# Regularity of Global Attractor for the Reaction-Diffusion Equation

Authors: Hong Luo
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

By using an iteration procedure, regularity estimates for
the linear semigroups, and a classical existence theorem of global attractor, we prove
that the reaction-diffusion equation possesses a global attractor in Sobolev spaceHk for all k>0, which attracts any bounded subset of Hk(Ω) in the Hk-norm.


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## Body

## 1. Introduction

This paper is concerned with the following initial-boundary problem of reaction-diffusion systems involving an unknown functionu=u(x,t):
(1.1)∂u∂t=aΔu-g(u)inΩ×(0,∞),u=0,in∂Ω×(0,∞),u(x,0)=φ,inΩ,
where a>0 is a given constant. Δ is the Laplace operator. Ω denotes an open bounded set of Rn(n=1,2,3) with smooth boundary ∂Ω. g(s) is a polynomial on s∈R1, which is given by
(1.2)g(s)=∑k=1p‍aksk,
where p should be an odd number, that is, p=2m+1(m≥1), and
(1.3)ap>0.The reaction-diffusion systems (1.1) have been extensively studied during the last decades, one of the motivations being that such systems could account for phenomena occurring in living organisms. In 1952, Turing [1] proposed that a combination of chemical reaction and diffusion produces spatial patterns of chemical concentration, under certain conditions. Such patterns are of interest because they give a possible explanation for the development of pattern and form in developmental biology [2–4] and experimental chemical systems [5]. Schneider et al. [6–9] have studied existence of periodic travelling wave solutions and positive periodic solution of reaction-diffusion systems. In [10–17], asymptotic behaviour of the nonlinear reaction-diffusion equation, such as global attractors, inertial manifolds, and approximate inertial manifolds, has been studied.The global asymptotical behaviors of solutions and existence of global attractors are important for the study of the dynamical properties of general nonlinear dissipative dynamical systems. So, many authors are interested in the existence of global attractors such as [12–19]. As for the reaction-diffusion equation (1.1), the existence of global solutions and global attractors in L2(Ω) has been proved by Temam [16], Marion [18, 19], and Zhong et al. [17]. For convenience, we introduce the main results as follows.Lemma 1.1.
Under the conditions (1.2) and (1.3) for φ∈H, the following three claims hold.(1)
Equation (1.1) has a unique global weak solution u∈C((0,∞);H)⋂L2((0,T);H1/2), for T>0;(2)
Equation (1.1) has a unique strong solution u∈C([0,T);H1/2)⋂L2((0,T);H1) for any T>0;(3)
Equation (1.1) has a global attractor 𝒜⊂H, which attracts any bounded set of H in the H-norm.
Here the spacesH,H1/2 and H1 are defined as follows:
(1.4)H=L2(Ω),H1/2=H01(Ω),H1=H01(Ω)⋂H2(Ω).In this paper, we shall use the regularity estimates for the linear semigroups, combining with the classical existence theorem of global attractors, to prove that the reaction-diffusion equation possesses, in anykth differentiable function spaces Hk(Ω), a global attractor, which attracts any bounded set of Hk(Ω) in Hk-norm. The basic idea is an iteration procedure, which is from recent books and papers [20–24].


## 2. Preliminaries

LetX and X1 be two Banach spaces and X1⊂X a compact and dense inclusion. Consider the abstract nonlinear evolution equation defined on X, given by(2.1)dudt=Lu+G(u),u(x,0)=u0,
where u(t) is an unknown function, L:X1→X a linear operator, and G:X1→X a nonlinear operator.A family of operatorsS(t):X→X(t≥0) is called a semigroup generated by (2.1) provided that S(t) satisfies the properties:(1)
S(t):X→X is a continuous mapping for any t≥0;(2)
S(0)=id:X→X the identity;(3)
S(t+s)=S(t)·S(s)forallt,s≥0, and the solution of (2.1) can be expressed as
(2.2)u(t,u0)=S(t)u0.Next, we introduce the concepts and definitions of invariant sets, global attractors,ω-limit sets for the semigroup S(t).Definition 2.1.
LetS(t) be a semigroup defined on X. A set Σ⊂X is called an invariant set of S(t) if S(t)Σ=Σforallt≥0. An invariant set Σ is an attractor of S(t) if Σ is compact, and there exists a neighborhood U⊂X of Σ such that for any u0∈U:
(2.3)inf⁡v∈Σ⁡‖S(t)u0-v‖X⟶0,ast⟶∞.
