ) = 0, θ|t =0 = ∆u(0) − f (u(0)) + g , θ |∂ Ω = (4.6) Multiplying (4.6) by θ (t ) and combining with the assumption (1.2) and the Hölder inequality and the Poincaré inequality, we deduce that ∂t |θ (t )|22 + 2η|θ (t )|22 + η1 |∇θ (t )|22 − |θ (t )|22 0, (4.7) where η, η1 are positive constants Combining with Lemma 3.3, integrating (4.7) over [t1 , t2 ], we have t2 t1 |∂t u(s)|2H (Ω ) ds C and complete the proof of Corollary 4.1 Now, we have Theorem 4.1 Let the assumptions (1.2) and (1.3) hold and g ∈ L2 (Ω ) Then, for the semigroup S (t ) generated by Eq (1.1) possesses an exponential attractor M1 in H01 (Ω ) Moreover, these exponential attractors can be chosen such that dimF M1 , H01 (Ω ) C1 (4.8) and there exists a constant β1 > and δ1 > such that, for every bounded subset B of L (Ω ), there exists a constant C1 = C1 (B) such that distH (Ω ) (S (t )B, M1 ) C1 e−β1 t (4.9) for t > δ1 > By Lemma 4.1 and Corollary 4.1, applying the general method as in [3,15], we obtain the result of Theorem 4.1 Exponential attractor in L 2p−2 (Ω ) with g ∈ G In this section, we construct an exponential attractor in L2p−2 (Ω ) for Eq (1.1) with g ∈ G We begin with the following remark Remark 5.1 Since G ⊂ L2 (Ω ) ⊂ H −1 (Ω ), Theorems 3.1 and 4.1 holds true as well as for g ∈ G That is, the semigroup S (t ) has an exponential attractor in L2 (Ω ) for g ∈ G Due to Theorem 2.1, in order to obtain an exponential attractor in L2p−2 (Ω ), we need an asymptotic a priori estimate for u(t ) in L2p−2 (Ω ), i.e., to prove the condition (ii) of Theorem 2.1 That is the following theorem > and any bounded subset B ⊂ L2 (Ω ), Theorem 5.1 Let the assumptions (1.2) and (1.3) hold and let g ∈ G Then, for any there exist positive constants M = C1 Ω (|u(t )| M ) |u(t )|2p−2 γ 2−γ and T = T (B), such that 2−(2p−1)/(p−1) where the constant C1 is independent of for any u0 ∈ B as t T, (5.1) and B, γ ∈ (2, +∞) depending on g Remark 5.2 In Theorem 5.1, T depends only on B but not on exponential attractor in L2p−2 (Ω ) possible and M = C1 γ 2−γ , which makes our construction of an In order to prove Theorem 5.1, we need the following proposition which establishes the asymptotical a priori estimate in L2 (Ω ) for ut (s) 762 Y Zhong, C Zhong / Nonlinear Analysis 71 (2009) 751–765 Proposition 5.1 Let the assumptions of Theorem 5.1 hold Then, for the same , B ⊂ L2 (Ω ), γ and T = T (B) as in Theorem 5.1, there exist M = C3 · Ω (|u| M ) γ such that 2−γ |ut (s)|2 C4 for u0 ∈ B and any s where the constants C3 , C4 are independent of T (5.2) and B, ut (s) = d dt (S (t )u0 )|t =s Before proving Proposition 5.1, we give the following lemma, which in fact shows that T = T (B) in Theorem 5.1 and Proposition 5.1 depends only on B but not on , moreover it can be obtained from the following lemma Lemma 5.1 For any Ω > and any bounded subset B ⊂ L2 (Ω ), there exist T = T (B) |θ2 (t )|2 < C and N = N ( , B) ∈ N, such that for any u0 ∈ B, (5.3) provided that t T (B) > and m N, where θ2 = (I − Pm )θ = (I − Pm )ut , the constant C is independent of the canonical projector on Hm = span{ω1 , · · · ωm } and I be the identity operator and B, Pm is Proof By differentiating Eq (1.1) by t and denoting θ (t ) = ∂t u(t ), we have ∂t θ (t ) − ∆θ (t ) + f (u(t ))θ (t ) = (5.4) Multiplying (5.4) by θ2 (t ) and integrating over Ω , we have d dt |θ2 |22 + |∇θ2 |22 Ω |f (u)θ||θ2 |; hence, d dt |θ2 |22 + λm |θ2 |22 Ω |f (u)θ||θ2 |, (5.5) where λm is the m-eigenvalue of −∆ with Dirichlet boundary condition On the other hand, from