On Asymptotic Behavior for Reaction Diffusion Equation with Small Time Delay

and Applied Analysis 3 Definition 2.1. A function u x, t is called a weak solution of 2.1 if and only if i u ∈ L2 0, T ;H1 0 Ω , with u′ ∈ L2 0, T ;H−1 Ω , ii u| −τ,0 u0 ∈ L2 Ω , iii ∫T 0 〈ut, φ〉 Du,Dφ dt ∫T 0 f u t , u t − τ , φ dt, for each φ ∈ L2 0, T ;H1 0 Ω . Here 〈, 〉 and , denote the pair of H−1 Ω and H1 0 Ω , the inner product in L2 Ω , respectively. Next we will give two very important lemmas many times used in the proof of two theorems. Lemma 2.2. If {un} is bounded in L2 −τ, T ;H1 0 Ω ∩L∞ −τ, T ;L2 Ω , then {f un t , un t−τ } is bounded in L2 0, T ;L2 Ω . Proof. Let a N − 2 /N ∈ 0, 1 , and because ρ 1 2/N, 2ρ a 2N/ N − 2 2 1 − a . Before testing the boundedness of ‖f‖L2 0,T ;L2 Ω , we firstly estimate ‖un‖L2ρ Ω ‖un‖ L2ρ Ω ∫ Ω |un| 2N/ N−2 · |un| 1−a dx ≤ [∫ Ω |un| N−2 dx ]a[∫ Ω |un|dx ]1−a ‖un‖ N−2 L2N/ N−2 Ω · ‖un‖ 2 1−a L2 Ω ≤ C1‖un‖ N−2 H1 0 Ω C1‖un‖ 2 H1 0 Ω . 2.4 Here we utilize the Hölder inequality, the fact ofH1 0 Ω ⊂ L2N/ N−2 continuously and {un} is bounded in L∞ −τ, T ;L2 Ω . So ∫T 0 ∫ Ω |un x, t |dx dt ≤ C1‖un x, t ‖2L2 −τ,T ;H1 0 Ω , ∫T 0 ∫ Ω |un x, t − τ |dx dt t−τ s ∫T−τ −τ ∫ Ω |un x, s |dx ds ≤ ∫T −τ ∫ Ω |un x, t |dx dt ≤ C1‖un x, t ‖2L2 −τ,T ;H1 0 Ω . 2.5 In view of 2.2 , we can easily see ∣∣f un t , un t − τ ∣∣2 ≤ C [ 1 |un t | |un t − τ | ] . 2.6 Integrating the above inequality with t and x, we complete the proof. Remark 2.3. If {un} is bounded in L∞ −τ, T ;H1 0 Ω , we can also get the same conclusion. The underlying lemma is the famous Aubin-Lions lemma. We only give the statement of the lemma. 4 Abstract and Applied Analysis Lemma 2.4. Let X0, X, and X1 be three Banach spaces with X0 ⊆ X ⊆ X1. Suppose that X0 is compactly embedded in X and X is continuously embedded in X1. Suppose also that X0 and X1 are reflexive spaces. For 1 < p, q < ∞, let W {u ∈ L 0, T ;X0 |u′ ∈ L 0, T ;X1 }. Then the embedding of W into L 0, T ;X is also compact. Finally we give the definition of equilibrium solution of 1.2 and omega limit setω u , where u x, t is a bounded solution of 1.1 . Selecting H1 0 Ω as our phase space, we denote by ω u the limit set ω u { v | there exists tn −→ ∞ such that ‖u ·, tn − v‖H1 0 Ω −→ 0 } . 2.7 As usual, an equilibrium solution of 1.2 is defined as a solution which does not depend on t; the equilibrium states are thus the functions u ∈ H1 0 Ω ∩ H2 Ω satisfying the elliptic boundary value problem −Δu f u, u in Ω, u 0 on Ω 2.8 in the weak sense. Let each equilibrium be isolated and let u ·, t be the bounded complete solution of 1.2 . Then we have lim t→−∞ u ·, t E1, lim t→ ∞ u ·, t E2 2.9 for some equilibrium E1 and E2 with V E1 > V E2 , where V is the Lyapunov function 1.3 . A complete solution of 1.2 means a solution u ·, t defined on −∞, ∞ . Now we will introduce our main results. 3. Main Results In this section, we will prove two theorems. One is the existence of global solution. The other is our core, Theorem 3.2. Theorem 3.1. For given τ > 0, u0 ∈ L2 Ω , problem 2.1 has a global weak solution. Proof. We will use classical Galerkin’s method to build a weak solution of 2.1 . Consider the approximate solution um t of the form um t m ∑ k 1 umk t ωk, 3.1 Abstract and Applied Analysis 5 where {ωk}k 1 is an orthogonal basis ofH1 0 Ω and {ωk}k 1 is an orthonormal basis of L2 Ω . We get um from solving the following ODES: ( um,ωk ) Dum,Dωk ( f um t , um t − τ , ωk ) 0 < t ≤ T, k 1, 2, . . . m , umk t u0, ωk −τ ≤ t ≤ 0, k 1, 2, . . . m . 3.2and Applied Analysis 5 where {ωk}k 1 is an orthogonal basis ofH1 0 Ω and {ωk}k 1 is an orthonormal basis of L2 Ω . We get um from solving the following ODES: ( um,ωk ) Dum,Dωk ( f um t , um t − τ , ωk ) 0 < t ≤ T, k 1, 2, . . . m , umk t u0, ωk −τ ≤ t ≤ 0, k 1, 2, . . . m . 3.2 According to standard existence theory of ODES, we can obtain the local existence of um. Next we will establish some priori estimates for um. Multiplying 2.1 by um and integrating over Ω, we have 1 2 d dt ‖um‖L2 Ω ‖um‖H1 0 Ω ∫ Ω f um t , um t − τ um dx. 3.3 Because of 2.3 and the Cauchy inequality, we can get d dt ‖um‖L2 Ω 2‖um‖H1 0 Ω ≤ C ′ 1‖um‖L2 Ω C′ 2. 3.4 Getting rid of the term 2‖um‖H1 0 Ω , from the differential form of Gronwall’s inequality, we yield the estimate max 0≤t≤T ‖um‖L2 Ω ≤ C1‖u0‖L2 Ω C2. 3.5 Returning once more to inequality 3.4 , we integrate from 0 to T and employ the inequality above to find ‖um‖2L2 0,T ;H1 0 Ω ≤ C1‖u0‖ 2 L2 Ω C2. 3.6 Multiplying 2.1 by um and then integrating over Ω, we have ∥u′m ∥∥2 L2 Ω ∫