In this case, we say thatΣ attracts U. Especially, if Σ attracts any bounded set of X, Σ is called a global attractor of S(t).
For a setD⊂X, we define the ω-limit set of D as follows:
(2.4)ω(D)=⋂s≥0‍⋃t≥s‍S(t)D¯,
where the closure is taken in the X-norm. The following Lemma 2.2 is the classical existence theorem of global attractor by Temam [16].Lemma 2.2.
LetS(t):X→X be the semigroup generated by (2.1). Assume that the following conditions hold:(1)
S(t) has a bounded absorbing set B⊂X, that is, for any bounded set A⊂X there exists a time tA≥0 such that S(t)u0∈B for  all u0⊂A and t>tA;(2)
S(t) is uniformly compact, that is, for any bounded set U⊂X and some T>0 sufficiently large, the set ⋃t≥T‍S(t)U¯ is compact in X.
Then, that theω-limit set 𝒜=ω(B) of B is a global attractor of (2.1), and 𝒜 is connected providing B is connected.
Note that we used to assume that the linear operatorL in (2.1) is a sectorial operator which generates an analytic semigroup etL. It is known that there exists a constant λ≥0 such that L-λI generates the fractional power operators ℒα and fractional order spaces Xα for α∈R1, where ℒ=-(L-λI). Without loss of generality, we assume that L generates the fractional power operators ℒα and fractional order spaces Xα as follows:
(2.5)Lα=(-L)α:Xα⟶X,α∈R1,
where Xα=D(ℒα) is the domain of ℒα. By the semigroup theory of linear operators (Pazy [25]), we know that Xβ⊂Xα is a compact inclusion for any β>α.
Thus, Lemma2.2 can be equivalently expressed in the following Lemma 2.3 [24].Lemma 2.3.
Letu(t,u0)=S(t)u0(u0∈X,t≥0) be a solution of (2.1) and S(t) the semigroup generated by (2.1). Let Xα be the fractional order space generated by L. Assume(1)
for someα≥0, there is a bounded set B⊂Xα; for any u0∈Xα, there exists tu0>0 such that
(2.6)u(t,u0)∈B,∀t>tu0;(2)
there is aβ>α, for any bounded set U⊂Xβ there are T>0 and C>0 such that
(2.7)‖u(t,u0)‖Xβ≤C,∀t>T,u0∈U.
Then (2.1) has a global attractor 𝒜⊂Xα, which attracts any bounded set of Xα in the Xα-norm.
For sectorial operators, we also have the following properties, which can be found in [25].Lemma 2.4.
LetL:X1→X be a sectorial operator which generates an analytic semigroup T(t)=etL. If all eigenvalues λ of L satisfy Reλ<-λ0 for some real number λ0>0, then for ℒα(ℒ=-L) we have (1)
T(t):X→Xα is bounded for all α∈R1 and t>0;(2)
T(t)ℒαx=ℒαT(t)x  for  all  x∈Xα;(3)
for eacht>0,ℒαT(t):X→X is bounded, and
(2.8)‖LαT(t)‖≤Cαt-αe-δt,where someδ>0,Cα>0 is a constant only depending on α;(4)
theXα-norm can be defined by
(2.9)‖x‖Xα=‖Lαx‖X;(5)
ifℒ is symmetric, for any α,β∈R1, we have
(2.10)〈Lαu,v〉H=〈Lα-βu,Lβv〉H.


## 3. Main Results

LetH and H1 be the spaces defined as in (1.4). We define the operators L:H1→H and G:H1→H by
(3.1)Lu=aΔu,Gu=-g(u),
where g(u) is the same as one of (1.2). Thus, the reaction-diffusion equation (1.1) can be written in the abstract form (2.1). It is well known that the linear operator L:H1→H given by (3.1) is a sectorial operator and ℒ=-L. The space H1 is the same as (1.4), H1/2 is given by H1/2= closure of H1 in H1(Ω), and Hk=H2k(Ω)∩H1 for k≥1.The main result in this paper is given by the following theorem, which provides the existence of global attractors of the reaction-diffusion equation (1.1) in any kth order space Hk.Theorem 3.1.
Let the functiong be a polynomial of order p(3.2)g(u)=∑k=1p‍akuk,p=2m+1(m≥1,m∈N),
with leading coefficient
(3.3)ap>0.
Assumep=3 for n=3. Then, for any α≥0 (1.1) has a global attractor 𝒜 in Hα, and 𝒜 attracts any bounded set of Hα in the Hα-norm.Proof.
