# Exponential Attractors for Parabolic Equations with Dynamic Boundary Conditions

Authors: Zhao-hui Fan
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We study exponential attractors for semilinear parabolic equations with dynamic boundary conditions in bounded domains. First, we give the existence of the exponential attractor inL2(Ω)×L2(Γ) by proving that the corresponding semigroup satisfies the enhanced flattering property. Second, we apply asymptotic a priori estimate and obtain the exponential attractor in Lp(Ω)×Lq(Γ). Finally, we show the exponential attractor in (H1(Ω)∩Lp(Ω))×Lq(Γ).


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## Body

## 1. Introduction

Parabolic equations with dynamical boundary conditions have strong backgrounds in mathematical physics. They arise in the heat transfer theory in a solid in contact with moving fluid, thermoelasticity, diffusion phenomena, heat transfer in two medium, problems in fluid dynamic, and so forth. At present, there are many monographs in the whole world (see [1–13]). Several approaches have been used for these equations, like the theory of semigroup, with Bessel potential and Besov space, and the variational setting. In particular, we are devoted to the long-time behavior of the solutions. For instance, In [1], the authors showed existence of pullback attractors. In [8, 9], the authors gave well posedness and global attractors in Lp(Ω)×Lp(Γ). In [12, 13]; the authors obtained uniform attractors and some asymptotic regularity of global attractors in (H1(Ω)∩Lp(Ω))×Lq(Γ). An exponential attractor, in contrast to a global attractor (or a uniform attractor), enjoys a uniform exponential rate of convergence of its solution. Because of this, exponential attractors possess more practical property. But to our knowledge, it does not seem to be in the literature any study of the existence of exponential attractors for this kind of equations.This paper is concerned with existence of exponential attractors for the following reaction-diffusion equation with dynamic boundary condition(1)ut-Δu+f(u)=h(x)inΩ,ut+∂u∂ν+g(u)=ρ(x)onΓ,u(0,x)=u0(x),inΩ-,
where Ω⊂ℝN, N≥1, is a bounded domain with a smooth boundary Γ. Here ν is the outer unit normal on Γ.In [14], the authors established some necessary and sufficient conditions for the existence of exponential attractors for continuous and norm-to-weak continuous semigroup and provided a new method for proving the existence of exponential attractors by combining with the flattering property. Motivated by some ideas in [14–16], we combine asymptotic a prior estimate with the enhanced flattening property and show sufficient and necessary existence of exponential attractors in uniformly convex Banach spaces. As an application, we prove the existence of exponential attractors for the reaction-diffusion equation with dynamic boundary condition.This paper is organized as follows. In Section2, we recall some basic results and then give our theorems, that is, Theorems 5, 6, and 9 and the solution semigroup corresponding to (1). In Section 3, we obtain the exponential attractor in L2(Ω)×L2(Γ) for weak solutions, then combine asymptotical a prior estimate and show the exponential attractor in Lp(Ω)×Lq(Γ). Finally, we derive the existence of the exponential attractor in the space (H1(Ω)∩Lp(Ω))×Lq(Γ).In the following, the constantsCi,  i=0,1,2,… will always denote generic constants different in various occurrences. The symbol C(s) will denote a positive constant dependent of s. We will write ΩT for Ω×(0,T),ΓT for Γ×(0,T),(·,·) for the inner product in L2(Ω) and 〈·,·〉 for the inner product in L2(Γ). For convenience, we denote the norm of u by |u|Ω in the space L2(Ω) and |u|Γ in the space L2(Γ).