the assumption (1.3) and Lemma 3.3 and the fact that the solution semigroup has a bound absorbing set B2 in L2p−2 (Ω ), we know that |f (u)θ| Ω Ω |f (u)| p−2 p−1 p−1 p−2 2(p−1) p−1 M0 , |θ | Ω for any u0 ∈ B (5.6) provided that t T0 (B), where the constant M0 is independent of B and the constant T0 depends on B and p but independent of Therefore, combining (5.5) and (5.6), using the Hölder inequality again, we can deduce that d dt if t |θ2 |22 + λm |θ2 |22 C T0 , which shows that |θ2 (t )|22 |θ2 (T0 )|22 e−λm (t −T0 ) + C λm C e−λm (t −T0 ) + λm By the property of eigenvalue λm → ∞ as m → ∞, and choosing T = T (B) = T0 (B) + and m to be large enough, we can obtain the result of Lemma 5.1 and complete the proof of Lemma 5.1 The following lemma is about the property of a bounded absorbing set, see [17] Lemma 5.2 Lemma 5.2 Let {S (t )}t be a semigroup on Lq (Ω )(q > 0) and {S (t )}t have a bounded absorbing set B0 in Lq (Ω ) Then for any δ > and any bounded subset B ⊂ L (Ω ), there exist positive constants T = TB and M = M (δ) = M0 · δ q m(Ω (|S (t )u0 | M )) where M0 = supu∈B0 u Lq (Ω ) δ for any u0 ∈ B and t − 1q such that T, and m(E ) denotes the Lebesgue measure of E ⊂ Ω and Ω (|u| M) {x ∈ Ω ||u(x)| M } Now we begin to prove Proposition 5.1 Proof of Proposition 5.1 From Lemma 3.3, it follows that the trajectories ut (s) has a bounded absorbing set B1 in L2 (Ω ) Combining with Lemma 5.1 together with the Condition (C) (see [12]), therefore, for any > 0, S (t )B1 has a finite -net Y Zhong, C Zhong / Nonlinear Analysis 71 (2009) 751–765 in L2 (Ω ) with centers (ut )1 , (ut )2 , , (ut )k for t (ut )i (1 i k) satisfying |ut (s) − (ut )i |2 < Ω 763 T (B) > That is for any ut (s) with s T (B), we can find some , (5.7) where (ut )i ∈ L2 (Ω ), i = · · · k are independent of time s and T (B) independent of given by Lemma 5.1 At the same time, for the fixed > 0, applying the same argument as Remark 2.1.a and combining with Lemma 3.3, therefore, there is a δ = (ut )i , i k, we have |(ut )i |2 M1 2γ 2γ γ −2 , · where M1 = max1 i k |(ut )i |Lγγ−(2Ω ) and γ can be chosen as in g, such that for each , (5.8) E provided that m(E ) < δ(E ⊂ Ω ) On the other hand, from Lemma 5.2 together with the existence of a bounded absorbing set B2 in L2 (Ω ), therefore, 2γ we can choose TB = T (B) in Lemma 5.2, and for the δ = M1 · γ −2 given above, there exists M = M2 · δ − , such that m(Ω (|u| M )) < δ holds for any u0 ∈ B and t T (B), where M2 = supu∈B2 |u|2 Therefore, we have Ω (|u| M ) |ut (s)|2 = Ω (|u| M ) |ut (s) − (ut )i + (ut )i |2 2 Ω (|u| M ) 22+1 C4 |ut (s) − (ut )i |2 + 22 Ω (|u| M ) |(ut )i |2 2 and M = M2 δ − 12 = M2 · M1 − 12 2γ γ −2 · γ = C3 · 2−γ , γ > Thus, Proposition 5.1 is proved Thanks to above results, now we begin to prove Theorem 5.1 > (small enough), from Proposition 5.1 and the properties of g , f , there exist > 0, such that the following estimates are valid for any u0 ∈ B and t T (B): Proof of Theorem 5.1 For any fixed γ M = C5 · 2−γ m(Ω (|u(t )| 2γ γ −2 M )) < δ = C · and Ω (|u(t )| M ) |g |2 < (5.9) and f ( s) for any s Let Ω1 = Ω (u(t ) Ω1 (u − M ) M, f (s) M ) and Ω2 = Ω (u(t ) p−1 · ut + (p − 1) Ω1 −M for any s 2M ) Multiplying (1.1) by [(u − M )+ ]p−2 · (u − M )+ , we have (u − M ) p−2 |∇ u| + Ω1 From estimate (5.9) and Proposition 5.1, we have Ω1 f (u)(u − M )p−1 C Therefore, we have Ω2 f (u)up−1 · 2p−1 Ω2 Ω1 C f (u)up−1 1− M f (u)(u − M )p−1 u p−1 f (u)(u − M ) p−1 |g | Ω1 · Ω1 (u − M ) 2p−2 764 Y Zhong, C Zhong / Nonlinear Analysis 71 (2009) 