From Lemma1.1, we know that the solution of system (1.1) is a weak solution for any φ∈H. Hence, the solution u(t,φ) of system (1.1) can be written as
(3.4)u(t,φ)=etLφ+∫0t‍e(t-τ)LG(u)dτ.
By (3.1), we rewrite (3.4) as
(3.5)u(t,φ)=etLφ-∫0t‍e(t-τ)Lg(u)dτ.
Next, according to Lemma2.3, we prove Theorem 3.1 in the following six steps.
Step 1. We prove that for any bounded set U⊂H1/2 there is a constant C>0 such that the solution u(t,φ) of system (1.1) is uniformly bounded by the constant C for any φ∈U and t≥0. To do that, we firstly check that system (1.1) has a global Lyapunov function as follows: (3.6)F(u)=∫Ω‍(12|∇u|2+f(u))dx,
where(3.7)f(z)=∫0z‍g(z)dz=∑k=1p‍1k+1akzk+1.
In fact, ifu(t,·) is a weak solution of system (1.1), we have
(3.8)ddtF(u(t,φ))=〈DF(u),dudt〉H.
By (3.1) and (3.6), we get
(3.9)dudt=Lu+G(u)=-DF(u).
Hence, it follows from (3.8) and (3.9) that
(3.10)dF(u)dt=〈DF(u),-DF(u)〉H=-‖DF(u)‖H2,
which implies that (3.6) is a Lyapunov function.
Integrating (3.10) from 0 to t gives
(3.11)F(u(t,φ))=-∫0t‍‖DF(u)‖H2dt+F(φ).
Using (1.2) and (3.6), we have
(3.12)F(u)=∫Ω‍(12|∇u|2+f(u))dx=∫Ω‍(12|∇u|2+1p+1apup+1+∑k=1p-1‍1k+1akuk+1)dx≥∫Ω‍(12|∇u|2+1p+1apup+1-∑k=1p-1‍1k+1|ak||u|k+1)dx≥∫Ω‍(12|∇u|2+1p+1apup+1-∑k=1p-1‍1k+1|ak|(ε|u|p+1+ε-(k+1)/(p-k)))dx=∫Ω‍(12|∇u|2+1p+1apup+1-ε(∑k=1p-1‍1k+1|ak|)|u|p+1-∑k=1p-1‍1k+1|ak|ε-(k+1)/(p-k))dx.
Choosing ε such that ε(∑k=1p-1‍(1/(k+1))|ak|)=(1/2(p+1))ap, and noting that p is an odd number, that is, p=2m+1(m≥1), we get
(3.13)F(u)≥∫Ω‍(12|∇u|2+1p+1apup+1-12(p+1)ap|u|p+1-∑k=1p-1‍1k+1|ak|ε-(k+1)/(p-k))dx=∫Ω‍(12|∇u|2+12(p+1)ap|u|p+1-∑k=1p-1‍1k+1|ak|ε-(k+1)/(p-k))dx≥C1∫Ω‍(|∇u|2+|u|p+1)dx-C2.
Combining with (3.11) yields
(3.14)C1∫Ω‍(|∇u|2+|u|p+1)dx-C2≤-∫0t‍‖DF(u)‖H2dt+F(φ),C1∫Ω‍(|∇u|2+|u|p+1)dx+∫0t‍‖DF(u)‖H2dt≤F(φ)+C2,∫Ω‍(|∇u|2+|u|p+1)dx≤C,∀t≥0,φ∈U,
which implies
(3.15)‖u(t,φ)‖H1/2≤C,∀t≥0,φ∈U⊂H1/2.
where C1,C2, and C are positive constants. C only depends on φ.
Step 2. We prove that for any bounded set U⊂Hα(1/2≤α<1) there exists C>0 such that
(3.16)‖u(t,φ)‖Hα≤C,∀t≥0,φ∈U,α<1.
By H1/2↪L2p(Ω), we have
(3.17)‖g(u)‖H2=∫Ω‍|g(u)|2dx=∫Ω‍|∑k=1p‍akuk|2dx≤∫Ω‍[ap|u|p+∑k=1p-1‍|ak|(ε|u|p+ε-k/(p-k))]2dx≤C(∫Ω‍|u|2pdx+1)≤C(‖u‖H1/22p+1),
which implies  that g:H1/2→H is bounded.