Introduction
With delay systems appearing frequently in science, engineering, physics, biology, economics, and so forth, many authors have recently devoted their interests to the effect of small delays on the dynamics of some system.This problem is relatively well understood for linear systems, including both finite-dimensional and infinite-dimensional situations, see 1-5 .However, for nonlinear systems, the problem is much more complicated, but there are some very nice results in 6-10 .
In this paper, we consider the following scalar reaction-diffusion equation with a time delay It is proved in 11-13 that for such diffusion equation without delay, Abstract and Applied Analysis subject to homogeneous boundary conditions, all globally defined bounded solutions must approach the set of equilibria as t tends to infinity.This depends heavily on the fact that 1.2 is a gradient system with the Lyapunov function where F is a primitive of f.It is well known that solutions of 1.1 will typically oscillate in t as t → ∞ if the delay is not sufficiently small.However, we will point out such interesting result that oscillations do not happen for sufficiently small delay.Specifically we obtain the conclusion that for given R, ε > 0 there exists a sufficiently small τ > 0 such that any solution of 1.1 satisfying lim sup t → ∞ u x, t H 1 0 Ω ≤ R will ultimately enter and stay in the ε− neighborhood of some equilibrium.
As a matter of fact, for the finite-dimensional situation, in 6 Li and Wang considered the general nonlinear gradient system with multiple small time delays Making use of the Morse structure of invariant sets of gradient systems, he obtained a similar result.Following this idea, we investigate 1.1 in the infinite-dimensional situation.The difference between the two situations is very great.For example, under the finite-dimensional situation there must exist convergent subsequence for any bounded sequence.This is not correct in the infinite-dimensional situation.We only have weak compactness.In other words, bounded sequences in a reflexive Banach space are weakly precompact.In order to overcome this difficulty, we apply the famous Aubin-Lions lemma 14 .

Preliminaries
In this paper, we assume Ω to be an open, bounded subset of R N and τ to be a positive parameter the delay .Consider the following scalar delayed initial boundary value problem: where the nonlinear f : R 2 → R is assumed to be continuous and to satisfy

2.3
Here C, C 1 , and C 2 are all constants, ρ 1 2/N.Firstly we will give the definition of weak solution for 2.1 .
for each ϕ ∈ L 2 0, T; H 1 0 Ω .Here , and , denote the pair of H −1 Ω and H 1 0 Ω , the inner product in L 2 Ω , respectively.Next we will give two very important lemmas many times used in the proof of two theorems.