751–765 Noticing m(Ω2 ) C (by (5.9)) and the assumption (1.3), the inequality above implies that: u2p−2 C T (B) as t Ω2 (5.10) Taking |(u + M )− |p−2 · (u + M )− as the test function, we have in the same fashion as above that Ω (u(t ) −2M ) |u(t )|2p−2 C T (B) as t (5.11) Combining (5.10) and (5.11) and choosing appropriate constants C3 , C4 , C5 from above, we obtain the estimate (5.1) and complete the proof of Theorem 5.1 In particular, for the bounded absorbing set B2 , we have Corollary 5.1 Let the assumptions of Theorem 5.1 hold Then, for any M =C· γ 2−γ > and the set B2 , there exist positive constants and T = T (B2 ), such that Ω (|u(t )| M ) |u(t )|2p−2 2−(2p−1)/(p−1) for any u0 ∈ B2 as t where the constant C depends on B2 but independent of T (B2 ), (5.12) Corollary 5.2 Let the assumptions of Theorem 5.1 hold Then, the semigroup S (t ) is uniformly Hölder continuous on [0, T (B2 )]× B2 in the topology of L2p−2 (Ω ), i.e., |S (t1 )u10 − S (t2 )u20 |2p−2 for ui0 ∈ B2 and t1 , t2 C |u10 − u20 |2p−2 + |t1 − t2 | (5.13) T (B2 ) where the constant C depends on B2 , T (B2 ) Proof First, In order to prove the Lipschitz continuity with respect to the initial condition, multiplying Eq (3.18) by |v|2p−4 v (where v = u1 (t ) − u2 (t )) and integrating over Ω , we have d 2p − dt −2 2p−4 2p−4 |v|2p v 2p−2 + (2p − 3)|v|2p−4 |∇v|2 = −l(t )v, |v| (5.14) Combining with the assumption (1.2), we have d 2p − dt −2 2p−4 |v|2p 2p−2 + (2p − 3)|v|2p−4 |∇v| 2p−4 Since (2p − 3)|v|2p−4 |∇v|2 > for p |v(t )|2p−2 −2 |v|2p 2p−2 (5.15) and using the Gronwall inequality, we have e(2p−2) |v(0)|2p−2 (5.16) which implies that V (t ) possesses the Lipschitz property with respect to the initial condition Secondly, we need to prove the Hölder continuity with respect to t, we note that the solution u(t ) ∈ C ([0, T ], L2 (Ω )) (see [17]), we have u(t1 ) − u(t2 ) = t2 ∂t u(s)ds (5.17) t1 which implies that |u(t1 ) − u(t2 )|L2p−2 (Ω ) = t2 ∂t u(s)ds t1 t2 2p−2 |∂t u(s)|2p−2 ds t1 t2 |t1 − t2 | |∂t u(s)|22p−2 ds (5.18) t1 From Lemma 3.3, therefore, inserting estimates (3.30) into the right-hand side of inequality (5.18), we easily obtain that |u(t1 ) − u(t2 )|L2p−2 (Ω ) C1 |t1 − t2 | (5.19) Y Zhong, C Zhong / Nonlinear Analysis 71 (2009) 751–765 765 where C1 depends on B2 and T (B2 ) Let C = max(C1 , e(2p−2) ), combining estimates (5.16) and (5.19), we easily obtain estimate (5.13) and prove Corollary 5.2 Now, we have Theorem 5.2 Let the assumptions (1.2) and (1.3) hold and g ∈ G Then, the semigroup S (t ) generated by Eq (1.1) possesses an exponential attractor M2 in L2p−2 (Ω ) Moreover, these exponential attractors can be chosen such that dimF (M2 , L2p−2 (Ω )) C2 , (5.20) and there exists a constant β2 > and δ2 > such that, for every bounded subset B of L (Ω ), there exists a constant C2 = C2 (B) such that distL2p−2 (Ω ) (S (t )B, M2 ) C2 e−β2 t (5.21) for t > δ2 > Applying Theorem 2.1 together with Corollary 5.1, we can prove the existence of exponential attractor in L2p−2 (Ω ) for discrete semigroup And then, using the Standard manner (e.g., see [3, chap 3]) together with Corollary 5.2, and we can obtain the continuous case and finish the proof of Theorem 5.2 Remark 5.3 For g ∈ H −1 (Ω ) denoted by Di f i + h(x)(= Σin=0 Di f i + h(x)), where f i , h ∈ L2 (Ω )(i = 1, n) If fi , h ∈ G Then, using our method, we can obtain an exponential attractor in Lp (Ω ) for (1.1) with g ∈ H −1 (Ω ) Acknowledgments This work is supported in part by the NSFC Grant (10771089) and Trans-Century