Hence, it follows from (2.9) and (3.5) that
(3.18)‖u(t,φ)‖Hα=‖etLφ+∫0t‍e(t-τ)Lg(u)dτ‖Hα≤‖φ‖Hα+∫0t‍‖Lαe(t-τ)Lg(u)‖Hdτ≤‖φ‖Hα+∫0t‍‖Lαe(t-τ)L‖‖g(u)‖Hdτ≤‖φ‖Hα+C∫0t‍‖Lαe(t-τ)L‖(‖u‖H1/22p+1)dτ≤‖φ‖Hα+C∫0t‍τβe-δtdτ≤C,∀t≥0,φ∈U⊂Hα,
where β=α(0<β<1). Hence, (3.16) holds.
Step 3. We prove that for any bounded set U⊂Hα(1≤α<3/2) there exists C>0 such that
(3.19)‖u(t,φ)‖Hα≤C,∀t≥0,φ∈U⊂Hα,α<32.
In fact, by the embedding theorems of fractional order spaces [25]:
(3.20)Hα↪C0(Ω)∩H1(Ω),α≥34,
we have
(3.21)‖g(u)‖H1/22=∫Ω‍|∇g(u)|2dx=∫Ω‍|∇(∑k=1p‍akuk)|2dx≤∫Ω‍|∑k=1p‍kaku(k-1)∇u|2dx≤∫Ω‍[pap|u|p-1+∑k=1p-1‍k|ak|(ε|u|p-1+ε-(k-1)/(p-k))]2|∇u|2dx≤C∫Ω‍(|u|2p-2+1)|∇u|2dx≤C∫Ω‍(sup⁡x∈Ω⁡|u|2p-2+1)|∇u|2dx≤C(‖u‖Hα2p-2+1)‖u‖H1/22≤C(‖u‖Hα2p-2+1)‖u‖Hα2,
which implies(3.22)g:Hα⟶H1/2isboundedforα≥34.
Therefore, it follows from (3.16) and (3.22) that
(3.23)‖g(u)‖H1/2<C,∀t≥0,φ∈U⊂Hα,34≤α<1.
Then, by using same method as that in Step 2, we get from (3.23) that
(3.24)‖u(t,φ)‖Hα=‖etLφ+∫0t‍e(t-τ)Lg(u)dτ‖Hα≤‖φ‖Hα+∫0t‍‖Lαe(t-τ)Lg(u)‖Hdτ≤‖φ‖Hα+∫0t‍‖Lαe(t-τ)L‖‖g(u)‖Hdτ≤‖φ‖Hα+C∫0t‍‖Lα-(1/2)e(t-τ)L‖‖g(u)‖H1/2dτ≤‖φ‖Hα+C∫0t‍τ-βe-δtdτ≤C,∀t≥0,φ∈U⊂Hα,
where β=α-1/2(0<β<1). Hence, (3.19) holds.
Step 4. We prove that for any bounded set U⊂Hα(3/2≤α<2) there exists C>0 such that
(3.25)‖u(t,φ)‖Hα≤C,∀t≥0,φ∈U⊂Hα,α<2.
In fact, by the embedding theorems of fractional order spaces [25]:
(3.26)H2↪W1,6↪W1,4,Hα↪C0(Ω)∩H2(Ω),α≥1,
we have(3.27)‖g(u)‖H12=∫Ω‍|Δg(u)|2dx=∫Ω‍|Δ(∑k=1p‍akuk)|2dx=∫Ω‍|∑k=2p‍[k(k-1)aku(k-2)(∇u)2+kakuk-1Δu]+a1Δu|2dx≤∫Ω‍{p(p-1)ap|u|p-2(∇u)2+pap|u|p-1Δu+∑k=2p-1‍[k(k-1)ak(ε|u|p-2+ε-(k-2)/(p-k))(∇u)2+kak(ε|u|p-1+ε-(k-1)/(p-k))Δu]+a1|Δu|}2dx≤C∫Ω‍(|u|p-2|∇u|2+|∇u|2+|u|p-1|Δu|+|Δu|)2dx≤C∫Ω‍(|u|2p-4|∇u|4+|∇u|4+|u|2p-2|Δu|2+|Δu|2)dx≤C∫Ω‍(sup⁡x∈Ω⁡|u|2p-4|∇u|4+|∇u|4+sup⁡x∈Ω⁡|u|2p-2|Δu|2+|Δu|2)dx≤C(|u|Hα2p-4‖u‖W1,44+‖u‖W1,44+|u|Hα2p-2‖u‖H22+‖u‖H22)≤C(|u|Hα2p-4‖u‖H24+‖u‖H24+|u|Hα2p-2‖u‖Hα2+‖u‖Hα2)≤C(|u|Hα2p+‖u‖Hα4+‖u‖Hα2),
which implies that
(3.28)g:Hα⟶H1isboundedforα≥1.