## 2. Preliminary

### 2.1. The Basic Results and Theorems

LetX be a complete metric space and a one-parameter family of mappings S(t): let X→X(t≥0) be a semigroup. Here we omit the definitions of continuous or norm-to-weak semigroups, dynamical systems, global attractors, and exponential attractors (see [15–20]).Definition 1.
LetX be a metric space and B be a bounded subset of X. The Kuratowski measure of noncompactness α(B) of B is defined as
(2)α(B)=inf{δ>0∣Badmitsafinitecoverbysetsofdiameter≤δy}.Theorem 2 (see [14]).
Assume thatB is a bounded absorbing set for discrete dynamical system S(n) in X; then the following are equivalent.(1)
The measure of noncompactness is exponentially decaying for dynamical system(S(n),X), that is, there exist k,l>0 such that α(⋃m>n‍S(m)B)≤ke-ln.(2)
ForS(n), there exist exponential attractors.Theorem 3 (see [14]).
Assume that B is a bounded absorbing set forS(t) in X; then the following are equivalent.(1)
The measure of noncompactness is exponentially decaying for dynamical system(S(t),X), that is, there exist k,l>0 such that α(⋃s>t‍S(s)B)≤ke-lt.(2)
ForS(t), there exist exponential attractors.Definition 4 (see [14] (Enhanced Flattening Property)).
LetX be a uniformly convex Banach space; for any bounded set B of X, there exist k,l>0 and T>0, and a finite dimension subspace X1 of X, such that(1)
Pm(⋃s≥t‍S(s)B) is bounded and(2)
||(I-Pm)(⋃s≥t‍S(s)x)||≤ke-lt+k(m), for all x∈B,for allt≥T. Here Pm:X→X1 is a bounded projector, m is the dimension of X1, ||·|| denotes the norm in X, and k(s) is a real-valued function satisfying lims→∞k(s)=0.Inspired by [14, 21], we easily obtain the following.Theorem 5.
LetX be a uniformly convex Banach space and {S(t)}t≥0 be a continuous or norm-to-weak continuous semigroup in X. Then the following conditions are equivalent.(1)
The measure of noncompactness is exponentially decaying for dynamical system(S(t),X), that is, there exist k,l>0 such that α(⋃s>t‍S(s)B)≤ke-lt.(2)
S(t) satisfies the enhanced flattening property.Proof.
(
1
⇒
2
) On account of condition (1), for any bounded subset B of X and for any ε>0, there exist t=t(B) such that
(3)α(⋃s>t‍S(s)B)≤ke-lt.
Namely, there exist a finite number of subset A1,A2,…,An with diameter less than ke-lt, such that
(4)⋃s>t‍S(s)B⊂∪i=1nAi.
Let xi∈Ai, then
(5)⋃s>t‍S(s)B⊂∪i=1nN(xi,ke-lt).
Let X1=span{x1,x2,…,xn}, since X is uniformly convex, there exist a projection P:X→X1, such that for any x∈X, ||x-Px||=dist(x,X1). Hence,
(6)∥(I-P)S(t)x∥≤ke-lt<ke-lt+k(m).(2⇒1) have been shown in Theorem 4.3 of [14], so we omitted it here.By Theorem5, we can deduce the following.Theorem 6.
LetX be a uniformly convex Banach space and {S(t)}t≥0 be a continuous or norm-to-weak continuous semigroup in X. Then, for dynamical system (S(t),X), there exist exponential attractors in X if and only if(1)
there is a bounded absorbing setB⊂X, and(2)
S(t) satisfies the enhanced flattening property.In addition, we use later the following theorem about global attractors.Theorem 7 (see [15, Corollary 5.7]).
Let{S(t)}t≥0 be a semigroup on Lp(Ω), (p≥1), be a continuous or weak continuous semigroup on Lq(Ω) for some q≤p, and have a global attractor in Lq(Ω). Then, {S(t)}t≥0 has a global attractor in Lp(Ω) if and only if(1)
{S(t)}t≥0 has a bounded absorbing set B0 in Lp(Ω), and(2)
for anyε>0 and any bounded subset B⊂Lp(Ω), there exist positive constants M=M(,B) and T=T(,B), such that