Training Programme Foundation for the Talents by the Ministry of Education References [1] A Babin, B Nicolaenko, Exponential attractors of reaction–dissfusion systems in an unbounded domain, J Dynam Differential Equations (4) (1995) 567–590 [2] L Dung, B Nicolaenko, Exponential attractors in Banach spaces, J Dynam Differential Equations 13 (4) (2001) 791–806 [3] A Eden, C Foias, B Nicolaenko, R Teman, Exponential attractors for dissipative evolution equations, in: Research in Applied Mathematics, vol 37, John-Wiley, New York, 1994 [4] A Eden, C Foias, V Kalantarov, A remark on two constructions of exponential attractors for α -contractions, J Dynam Differential Equations 10 (1) (1998) 37–45 [5] M Efendiev, S Zelik, Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems, Proc Roy Soc Edinburgh 135A (2005) 703–730 [6] M Efendiev, A Miranville, S Zelik, Exponential attractors for a singularly perturbed Cahn–Hilliard system, Math Nachr 272 (2004) 11–31 [7] M Efendiev, A Miranville, S Zelik, Exponential attractors for a nonlinear reaction diffusion system in R3 , C R Acad Sci Paris, Sér I 330 (2000) 713–718 [8] M Efendiev, A Miranville, S Zelik, Infinite dimensional exponential attractors for a non-autonomous reaction–diffusion system, Math Nachr 248/249 (2003) 72–96 [9] M Efendiev, A Miranville, The dimension of the global attractors for dissipative reaction–diffusion systems, Appl Math Lett 16 (2003) 351–355 [10] A Miranville, Finite dimensional global attractor for a class of doubly nonlinear parabolic equations, CEJM (1) (2006) 163–182 [11] J Málek, D Pra˘zák, Large time behavior via the method of -trajectories, J Differential Equations 181 (2002) 243–279 [12] Q.F Ma, S.H Wang, C.K Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ Math J 51 (6) (2002) [13] D Prazak, A necessary and sufficient condition for the existence of an exponential attractor, CEJM (2003) 411–417 [14] J.C Robinson, Infinite-Dimensional Dynamical Systems an Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, Cambridge, 2001 [15] C.Y Sun, M.H Yang, Exponential attractors for the strongly damped wave equations (submitted for publication) [16] R Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1997 [17] C.K Zhong, M.H Yang, C.Y Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction–diffusion equations, J Differential Equations 223 (2006) 367–399 [18] S Zelik, The attractor for a nonlinear reaction–diffusion systm with a supercritical nonlinearity and its dimension, Rend Accad Naz Sci XL Mem Mat Appl (2000) [...]... bounded absorbing set B1 in L2 (Ω ) Combining with Lemma 5.1 together with the Condition (C) (see [12]), therefore, for any > 0, S (t )B1 has a finite -net Y Zhong, C Zhong / Nonlinear Analysis 71 (2009) 751–765 in L2 (Ω ) with centers (ut )1 , (ut )2 , , (ut )k for t (ut )i (1 i k) satisfying |ut (s) − (ut )i |2 < Ω 2 763 T (B) > 0 That is for any ut (s) with s T (B), we can find some , (5.7) where... attractors for the strongly damped wave equations (submitted for publication) [16] R Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1997 [17] C.K Zhong, M.H Yang, C.Y Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction diffusion equations, J Differential Equations 223 (2006) 367–399 [18] S... Banach spaces, J Dynam Differential Equations 13 (4) (2001) 791–806 [3] A Eden, C Foias, B Nicolaenko, R Teman, Exponential attractors for dissipative evolution equations, in: Research in Applied Mathematics, vol 37, John-Wiley, New York, 1994 [4] A Eden, C Foias, V