Therefore, it follows from (3.19) and (3.28) that
(3.29)‖g(u)‖H1<C,∀t≥0,φ∈U⊂Hα,1≤α<32.
Then, we get from (3.29) that
(3.30)‖u(t,φ)‖Hα=‖etLφ+∫0t‍e(t-τ)Lg(u)dτ‖Hα≤‖φ‖Hα+∫0t‍‖Lαe(t-τ)Lg(u)‖Hdτ≤‖φ‖Hα+∫0t‍‖Lαe(t-τ)L‖‖g(u)‖Hdτ≤‖φ‖Hα+∫0t‍‖Lα-1e(t-τ)L‖‖g(u)‖H1dτ≤‖φ‖Hα+C∫0t‍τ-βe-δtdτ≤C,∀t≥0,φ∈U⊂Hα,
where β=α-1(0<β<1). Hence, (3.25) holds.
Step 5. We prove that for any bounded set U⊂Hα(α≤0) there exists C>0 such that
(3.31)‖u(t,φ)‖Hα≤C,∀t≥0,φ∈U⊂Hα,α≥0.
In fact, by the embedding theorems of fractional order spaces [25]:
(3.32)H3↪W2,6↪W1,6,H3↪W2,6↪W2,4,Hα↪C0(Ω)∩H3(Ω),α≥32,
we have
(3.33)‖g(u)‖H3/22=∫Ω‍|∇Δg(u)|2dx=∫Ω‍|∇Δ(∑k=1p‍akuk)|2dx=∫Ω‍(∑k=2p[k(k-1)(k-2)ak|u|k-3|∇u|3+3k(k-1)ak|u|k-2|∇u||Δu|‍+kak|u|k-1|∇u||u|k-3|∇u|3]+a1|∇u||Δu|∑k=1p‍)2dx≤∫Ω‍{p(p-1)(p-2)ap|u|p-3|∇u|3+3p(p-1)ap|u|p-2|∇u||Δu|+pap|u|p-1|∇u|+∑k=3p-1[k(k-1)(k-2)ak(ε|u|p-3+ε-(k-3)/(p-k))|∇u|3‍+3k(k-1)ak(ε|u|p-2+ε-(k-2)/(p-k))|∇u||Δu|+kak(ε|u|p-1+ε-(k-1)/(p-k))|∇Δu|(ε|u|p-3+ε-(k-3)/(p-k))|∇u|3]+6a2|∇u||Δu|+2a2|u||∇u|+a1|∇Δu||∇u|3(p-2)}2dx≤C∫Ω‍(|u|p-3|∇u|3+|∇u|3+|u|p-2|∇u||Δu|+|∇u||Δu|+|u|p-1|∇Δu|+|u||∇u|+|∇Δu||u|p-3|∇u|3)2dx≤C∫Ω‍(|u|2p-6|∇u|6+|∇u|6+|u|2p-4|∇u|2|Δu|2+|∇u|2|Δu|2+|u|2p-2|∇Δu|2+|u|2|∇u|2+|∇Δu|2)dx≤C∫Ω‍(sup⁡x∈Ω⁡|u|2p-6|∇u|6+|∇u|6+sup⁡x∈Ω⁡|u|2p-4|∇u|2|Δu|2+|∇u|2|Δu|2+sup⁡x∈Ω⁡|u|2p-2|∇Δu|2+|u|2|∇u|2+|∇Δu|2)dx≤C(‖u‖Hα2p-6‖u‖W1,66+‖u‖W1,66+‖u‖Hα2p-4‖u‖W1,22‖u‖W2,22+‖u‖W2,44+‖u‖Hα2p-2‖u‖W3,22+‖u‖2‖u‖W1,22+‖u‖W3,22)≤C(‖u‖Hα2p-6‖u‖Hα6+‖u‖Hα2p-4‖u‖W2,44+‖u‖Hα2p-2|u|H32+‖u‖H36+‖u‖W2,44+∥u∥H32)≤C(‖u‖Hα2p+‖u‖Hα2p-4‖u‖Hα4+‖u‖Hα2p-2‖u‖Hα2+‖u‖Hα6+‖u‖Hα4+‖u‖Hα2)≤C(‖u‖Hα2p+‖u‖Hα6+‖u‖Hα4+‖u‖Hα2),
which implies  that
(3.34)g:Hα⟶H3/2isboundedforα≥32.