2.4
Here we utilize the H ölder inequality, the fact of

2.5
In view of 2.2 , we can easily see Integrating the above inequality with t and x, we complete the proof.
Remark 2.3.If {u n } is bounded in L ∞ −τ, T; H 1 0 Ω , we can also get the same conclusion.The underlying lemma is the famous Aubin-Lions lemma.We only give the statement of the lemma.
Lemma 2.4.Let X 0 , X, and X 1 be three Banach spaces with X 0 ⊆ X ⊆ X 1 .Suppose that X 0 is compactly embedded in X and X is continuously embedded in X 1 .Suppose also that X 0 and X 1 are reflexive spaces.For 1 < p, q < ∞, let W {u ∈ L p 0, T; X 0 |u ∈ L q 0, T ; X 1 }.Then the embedding of W into L p 0, T; X is also compact.
Finally we give the definition of equilibrium solution of 1.2 and omega limit set ω u , where u x, t is a bounded solution of 1.1 .Selecting H 1 0 Ω as our phase space, we denote by ω u the limit set As usual, an equilibrium solution of 1.2 is defined as a solution which does not depend on t; the equilibrium states are thus the functions u ∈ H 1 0 Ω ∩ H 2 Ω satisfying the elliptic boundary value problem in the weak sense.
Let each equilibrium be isolated and let u •, t be the bounded complete solution of 1.2 .Then we have lim for some equilibrium E 1 and E 2 with V E 1 > V E 2 , where V is the Lyapunov function 1.3 .A complete solution of 1.2 means a solution u •, t defined on −∞, ∞ .Now we will introduce our main results.

Main Results
In this section, we will prove two theorems.One is the existence of global solution.The other is our core, Theorem 3.2.
Proof.We will use classical Galerkin's method to build a weak solution of 2.1 .Consider the approximate solution u m t of the form We get u m from solving the following ODES: According to standard existence theory of ODES, we can obtain the local existence of u m .Next we will establish some priori estimates for u m .Multiplying 2.1 by u m and integrating over Ω, we have 1 2 Because of 2.3 and the Cauchy inequality, we can get Getting rid of the term 2 u m 2 H 1 0 Ω , from the differential form of Gronwall's inequality, we yield the estimate 3.5 Returning once more to inequality 3.4 , we integrate from 0 to T and employ the inequality above to find Multiplying 2.1 by u m and then integrating over Ω, we have Using the Cauchy inequality and Lemma 2.2, we get

3.8
Again from the differential form of Gronwall's inequality, we integrate from 0 to T

3.10
According to estimates 3.6 , 3.10 , Lemma 2.2, and weak compactness, we see that

3.11
Here a subsequence of Next fix an integer N 0 and choose a function v ∈ C 1 0, T ; H 1 0 Ω having the form where {d k } N 0 k 1 are given smooth functions.Choosing m ≥ N 0 and multiplying 3.2 by d k t sum k 1, 2, . . ., N 0 , and then integrating with respect to t, we can find Recalling 3.11 and 3.12 and passing to weak limits, we get Because functions of the form v t are dense in L 2 0, T; H 1 0 Ω , so the above equality holds for all functions v ∈ L 2 0, T; H 1 0 Ω .Lastly we will show u| −τ,0 u 0 ∈ L 2 Ω .Notice that for each v ∈ C 1 0, T ; H 1 0 Ω with v T 0 we get the following from 3.15 : Similarly, from 3.14 , we deduce 3.17 In view of 3.2 , u m 0 → u 0 in L 2 Ω ; once again employ 3.11 and 3.12 to find

3.18
As v 0 is arbitrary, so we get the result u 0 u 0 .Since for t ∈ −τ, 0 , u m t → u 0 in L 2 Ω , we can obtain the result.As for T being arbitrary, we see the global existence of 2.1 .Theorem 3.2.Assume that each equilibrium of 1.2 is isolated.Let R, ε > 0 be given arbitrarily.Then there exists a sufficiently small τ > 0 such that any solution of 1.1 with lim sup t → ∞ u •, t H 1 0 Ω ≤ R will eventually enter and stay in the ε− neighborhood of some equilibrium.
Proof.Here we select H 1 0 Ω as our phase space.For simplicity, we will verify the correctness of the conclusion for such bounded solutions u x, t of 1.1 as u •, t H 1 0 Ω ≤ R for all t ∈ 0, ∞ .That is to say they are in B R .
Assume there are n equilibria of 1.2 {E 1 , . . ., E n }, ordered by where V is the Lyapunov function 1.3 .We will follow two steps to prove our result.
Step 1.We firstly verify that for any δ > 0, there exists a sufficiently small τ > 0 such that for any solution u x, t of 1.1 in B R .In order to prove 3.19 , we proceed by contradiction, which is used repeatedly in the following proof.Assume that there was a decreasing sequence τ k → 0 and a corresponding solution sequence u k of 1.1 in B R satisfying for all 1 ≤ j ≤ n and k ∈ N. According to the definition of ω u , for each k we can take a t k > 0 such that for