(7)∫Ω(|S(t)u0|≥M)‍|S(t)u0|p<εforanyu0∈B,t≥T,where Ω(|u|≥M)={x∈Ω||u(x)|≥M}.We give the following lemma concerning the covering of the set in two different topologies used later in the proof of Theorem9.Lemma 8 (see [15, Lemma 5.3]).
For anyε>0, the bounded subset B of Lp(Ω) has a finite ε-net in Lp(Ω) if there exists a positive constant M=M(ε), such that(1)
B has a finite (3M)(q-p)/q(ε/2)p/q-net in Lq(Ω);(2)
∫Ω(|u|≥M)‍|u|p≤2-2p+2ε, for any u0∈B0.Inspired by [16], we give the subsequent theorem which describes our new technique to construct an exponential attractor in a stronger topological space.Theorem 9.
Assume thatp>q>0 and Ω⊂ℝn. Let S(t) be a continuous or norm-to-weak continuous semigroup on Lp(Ω) and Lq(Ω) and B0 be a positively invariant bounded absorbing set in Lp(Ω). If the following conditions hold true:(1)
S(t) has an exponential attractor in Lq(Ω);(2)
for anyε>0, there exist positive constant M=M(ε) such that
(8)∫Ω(|S(t)u0|≥M)‍|S(t)u0|p≤2-2p+2ε,foranyu0∈B0,then S(t) has an exponential attractor in Lp(Ω).Proof.
TakeT>0, and let Sn=S(nT); obviously Sn is a discrete dynamical system. On account of Condition (1), for Sn there exists an exponential attractor ℳ in Lq(Ω). By Theorem 2, we find that there exist k,l>0 such that α(⋃m>n‍S(m)B)≤ke-ln. By the definition of the measure of noncompactness, for all n∈N, there exist finite points xni∈S(n)B such that S(n)B⊂∪i=1Mn(BLq(Ω)(xni,ke-ln)∩S(n)B). Then, there exist n0∈ℕ and M1=Cεγ/(q-γ)(γ>q) such that
(9)ke-ln0≤(3M1)(q-p)/q(ε2)p/q≤ke-l(n0-1).
From Lemma 8 and Condition (2), it follows that S(n0)(B) has a finite ε-net in Lp(Ω).
Letε be replaced by ε/2, there n1∈ℕ and M2=C(ε/2)γ/(q-γ)(γ>q) such that
(10)ke-l(n0+n1)≤(3M2)(q-p)/q(ε22)p/q≤ke-l(n0+n1-1).
From Lemma 8 and Condition (2), it follows that S(n0+n1)(B) has a finite ε/2-net in Lp(Ω).
By induction, letε/2i-1 be replaced by ε/2i, there ni∈ℕ and Mi+1=C(ε/2i)γ/(q-γ)(γ>q) such that
(11)ke-l(n0+n1+⋯+ni)≤(3Mi+1)(q-p)/q(ε2i+1)p/q≤ke-l(n0+n1+⋯+ni-1).
From Lemma 8 and Condition (2), it follows that S(n0+n1+⋯+ni)(B) has a finite ε/2i-net in Lp(Ω). Note that
(12)(3Mi+1)(q-p)/q(ε/2i+1)p/q(3Mi)(q-p)/q(ε/2i)p/q=(Mi+1/Mi)(q-p)/q2p/q=((ε/2i)γ/(q-γ)/(ε/2i-1)γ/(q-γ))(q-p)/q2p/q=(12)γ(p-q)/q(γ-q)+p/q<1.
So, the ratio of (3Mi+1)(q-p)/q(ε/2i+1)p/q and (3Mi)(q-p)/q(ε/2i)p/q is a constant for given γ, p, q, and ε. In fact,
(13)(3Mi+1)(q-p)/q(ε2i+1)p/q=(12)p/q(3C)(q-p)/qε(γ(p-q)/q(γ-q))+p/q×(12)((γ(p-q)/q(γ-q))+p/q)i.
Substituting (13) into (11), we deduce easily that
(14)ε2i≤2(3Mi+1)(p-q)/qkq/pe-ql(n0+n1+⋯+ni-1)/p=2(3C)(q-p)/qεγ(p-q)/q(γ-q)(12)(γ(p-q)/q(γ-q))i×kq/pe-ql(n0+n1+⋯+ni-1)/p.
Choose that 2(3C)(q-p)/qεγ(p-q)/q(γ-q)kq/peql/p=k′≥2(3C)(q-p)/qεγ(p-q)/q(γ-q)(1/2)(γ(p-q)/q(γ-q))ikq/peql/p and l′=ql/p, we know that S(n0+n1+⋯+ni)(B) has a finite k′e-l′(n0+n1+⋯+ni)-net in Lp(Ω).
Combining (9), (10) (11), and (13), we can choose n1=n2=⋯=ni=[(1/l)(γ(p-q)/q(γ-q)+p/q)ln2]+1, where [a] is the integer part of a. Denoted by l0=[(1/l)(γ(p-q)/q(γ-q)+p/q)ln2]+1, we have S(n0+n1+⋯+ni)(B)=S(n0+il0) has a finite k′e-l′(n0+il0)-net in Lp(Ω). By Theorem 2, an exponential attractor exists in Lp(Ω) for discrete semigroup S(il0). Using the same argument as in [18], it is easy to deduce that an exponential attractor ℳ exists in Lp(Ω) for discrete semigroup S(n). Let ℳ0=∪0≤s≤TS(s)ℳ. As an repetition of the general method developed by Li et al. [14], we can show that ℳ0 is the exponential attractor for dynamical system (S(t),X).