Kalantarov, A remark on two constructions of exponential attractors for α -contractions, J Dynam Differential Equations 10 (1) (1998) 37–45... Exponential attractors for a nonlinear reaction diffusion system in R3 , C R Acad Sci Paris, Sér I 330 (2000) 713–718 [8] M Efendiev, A Miranville, S Zelik, Infinite dimensional exponential attractors for a non-autonomous reaction diffusion system, Math Nachr 248/249 (2003) 72–96 [9] M Efendiev, A Miranville, The dimension of the global attractors for dissipative reaction diffusion systems, Appl Math Lett 16... Lemma 4.1 and Corollary 4.1, applying the general method as in [3,15], we obtain the result of Theorem 4.1 5 Exponential attractor in L 2p−2 (Ω ) with g ∈ G In this section, we construct an exponential attractor in L2p−2 (Ω ) for Eq (1.1) with g ∈ G We begin with the following remark Remark 5.1 Since G ⊂ L2 (Ω ) ⊂ H −1 (Ω ), Theorems 3.1 and 4.1 holds true as well as for g ∈ G That is, the semigroup... )|2p−2 −2 |v|2p 2p−2 (5.15) 2 and using the Gronwall inequality, we have e(2p−2) |v(0)|2p−2 (5.16) which implies that V (t ) possesses the Lipschitz property with respect to the initial condition Secondly, we need to prove the Hölder continuity with respect to t, we note that the solution u(t ) ∈ C ([0, T ], L2 (Ω )) (see [17]), we have u(t1 ) − u(t2 ) = t2 ∂t u(s)ds (5.17) t1 which implies that |u(t1... C2 (B) such that 2 distL2p−2 (Ω ) (S (t )B, M2 ) C2 e−β2 t (5.21) for t > δ2 > 0 Applying Theorem 2.1 together with Corollary 5.1, we can prove the existence of exponential attractor in L2p−2 (Ω ) for discrete semigroup And then, using the Standard manner (e.g., see [3, chap 3]) together with Corollary 5.2, and we can obtain the continuous case and finish the proof of Theorem 5.2 Remark 5.3 For g ∈... exponential attractor in Lp (Ω ) for (1.1) with g ∈ H −1 (Ω ) Acknowledgments This work is supported in part by the NSFC Grant (10771089) and Trans-Century Training Programme Foundation for the Talents by the Ministry of Education References [1] A Babin, B Nicolaenko, Exponential attractors of reaction–dissfusion systems in an unbounded domain, J Dynam Differential Equations 7 (4) (1995) 567–590 [2] L Dung,... = 0, θ|t =0 = ∆u(0) − f (u(0)) + g , θ |∂ Ω = 0 (4.6) Multiplying (4.6) by θ (t ) and combining with the assumption (1.2) and the Hölder inequality and the Poincaré inequality, we deduce that ∂t |θ (t )|22 + 2η|θ (t )|22 + η1 |∇θ (t )|22 − 2 |θ (t )|22 0, (4.7) where η, η1 are positive constants Combining with Lemma 3.3, integrating (4.7) over [t1 , t2 ], we have t2 t1 |∂t u(s)|2H 1 (Ω ) ds C 0 and... fixed > 0, applying the same argument as Remark 2.1.a and combining with Lemma 3.3, therefore, there is a δ = (ut )i , 1 i k, we have |(ut )i |2 2 1 M1 2γ 2γ γ −2 , · where M1 = max1 i k |(ut )i |Lγγ−(2Ω ) and γ can be chosen as in g, such that for each , (5.8) E provided that m(E ) < δ(E ⊂ Ω ) On the other hand, from Lemma 5.2 together with the existence of a bounded absorbing set B2 in L2 (Ω ), therefore, ...752 Y Zhong, C Zhong / Nonlinear Analysis 71 (2009) 751–765 At the same time, they also proved the existence... review the concept of an exponential attractor(see [2,3,5,6,8,13] for a detailed exposition) Y Zhong, C Zhong / Nonlinear Analysis 71 (2009) 751–765 753 Definition 2.2 Let E be a metric space and... also true by the same argument in the following proof of Theorem 2.1 R , 2i the other 754 Y Zhong, C Zhong / Nonlinear Analysis 71 (2009) 751–765 Now, we prove Theorem 2.1 Proof of Theorem 2.1