3.21
Let u k t u k t t k for t ≥ 0. It is easy to see u k is the weak solution of

3.22
Next we will show there is a strong convergent subsequence of { u k } ∞ k 1 in L 2 t, t 1; H 1 0 Ω for t ≥ 0. Still denoting u k , we can also prove the limit u is in fact the weak solution of 1.2 .From the elliptic equation regularity theorem, we can multiply 3.22 by −Δ u k and integrate over Ω

3.23
Because of the remark in Section 2, we can get

3.24
Integrating from t to t 1, from the boundedness of u k •, t 2

3.25
So we prove that u is the weak solution of 1.2 .Considering 3.21 , we have the following estimate for u:

3.26
From the above inequality, we can surely know lim t → ∞ u − E j H 1 0 Ω / 0 for all 1 ≤ j ≤ n.However, because 1.2 is a gradient system, this contradicts the fact that lim t → ∞ u •, t E j for some E j .We obtain the correctness of 3.19 .
Step 2. We will complete the proof of the theorem that if τ is sufficiently small, then for any bounded solution u •, t of 1.1 there must exist a E j and sufficiently large T such that for t > T u •, t − E j H 1 0 Ω < ε.

3.27
Here we also adopt contradiction method to prove the result.If the desired conclusion was not correct, there would be a decreasing sequence τ k → 0 and a corresponding solution sequence u k of 1.1 in B R which does not satisfy 3.27 .
In view of τ k → 0, it is easy to infer that lim Without loss of generality, we can assume that for all k ≥ 1

3.29
Denote by j k the smallest j satisfying

3.30
It is easy to see that there exists a subsequence {k We will claim if j 1 < n, then there exists a δ 1 ∈ 0, ε and k * 1 such that for

3.31
Indeed, if the fact did not hold, there would be a subsequence of {k 1 i } ∞ i 1 for simplicity still denoted by {k According to the definition of j k and 3.32 , we can choose a sequence t i > 0 satisfying > ε, for t > t i , j < j 1 .

3.33
Now we define

3.35
From 3.33 and the definition of η i , it is clear to see

3.37
Obviously v i •, t is the weak solution of

3.38
Following the method above, we can also prove there is a strong convergent subsequence of and letting T lim sup η i − t i , the limit v defined on −T, ∞ is indeed the weak solution of 1.2 .
Next we will show that T ∞.In fact, if T < ∞, then v t can be well defined at t −T .In view of 3.32 , we see v −T E j 1 .Hence v t ≡ E j 1 for t ≥ −T .That is to say, By the definition of continuity, there exists t 1 < 0 such that Then there must be j ≥ j 1 .Otherwise for j < j 1 where we use 3.37 and the fact v i s → v s strongly in L 2 t, t 1; H 1 0 Ω .So it is impossible that for j < j 1 , lim t → ∞ v t E j .Lastly we need to verify lim t → −∞ v t E j 1 .Considering 3.36 , for any ε > 0 sufficiently small such that

3.46
Because v i → v strongly in L 2 − η i − t i , − η i − t i ε ; H 1 0 Ω , we easily get the result.In a word we conclude that lim

3.47
This obviously contradicts 2.9 .So we get the correctness of 3.31 .According the definition of j k 1 i , we can conclude that d E j , ω u k 1 i ≥ δ 1 for all k 1 i > k * 1 , 1 ≤ j ≤ j 1 .

3.48
For convenience we may assume that 3.48 holds for all k 1 i .Fix a 0 < δ 2 < δ 1 , and denote by j k

3.51
Repeating the same argument again and again, we finally get sequences Definition 2.1.A function u x, t is called a weak solution of 2.1 if and only if Thanks to 3.11 and Lemma 1.3 in 14 , one has t 1; H 2 Ω .Multiplying 3.22 by ∂ t u k and utilizing the same method above, we can also conclude that {∂ t u k } is bounded in L 2 t, t 1; L 2 Ω .Applying the Aubin-Lions lemma, we can conclude that there is a strong convergent subsequence From 3.48 we know j k1i > j 1 for all k 1 i .Similarly there are a subsequence{k 2 i } ∞ i 1 of {k 1 i } ∞ i 1and a j 2 ∈ j 1 , n such that j k 2 i j 2 for all k 2 i .Following the same process above, we can prove that if j 2 < n, then there exists a δ 2 ∈ 0, δ 2 and k * 2 > k * 1 such that for k 2 i > k * * 2 , 1 ≤ j ≤ j 2 .