### 2.2. The Solution Semigroup

We can write Problem (1) as an evolution for unknown u(x,t) in Ω and v(x,t) on Γ(15)wt+Aw+F=0,
with the compatibility v=γ(u(t))(γ is trace operator) for t>0 and λ>0, where
(16)w=(uv),A=(-Δ+λ0∂∂ν0),F=(f(u)-λu-h(x)g(v)-ρ(x)).
In the case Ω bounded, the operator A has a compact resolvent and its spectrum, denoted by σ(A)={μn}n⊂R+, forms an increasing sequence converging to infinity (see [11, Theorem 1.4]). Moreover, there exists an orthonormal basis in L2(Ω)×L2(Γ), {ωn}n, which are solutions of the eigenvalue problem
(17)-Δu+λu=μnu,x∈Ω,∂u∂ν=μnu,x∈Γ,
where μn⊂R+ forms an increasing sequence converging to infinity. We denote by Pn the orthonormal projector
(18)Pn:L2(Ω)×L2(Γ)→span{ω1,ω2,…,ωn},u=∑i=1∞‍αiωi↦un=∑i=1n‍αiωi.
So, we can perform the Galerkin truncation by using orthonormal basis mentioned above and guarantee the following existence and uniqueness (see [1, 8, 9, 12]).Theorem 10.
Assume that the functionsf,g∈𝒞1satisfying
(19)-C0+C1|s|p≤f(s)s≤C0+C2|s|p,p≥2,-C3+C4|s|q≤g(s)s≤C3+C5|s|q,q≥2,(20)f′(s)≥-l,g′(s)≥-m.
Then, Problem (1) has a unique weak solution, for any T>0, given (u0,γ(u0))∈L2(Ω)×L2(Γ) and h(x)∈L2(Ω), ρ(x)∈L2(Γ) there exists a solution u with
(21)u∈𝒞([0,T];L2(Ω)),u∈L2(0,T;H1(Ω))∩Lp(ΩT),u∈𝒞([0,T];L2(Γ)),u∈Lq(ΓT),
and (u0,γ(u0))↦(u(t),γ(u)) is continuous on L2(Ω)×L2(Γ).By the last theorem, we can define the operator semigroup{S(t)}t≥0 in L2(Ω)×L2(Γ) as follows:
(22)S(t)(u0,γ(u0)):L2(Ω)×L2(Γ)×ℝ+→L2(Ω)×L2(Γ),
which is continuous in L2(Ω)×L2(Γ).Furthermore, we obtained the bounded absorbing set for dynamical system(S(t),X) (see [8, 12]).Theorem 11.
Under the assumptions of Theorem10, the semigroup {S(t)}t≥0 have (L2(Ω)×L2(Γ),Lp(Ω)×Lq(Γ))-, (L2(Ω)×L2(Γ),H1(Ω)×Lq(Γ))-bounded absorbing sets, that is, for any bounded subset B⊂L2(Ω)×L2(Γ), there exists a positive constant T, which is only dependent on the L2(Ω)×L2(Γ)-norm of B, such that(23)∥u(t)∥Lp(Ω)≤R0forany(u0,γ(u0))∈B,t≥T,∥u(t)∥Lq(Γ)≤R0forany(u0,γ(u0))∈B,t≥T,∥∇u(t)∥L2(Ω)≤R0forany(u0,γ(u0))∈B,t≥T,
where R0 is a positive constant independent of B, u(t)=S(t)u0.