Therefore, it follows from (3.25) and (3.34) that
(3.35)‖g(u)‖H3/2<C,∀t≥0,φ∈U⊂Hα,32≤α<2.
Then, we get from (3.35) that
(3.36)‖u(t,φ)‖Hα=‖etLφ+∫0t‍e(t-τ)Lg(u)dτ‖Hα≤‖φ‖Hα+∫0t‍‖Lαe(t-τ)Lg(u)‖Hdτ≤‖φ‖Hα+C∫0t‍‖Lα-(3/2)e(t-τ)L‖‖g(u)‖H3/2dτ≤‖φ‖Hα+C∫0t‍τ-βe-δtdτ≤C,∀t≥0,φ∈U⊂Hα,
where β=α-3/2(0<β<1). Hence, (3.31) holds for 2≤α<5/2.
By doing the same procedures as Steps 1–4, we can prove that (3.31) holds for all α≥0.
Step 6. We show that for any α≥0, system (1.1) has a bounded absorbing set in Hα. We first consider the case of α=1/2.
It is well known that the reaction-diffusion equation possesses a global attractor inH space, and the global attractor of this equation consists of equilibria with their stable and unstable manifolds. Thus, each trajectory has to converge to a critical point. From (3.31) and (3.10), we deduce that for any φ∈H1/2 the solution u(t,φ) of system (1.1) converges to a critical point of F. Hence, we only need to prove the following two properties: (1)
F(u)→∞⇔∥u∥H1/2→∞;(2)
the setS={u∈H1/2∣DF(u)=0} is bounded.
Property (1) is obviously true, we now prove property (2) in the following. It is easy to check ifDF(u)=0, u is a solution of the following equation:
(3.37)aΔu-g(u)=0,u|∂Ω=0,
where g(u) is given by (1.2). Taking the scalar product of (3.37) with u, then we derive that
(3.38)∫Ω‍(|∇u|2+∑k=12m+1‍akuk+1)dx=0.
By (1.3) and (3.38), we have
(3.39)∫Ω‍(|∇u|2+a2m+1u2(m+1)-∑k=12m‍|ak||u|k+1)dx≤0.
Using Hölder inequality and the above inequality, we have(3.40)∫Ω‍(|∇u|2+u2(m+1))dx≤C,
where C>0 is a constant. Thus, property (2) is proved.
Now, we show that system (1.1) has a bounded absorbing set in Hα for any α≥1/2, that is, for any bounded set U⊂Hα there are T>0 and a constant C>0 independent of φ such that
(3.41)‖u(t,φ)‖Hα≤C,∀t≥T,φ∈U.
From the above discussion, we know that (3.41) holds as α=1/2. By (3.5), we have
(3.42)u(t,φ)=e(t-T)Lu(T,φ)-∫0t‍e(t-τ)Lg(u)dτ.
LetB⊂H1/2 be the bounded absorbing set of system (1.1), and T0>0 such that
(3.43)u(t,φ)∈B,∀t≥T0,φ∈U⊂Hα(α≥12).
It is well known that(3.44)‖etL‖≤Ce-tλ1,
where λ1>0 is the first eigenvalue of the equation:
(3.45)aΔu-g(u)=0,u|∂Ω=0.
Hence, for any given T>0 and φ∈U⊂Hα(α≥1/2), we have
(3.46)‖e(t-τ)Lu(t,φ)‖Hα=‖Lαe(t-τ)Lu(t,φ)‖H⟶0,ast⟶∞.
From (3.42)-(3.43) and Lemma 2.4 for any 1/2≤α<1, we get that
(3.47)‖u(t,φ)‖Hα≤‖e(t-T0)Lu(T0,φ)‖Hα+∫T0t‍‖Lαe(t-τ)Lg(u)‖dτ≤‖e(t-T0)Lu(T0,φ)‖Hα+C∫0t-T0‍τ-αe-λ1τdτ,
where C>0 is a constant independent of φ.
Then, we infer from (3.46) and (3.47) that (3.41) holds for all 1/2≤α<1. By the iteration method, we have that (3.41) holds for all α≥1/2.
Finally, this theorem follows from (3.31)–(3.41) and Lemma 2.3. The proof is completed.



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*Source: 917190-2012-07-12.xml*