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 ∫