## 2.1. The Basic Results and Theorems

LetX be a complete metric space and a one-parameter family of mappings S(t): let X→X(t≥0) be a semigroup. Here we omit the definitions of continuous or norm-to-weak semigroups, dynamical systems, global attractors, and exponential attractors (see [15–20]).Definition 1.
LetX be a metric space and B be a bounded subset of X. The Kuratowski measure of noncompactness α(B) of B is defined as
(2)α(B)=inf{δ>0∣Badmitsafinitecoverbysetsofdiameter≤δy}.Theorem 2 (see [14]).
Assume thatB is a bounded absorbing set for discrete dynamical system S(n) in X; then the following are equivalent.(1)
The measure of noncompactness is exponentially decaying for dynamical system(S(n),X), that is, there exist k,l>0 such that α(⋃m>n‍S(m)B)≤ke-ln.(2)
ForS(n), there exist exponential attractors.Theorem 3 (see [14]).
Assume that B is a bounded absorbing set forS(t) in X; then the following are equivalent.(1)
The measure of noncompactness is exponentially decaying for dynamical system(S(t),X), that is, there exist k,l>0 such that α(⋃s>t‍S(s)B)≤ke-lt.(2)
ForS(t), there exist exponential attractors.Definition 4 (see [14] (Enhanced Flattening Property)).
LetX be a uniformly convex Banach space; for any bounded set B of X, there exist k,l>0 and T>0, and a finite dimension subspace X1 of X, such that(1)
Pm(⋃s≥t‍S(s)B) is bounded and(2)
||(I-Pm)(⋃s≥t‍S(s)x)||≤ke-lt+k(m), for all x∈B,for allt≥T. Here Pm:X→X1 is a bounded projector, m is the dimension of X1, ||·|| denotes the norm in X, and k(s) is a real-valued function satisfying lims→∞k(s)=0.Inspired by [14, 21], we easily obtain the following.Theorem 5.
LetX be a uniformly convex Banach space and {S(t)}t≥0 be a continuous or norm-to-weak continuous semigroup in X. Then the following conditions are equivalent.(1)
The measure of noncompactness is exponentially decaying for dynamical system(S(t),X), that is, there exist k,l>0 such that α(⋃s>t‍S(s)B)≤ke-lt.(2)
S(t) satisfies the enhanced flattening property.Proof.
(
1
⇒
2
) On account of condition (1), for any bounded subset B of X and for any ε>0, there exist t=t(B) such that
(3)α(⋃s>t‍S(s)B)≤ke-lt.
Namely, there exist a finite number of subset A1,A2,…,An with diameter less than ke-lt, such that
(4)⋃s>t‍S(s)B⊂∪i=1nAi.
Let xi∈Ai, then
(5)⋃s>t‍S(s)B⊂∪i=1nN(xi,ke-lt).
Let X1=span{x1,x2,…,xn}, since X is uniformly convex, there exist a projection P:X→X1, such that for any x∈X, ||x-Px||=dist(x,X1). Hence,
(6)∥(I-P)S(t)x∥≤ke-lt<ke-lt+k(m).(2⇒1) have been shown in Theorem 4.3 of [14], so we omitted it here.By Theorem5, we can deduce the following.Theorem 6.
LetX be a uniformly convex Banach space and {S(t)}t≥0 be a continuous or norm-to-weak continuous semigroup in X. Then, for dynamical system (S(t),X), there exist exponential attractors in X if and only if(1)
there is a bounded absorbing setB⊂X, and(2)
S(t) satisfies the enhanced flattening property.In addition, we use later the following theorem about global attractors.Theorem 7 (see [15, Corollary 5.7]).
Let{S(t)}t≥0 be a semigroup on Lp(Ω), (p≥1), be a continuous or weak continuous semigroup on Lq(Ω) for some q≤p, and have a global attractor in Lq(Ω). Then, {S(t)}t≥0 has a global attractor in Lp(Ω) if and only if(1)
{S(t)}t≥0 has a bounded absorbing set B0 in Lp(Ω), and(2)
for anyε>0 and any bounded subset B⊂Lp(Ω), there exist positive constants M=M(,B) and T=T(,B), such that
(7)∫Ω(|S(t)u0|≥M)‍|S(t)u0|p<εforanyu0∈B,t≥T,where Ω(|u|≥M)={x∈Ω||u(x)|≥M}.We give the following lemma concerning the covering of the set in two different topologies used later in the proof of Theorem9.Lemma 8 (see [15, Lemma 5.3]).
For anyε>0, the bounded subset B of Lp(Ω) has a finite ε-net in Lp(Ω) if there exists a positive constant M=M(ε), such that(1)
B has a finite (3M)(q-p)/q(ε/2)p/q-net in Lq(Ω);(2)
∫Ω(|u|≥M)‍|u|p≤2-2p+2ε, for any u0∈B0.Inspired by [16], we give the subsequent theorem which describes our new technique to construct an exponential attractor in a stronger topological space.Theorem 9.
Assume thatp>q>0 and Ω⊂ℝn. Let S(t) be a continuous or norm-to-weak continuous semigroup on Lp(Ω) and Lq(Ω) and B0 be a positively invariant bounded absorbing set in Lp(Ω). If the following conditions hold true:(1)
S(t) has an exponential attractor in Lq(Ω);(2)
for anyε>0, there exist positive constant M=M(ε) such that
(8)∫Ω(|S(t)u0|≥M)‍|S(t)u0|p≤2-2p+2ε,foranyu0∈B0,then S(t) has an exponential attractor in Lp(Ω).Proof.
TakeT>0, and let Sn=S(nT); obviously Sn is a discrete dynamical system. On account of Condition (1), for Sn there exists an exponential attractor ℳ in Lq(Ω). By Theorem 2, we find that there exist k,l>0 such that α(⋃m>n‍S(m)B)≤ke-ln. By the definition of the measure of noncompactness, for all n∈N, there exist finite points xni∈S(n)B such that S(n)B⊂∪i=1Mn(BLq(Ω)(xni,ke-ln)∩S(n)B). Then, there exist n0∈ℕ and M1=Cεγ/(q-γ)(γ>q) such that
(9)ke-ln0≤(3M1)(q-p)/q(ε2)p/q≤ke-l(n0-1).
From Lemma 8 and Condition (2), it follows that S(n0)(B) has a finite ε-net in Lp(Ω).
Letε be replaced by ε/2, there n1∈ℕ and M2=C(ε/2)γ/(q-γ)(γ>q) such that
(10)ke-l(n0+n1)≤(3M2)(q-p)/q(ε22)p/q≤ke-l(n0+n1-1).
From Lemma 8 and Condition (2), it follows that S(n0+n1)(B) has a finite ε/2-net in Lp(Ω).
By induction, letε/2i-1 be replaced by ε/2i, there ni∈ℕ and Mi+1=C(ε/2i)γ/(q-γ)(γ>q) such that
(11)ke-l(n0+n1+⋯+ni)≤(3Mi+1)(q-p)/q(ε2i+1)p/q≤ke-l(n0+n1+⋯+ni-1).
From Lemma 8 and Condition (2), it follows that S(n0+n1+⋯+ni)(B) has a finite ε/2i-net in Lp(Ω). Note that
(12)(3Mi+1)(q-p)/q(ε/2i+1)p/q(3Mi)(q-p)/q(ε/2i)p/q=(Mi+1/Mi)(q-p)/q2p/q=((ε/2i)γ/(q-γ)/(ε/2i-1)γ/(q-γ))(q-p)/q2p/q=(12)γ(p-q)/q(γ-q)+p/q<1.
So, the ratio of (3Mi+1)(q-p)/q(ε/2i+1)p/q and (3Mi)(q-p)/q(ε/2i)p/q is a constant for given γ, p, q, and ε. In fact,
(13)(3Mi+1)(q-p)/q(ε2i+1)p/q=(12)p/q(3C)(q-p)/qε(γ(p-q)/q(γ-q))+p/q×(12)((γ(p-q)/q(γ-q))+p/q)i.
Substituting (13) into (11), we deduce easily that
(14)ε2i≤2(3Mi+1)(p-q)/qkq/pe-ql(n0+n1+⋯+ni-1)/p=2(3C)(q-p)/qεγ(p-q)/q(γ-q)(12)(γ(p-q)/q(γ-q))i×kq/pe-ql(n0+n1+⋯+ni-1)/p.
Choose that 2(3C)(q-p)/qεγ(p-q)/q(γ-q)kq/peql/p=k′≥2(3C)(q-p)/qεγ(p-q)/q(γ-q)(1/2)(γ(p-q)/q(γ-q))ikq/peql/p and l′=ql/p, we know that S(n0+n1+⋯+ni)(B) has a finite k′e-l′(n0+n1+⋯+ni)-net in Lp(Ω).
Combining (9), (10) (11), and (13), we can choose n1=n2=⋯=ni=[(1/l)(γ(p-q)/q(γ-q)+p/q)ln2]+1, where [a] is the integer part of a. Denoted by l0=[(1/l)(γ(p-q)/q(γ-q)+p/q)ln2]+1, we have S(n0+n1+⋯+ni)(B)=S(n0+il0) has a finite k′e-l′(n0+il0)-net in Lp(Ω). By Theorem 2, an exponential attractor exists in Lp(Ω) for discrete semigroup S(il0). Using the same argument as in [18], it is easy to deduce that an exponential attractor ℳ exists in Lp(Ω) for discrete semigroup S(n). Let ℳ0=∪0≤s≤TS(s)ℳ. As an repetition of the general method developed by Li et al. [14], we can show that ℳ0 is the exponential attractor for dynamical system (S(t),X).


## 2.2. The Solution Semigroup

We can write Problem (1) as an evolution for unknown u(x,t) in Ω and v(x,t) on Γ(15)wt+Aw+F=0,
with the compatibility v=γ(u(t))(γ is trace operator) for t>0 and λ>0, where
(16)w=(uv),A=(-Δ+λ0∂∂ν0),F=(f(u)-λu-h(x)g(v)-ρ(x)).
In the case Ω bounded, the operator A has a compact resolvent and its spectrum, denoted by σ(A)={μn}n⊂R+, forms an increasing sequence converging to infinity (see [11, Theorem 1.4]). Moreover, there exists an orthonormal basis in L2(Ω)×L2(Γ), {ωn}n, which are solutions of the eigenvalue problem
(17)-Δu+λu=μnu,x∈Ω,∂u∂ν=μnu,x∈Γ,
where μn⊂R+ forms an increasing sequence converging to infinity. We denote by Pn the orthonormal projector
(18)Pn:L2(Ω)×L2(Γ)→span{ω1,ω2,…,ωn},u=∑i=1∞‍αiωi↦un=∑i=1n‍αiωi.
So, we can perform the Galerkin truncation by using orthonormal basis mentioned above and guarantee the following existence and uniqueness (see [1, 8, 9, 12]).Theorem 10.
Assume that the functionsf,g∈𝒞1satisfying
(19)-C0+C1|s|p≤f(s)s≤C0+C2|s|p,p≥2,-C3+C4|s|q≤g(s)s≤C3+C5|s|q,q≥2,(20)f′(s)≥-l,g′(s)≥-m.
Then, Problem (1) has a unique weak solution, for any T>0, given (u0,γ(u0))∈L2(Ω)×L2(Γ) and h(x)∈L2(Ω), ρ(x)∈L2(Γ) there exists a solution u with
(21)u∈𝒞([0,T];L2(Ω)),u∈L2(0,T;H1(Ω))∩Lp(ΩT),u∈𝒞([0,T];L2(Γ)),u∈Lq(ΓT),
and (u0,γ(u0))↦(u(t),γ(u)) is continuous on L2(Ω)×L2(Γ).By the last theorem, we can define the operator semigroup{S(t)}t≥0 in L2(Ω)×L2(Γ) as follows:
(22)S(t)(u0,γ(u0)):L2(Ω)×L2(Γ)×ℝ+→L2(Ω)×L2(Γ),
which is continuous in L2(Ω)×L2(Γ).Furthermore, we obtained the bounded absorbing set for dynamical system(S(t),X) (see [8, 12]).Theorem 11.
Under the assumptions of Theorem10, the semigroup {S(t)}t≥0 have (L2(Ω)×L2(Γ),Lp(Ω)×Lq(Γ))-, (L2(Ω)×L2(Γ),H1(Ω)×Lq(Γ))-bounded absorbing sets, that is, for any bounded subset B⊂L2(Ω)×L2(Γ), there exists a positive constant T, which is only dependent on the L2(Ω)×L2(Γ)-norm of B, such that(23)∥u(t)∥Lp(Ω)≤R0forany(u0,γ(u0))∈B,t≥T,∥u(t)∥Lq(Γ)≤R0forany(u0,γ(u0))∈B,t≥T,∥∇u(t)∥L2(Ω)≤R0forany(u0,γ(u0))∈B,t≥T,
where R0 is a positive constant independent of B, u(t)=S(t)u0.


## 3. The Main Results

### 3.1. Exponential Attractors inL2(Ω)×L2(Γ)

Theorem 12.
Under the hypothesis of Theorem11, {S(t)}t≥0 has a (L2(Ω)×L2(Γ),L2(Ω)×L2(Γ))-exponential attractor.Proof.
Letu2=(I-Pn)u, where Pn is denoted by the orthonormal projector as mentioned before. Multiplying  (1)1  by u2 and integrating by parts, we get(24)12ddt(∥u2∥Ω2+∥u2∥Γ2)+∫Ω‍|∇u2|2+∫Ω‍f(u)u2+∫Γ‍g(u)u2=(h,u2)+〈ρ,u2〉.
Using (19), we can deduce that
(25)|u|p≥|u|2-C,|u|q≥|u|2-C,
for any p,q≥2. Therefore,
(26)∫Ω‍f(u)u2+∫Γ‍g(u)u2≥∫Ω‍|u2|p+∫Γ‍|u2|q-C(m(Ω),m(Γ)).
Thus, we know
(27)ddt(∥u2∥Ω2+∥u2∥Γ2)+μn(∫Ω‍|u2|2+∫Γ‍|u|2)≤C(m(Ω),m(Γ),|h|Ω,|ρ|Γ).
Applying the Gronwall-inequality, we have
(28)∥u2(t)∥Ω2+∥u2(t)∥Γ2≤e-μnt+Cμn.
Obviously, the enhanced flattening property holds true. By Theorem 6, we obtain the exponential attractor in L2(Ω)×L2(Γ).


### 3.2.(L2(Ω)×L2(Γ),Lr(Ω)×Lr(Γ))-Exponential Attractor

Ifr=min{p,q}, we easily showed asymptotic a prior estimate of the solution of (1) in Lr(Ω)×Lr(Γ) in order to obtain global and uniform attractors in Lp(Ω)×Lp(Γ), where r(≥2) is an integer (resp., see [8, 12]).Theorem 13.
Under the hypothesis of Theorem11, then for any ε>0 and any bounded subset B⊂L2(Ω)×L2(Γ), there exist two positive constants T=T(B,ε) and M=M(ε) such that
(29)∫Ω(|u|≥M)‍|u|r+∫Γ(|u|≥M)‍|u|r≤Cε,∀t≥T,∫Γ(|u|≥M)‍|u|r≤Cε,∀t≥T(u0,γ(u0))∈B,
where the constant C is independent of ε and B.After Theorem13, we obtain the subsequent result.Theorem 14.
Under the assumption of Theorem11, then the semigroup {S(t)}t≥0 generated by Problem (1) with initial data u0∈L2(Ω) and γ(u0)∈L2(Γ) has a (L2(Ω)×L2(Γ),Lr(Ω)×Lr(Γ))-exponential attractor ℳr, that is, ℳr is compact, invariant in Lr(Ω)×Lr(Γ), and attracts every bounded in L2(Ω)×L2(Γ) in the topology of Lr(Ω)×Lr(Γ).Proof.
By Theorems11 and 12, it is easily verified that {S(t)}t≥0 has an exponential attractor in Lr(Ω)×Lr(Γ) where we apply Theorem 9.Ifr>min{p,q}, the reasoning process mentioned earlier is not available. We note the following result in [13] after authors obtained regularity of global attractor.Lemma 15 (see [13, Corollary 3.1]).
Under the assumptions of Theorem11, the semigroup {S(t)}t≥0 has a compact global attractor 𝒜 in (H1(Ω)∩Lp(Ω))×Lq(Γ).By Theorem7 and Lemma 15, we easily obtain the following.Theorem 16.
Under the hypothesis of Theorem11, then for any ε>0 and any bounded subset B⊂L2(Ω)×L2(Γ), there exist two positive constants T=T(B,ε) and M=M(ε) such that
(30)∫Ω(|u|≥M)‍|u|p+∫Γ(|u|≥M)‍|u|q≤Cε,∀t≥T,∫Γ(|u|≥M)‍|u|q≤Cε,∀t≥T(u0,γ(u0))∈B,
where the constant C is independent of ε and B.Applying Theorem9 to Theorems 11 and 16, we have the following.Corollary 17.
Under the assumption of Theorem12, the semigroup {S(t)}t≥0 generated by Problem (1) with initial data u0∈L2(Ω) and γ(u0)∈L2(Γ) has a (L2(Ω)×L2(Γ),Lp(Ω)×Lq(Γ))-exponential attractor ℳpq.Remark 18.
In fact, for any integerr≤max{p,q}, we can obtain Theorem 16 and Corollary 17 by Lemma 15. In other words, if r≤min{p,q}, we can obtain Theorem 16 and Corollary 17 by Lemma 15 and need not use asymptotic a prior estimate. Here, we point this result obtained by the different procedure.On account of Theorem6 and Corollary 17, we have the following.Corollary 19.
There exists somem such that
(31)∥(I-Pm)u∥Lp(Ω)+∥(I-Pm)u∥Lq(Γ)≤ke-lt+k(m).Now, we give some a prior estimates aboutut.Lemma 20 (see [8, 12]).
Under the assumption of Theorem11, for any bounded subset B⊂L2(Ω)×L2(Γ), there exists a positive constant T1 which depends only on the L2(Ω)×L2(Γ)-norm of B such that
(32)∫Ω‍|ut(s)|2dx+∫Γ‍|ut(s)|2dx≤R1∀s≥T1,(u0,0)∈B,
where R1 is a positive constant which depends on M.We can easily obtain the following theorem.Theorem 21.
Under the assumption of Theorem11, the semigroup {S(t)}t≥0 generated by Problem (1) with initial data u0∈L2(Ω) and γ(u0)∈L2(Γ) satisfies the enhanced flattering in the space H1(Ω)×Lq(Γ).Proof.
We denoteu=u1+u2, where u1=Pnu,  u2=(I-Pn)u, and Pn has been introduced in Section 3.
Multiplying Problem (1) by u2, we can get
(33)12ddt∫Ω‍|u2|2+12ddt∫Ω‍|u2|2+∫Ω‍|∇u2|2+∫Ω‍f(u)u2+∫Γ‍g(u)u2=(h,u2)+〈ρ,u2〉.
By (25) and (26), we can deduce
(34)∫Ω‍|∇u2|2≤(|ut2|Ω+|h|Ω)|u2|Ω+(|ut2|Γ+|ρ|Γ)|u2|Γ+∥u∥Ωp-1∥u2∥Lp(Ω)+∥u∥Γq-1∥u2∥Lq(Γ).
By Theorems 11 and 12, Corollary 19, and Lemma 20, we know that S(t)u0 satisfies the enhanced flattering property in H1(Ω). On account of Theorem 6 and Corollary 19, we can deduce that Theorem 21 is valid.It is immediate by Theorem6, Corollary 19, and Theorem 21.Corollary 22.
Under the assumption of Theorem12, the semigroup {S(t)}t≥0 generated by Problem (1) with initial data (u0,γ(u0))∈L2(Ω)×L2(Γ) has a (L2(Ω)×L2(Γ),(H1(Ω)∩Lp(Ω))×Lq(Γ))-exponential attractor ℳ.


## 3.1. Exponential Attractors inL2(Ω)×L2(Γ)

Theorem 12.
Under the hypothesis of Theorem11, {S(t)}t≥0 has a (L2(Ω)×L2(Γ),L2(Ω)×L2(Γ))-exponential attractor.Proof.
Letu2=(I-Pn)u, where Pn is denoted by the orthonormal projector as mentioned before. Multiplying  (1)1  by u2 and integrating by parts, we get(24)12ddt(∥u2∥Ω2+∥u2∥Γ2)+∫Ω‍|∇u2|2+∫Ω‍f(u)u2+∫Γ‍g(u)u2=(h,u2)+〈ρ,u2〉.
Using (19), we can deduce that
(25)|u|p≥|u|2-C,|u|q≥|u|2-C,
for any p,q≥2. Therefore,
(26)∫Ω‍f(u)u2+∫Γ‍g(u)u2≥∫Ω‍|u2|p+∫Γ‍|u2|q-C(m(Ω),m(Γ)).
Thus, we know
(27)ddt(∥u2∥Ω2+∥u2∥Γ2)+μn(∫Ω‍|u2|2+∫Γ‍|u|2)≤C(m(Ω),m(Γ),|h|Ω,|ρ|Γ).
Applying the Gronwall-inequality, we have
(28)∥u2(t)∥Ω2+∥u2(t)∥Γ2≤e-μnt+Cμn.
Obviously, the enhanced flattening property holds true. By Theorem 6, we obtain the exponential attractor in L2(Ω)×L2(Γ).


## 3.2.(L2(Ω)×L2(Γ),Lr(Ω)×Lr(Γ))-Exponential Attractor

Ifr=min{p,q}, we easily showed asymptotic a prior estimate of the solution of (1) in Lr(Ω)×Lr(Γ) in order to obtain global and uniform attractors in Lp(Ω)×Lp(Γ), where r(≥2) is an integer (resp., see [8, 12]).Theorem 13.
Under the hypothesis of Theorem11, then for any ε>0 and any bounded subset B⊂L2(Ω)×L2(Γ), there exist two positive constants T=T(B,ε) and M=M(ε) such that
(29)∫Ω(|u|≥M)‍|u|r+∫Γ(|u|≥M)‍|u|r≤Cε,∀t≥T,∫Γ(|u|≥M)‍|u|r≤Cε,∀t≥T(u0,γ(u0))∈B,
where the constant C is independent of ε and B.After Theorem13, we obtain the subsequent result.Theorem 14.
Under the assumption of Theorem11, then the semigroup {S(t)}t≥0 generated by Problem (1) with initial data u0∈L2(Ω) and γ(u0)∈L2(Γ) has a (L2(Ω)×L2(Γ),Lr(Ω)×Lr(Γ))-exponential attractor ℳr, that is, ℳr is compact, invariant in Lr(Ω)×Lr(Γ), and attracts every bounded in L2(Ω)×L2(Γ) in the topology of Lr(Ω)×Lr(Γ).Proof.
By Theorems11 and 12, it is easily verified that {S(t)}t≥0 has an exponential attractor in Lr(Ω)×Lr(Γ) where we apply Theorem 9.Ifr>min{p,q}, the reasoning process mentioned earlier is not available. We note the following result in [13] after authors obtained regularity of global attractor.Lemma 15 (see [13, Corollary 3.1]).
Under the assumptions of Theorem11, the semigroup {S(t)}t≥0 has a compact global attractor 𝒜 in (H1(Ω)∩Lp(Ω))×Lq(Γ).By Theorem7 and Lemma 15, we easily obtain the following.Theorem 16.
Under the hypothesis of Theorem11, then for any ε>0 and any bounded subset B⊂L2(Ω)×L2(Γ), there exist two positive constants T=T(B,ε) and M=M(ε) such that
(30)∫Ω(|u|≥M)‍|u|p+∫Γ(|u|≥M)‍|u|q≤Cε,∀t≥T,∫Γ(|u|≥M)‍|u|q≤Cε,∀t≥T(u0,γ(u0))∈B,
where the constant C is independent of ε and B.Applying Theorem9 to Theorems 11 and 16, we have the following.Corollary 17.
Under the assumption of Theorem12, the semigroup {S(t)}t≥0 generated by Problem (1) with initial data u0∈L2(Ω) and γ(u0)∈L2(Γ) has a (L2(Ω)×L2(Γ),Lp(Ω)×Lq(Γ))-exponential attractor ℳpq.Remark 18.
In fact, for any integerr≤max{p,q}, we can obtain Theorem 16 and Corollary 17 by Lemma 15. In other words, if r≤min{p,q}, we can obtain Theorem 16 and Corollary 17 by Lemma 15 and need not use asymptotic a prior estimate. Here, we point this result obtained by the different procedure.On account of Theorem6 and Corollary 17, we have the following.Corollary 19.
There exists somem such that
(31)∥(I-Pm)u∥Lp(Ω)+∥(I-Pm)u∥Lq(Γ)≤ke-lt+k(m).Now, we give some a prior estimates aboutut.Lemma 20 (see [8, 12]).
Under the assumption of Theorem11, for any bounded subset B⊂L2(Ω)×L2(Γ), there exists a positive constant T1 which depends only on the L2(Ω)×L2(Γ)-norm of B such that
(32)∫Ω‍|ut(s)|2dx+∫Γ‍|ut(s)|2dx≤R1∀s≥T1,(u0,0)∈B,
where R1 is a positive constant which depends on M.We can easily obtain the following theorem.Theorem 21.
Under the assumption of Theorem11, the semigroup {S(t)}t≥0 generated by Problem (1) with initial data u0∈L2(Ω) and γ(u0)∈L2(Γ) satisfies the enhanced flattering in the space H1(Ω)×Lq(Γ).Proof.
We denoteu=u1+u2, where u1=Pnu,  u2=(I-Pn)u, and Pn has been introduced in Section 3.
Multiplying Problem (1) by u2, we can get
(33)12ddt∫Ω‍|u2|2+12ddt∫Ω‍|u2|2+∫Ω‍|∇u2|2+∫Ω‍f(u)u2+∫Γ‍g(u)u2=(h,u2)+〈ρ,u2〉.
By (25) and (26), we can deduce
(34)∫Ω‍|∇u2|2≤(|ut2|Ω+|h|Ω)|u2|Ω+(|ut2|Γ+|ρ|Γ)|u2|Γ+∥u∥Ωp-1∥u2∥Lp(Ω)+∥u∥Γq-1∥u2∥Lq(Γ).
By Theorems 11 and 12, Corollary 19, and Lemma 20, we know that S(t)u0 satisfies the enhanced flattering property in H1(Ω). On account of Theorem 6 and Corollary 19, we can deduce that Theorem 21 is valid.It is immediate by Theorem6, Corollary 19, and Theorem 21.Corollary 22.
Under the assumption of Theorem12, the semigroup {S(t)}t≥0 generated by Problem (1) with initial data (u0,γ(u0))∈L2(Ω)×L2(Γ) has a (L2(Ω)×L2(Γ),(H1(Ω)∩Lp(Ω))×Lq(Γ))-exponential attractor ℳ.



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*Source: 389863-2013-03-10.xml*