Pullback attractors for nonclassical diffusion equations

MINISTRY OF EDUCATION AND TRAINING HANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY TANG QUOC BAO TANG QUOC BAO MATHEMATICS AND INFORMATICS PULLBACK ATTRACTORS FOR NONCLASSICAL DIFFUSION EQUATIONS MASTER OF SCIENCE THESIS Mathematics and Informatics 2010B Hanoi - 2011 MINISTRY OF EDUCATION AND TRAINING HANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY TANG QUOC BAO PULLBACK ATTRACTORS FOR NONCLASSICAL DIFFUSION EQUATIONS Major: Mathematics and Informatics MASTER OF SCIENCE THESIS Mathematics and Informatics Supervisor: Dr.Cung The Anh Hanoi - 2011 Contents Acknowledgements Introduction Existence and uniqueness of solutions 1.1 Setting of the problem 1.2 Existence and uniqueness of solutions 6 Existence and upper semicontinuity of pullback attractors 13 2.1 Existence of pullback attractors 13 2.2 The upper semicontinuity of pullback attractors at ε = 30 Conclusions 38 Acknowledgements I wish to express my thanks to Dr Cung The Anh for suggesting the problem and for many stimulating conversations I’m also very thankful for Faculty of Applied Mathematics and Informatics, University of Science and Technology, where the thesis was written, for financial support Last, but not least, I am grateful to my family, my friends for their encouragement, which helps me very much in completing the thesis Introduction In this thesis we consider the Cauchy problem for a class of non-autonomous nonclassical diffusion equations of the form ut − ε∆ut − ∆u + f (x, u) + λu = g(x, t), x ∈ Rn , t > τ, u|t=τ = uτ (x), x ∈ Rn , (0.1) where λ > 0, ε ∈ [0, 1], the nonlinearity f and the external force g satisfy some specified conditions later Nonclassical diffusion equations arise as models to describe physical phenonmena, such as non-Newtonian flows, soil mechanics, and heat conduction (see e.g [1, 12, 18]) In the last few years, the existence and long-time behavior of solutions to nonclassical difussion equations has attracted the attention of many mathematicians Let us review some recent results on nonclassical diffusion equations For autonomous case, that is the case g independent of time t, in [25], the author considered equation (0.1) in a bounded domain Ω with ε = 1, the time-independent external force g ∈ L2 (Ω) and the nonlinearity f satisfy the following conditions lim sup |s|→∞ f (s) < λ1 , s (0.2) where λ1 is the first eigenvalue of the operator −∆ in Ω with Dirichlet condition, and |f (s)| ≤ C(1 + |s|4 ), (0.3) and |f (s)| ≤ C(1 + |s|γ ), γ < (0.4) The growth condition (0.4) of f is usually called subcritical case Under conditions (0.2)-(0.4), the author proved the existence of a global attractor in H01 (Ω) Also consider autonomous nonclassical diffusion equations in a bounded domain, the authors in [22] assumed that the initial uτ ∈ H (Ω) ∩ H01 (Ω) and the nonlinearity f satisfies the subcritical growth condition Under those assumptions, the authors proved that for each ε ∈ [0, 1], there exists a global attractor Aε ⊂ H (Ω) ∩ H01 (Ω) for problem (0.1) Moreover, they showed that when ε → 0, Aε tends to A0 in the sense that lim distH01 (Ω) (Aε , A0 ) = as ε → 0, ε→0 where distH01 (Ω) is the Hausdorff semi-distance in H01 (Ω) For non-autonomous case, under a Sobolev growth condition of f , the authors in [2] proved that there exists a pullback attractor Aε = {Aε (t) : t ∈ R} ⊂ H01 (Ω) of (0.1) for each ε ∈ [0, 1] and lim sup distL2 (Ω) (Aε (t), A0 (t)) = as ε → 0, ε→0 t∈I for any interval I ⊂ R We refer the reader to [14, 25, 15, 24] for other results It is noticed that all existing results are devoted in bounded domains The dynamic of nonclassical diffusion equations in unbounded domains is not well understood In this thesis, we consider the existence and long-time behavior of solutions to problem (0.1) in the case of unbounded domains, the nonlinearity of polynomial type, and the unbounded external force g depending on time t The main aims of this thesis are to prove the existence of pullback attractors Aˆε = {Aε (t) : t ∈ R}, ε ∈ [0, 1], in H (Rn ) ∩ Lp (Rn ) for problem (0.1) and to show the upper semicontinuity of Aˆε at ε = The results obtained in this thesis has been accepted for publication in the journal Communications on Pure and Applied Analysis [3] The existence of a pullback attractor for problem (0.1) (on the entire space Rn ) in the case ε = has been proved recently in [21] In the case ε > 0, since equation (0.1) contains the term −ε∆ut , it is different from the classical reaction-diffusion equation essentially For example, the reaction-diffusion equation has some kind of ”regularity”, e.g., although the initial datum only belongs to a weaker topology space, the solution will belong to a stronger topology space with higher regularity However, for problem (0.1) when ε > 0, because of −∆ut , if the initial datum uτ belongs to H (Rn ) ∩ Lp (Rn ), the solution u(t) with intial datum u(τ ) = uτ is always in H (Rn ) ∩ Lp (Rn ) and has no higher regularity, which is similar to hyperbolic equations This brings some difficulty in establishing the existence of pullback attractors for nonclassical diffusion equations On the other hand, notice that the domain Rn for (1.1) is unbounded, so Sobolev embeddings are no longer compact in this case This introduces a major obstacle for examining the asymptotic compactness of solutions To try to overcome these difficulties, we combine the method of tailestimates [20] and the asymptotic a priori estimate method [11] to prove the asymptotic compactness of the corresponding process We first use these methods to prove the existence of an (H (Rn ) ∩ Lp (Rn ), Lp (Rn ))-pullback attractor Then by verifying the condition (PDC) introduced in [7], we obtain the existence of a pullback attractor Aˆε in H (Rn ) ∩ Lp (Rn ) Next, we study the continuous dependence on ε of solutions to problem (0.1) as ε → Hence using an abstract result derived recently by Carvalho et al [5] and techniques similar to ones used in [2], we prove the upper semicontinuity of pullback attractors Aˆε in L2 (Rn ) at ε = The rest of the thesis is organized as follows In Chapter 1, we prove the well-posedness of equation (0.1), that is, the existence and uniqueness of solutions In Chapter 2, the existence and upper semi-continuity of pullback attractors are investigated In conclusion, we give some ways to extend the results Chapter Existence and uniqueness of solutions The aims of this Chapter is to prove the existence and uniqueness of solutions In Section 2.1, we give some hypothesis and a definition of weak solutions The existence of weak solutions is investigated in Section 2.2 1.1 Setting of the problem This thesis is concerned with the non-autonomous nonclassical diffusion equation ut − ε∆ut − ∆u + f (x, u) + λu = g(t, x), (1.1) u|t=τ = uτ , where ε ∈ [0, 1] and λ > To study problem (1.1), we assume the following conditions: (H1) The initial datum uτ ∈ H (Rn ) ∩ Lp (Rn ) is given; (H2) The nonlinearity f satisfies f (x, u)u ≥ α|u|p − φ1 (x), (1.2) |f (x, u)| ≤ β|u|p−1 + φ2 (x), (1.3) fu (x, u) ≥ −γ, (1.4) p where p ≥ 2, φ1 ∈ L1 (Rn ), φ2 ∈ Lq (Rn ), q = p−1 , are nonnegative functions, α, β, γ are positive constants, |φ1 (x)| ≤ C0 for all x ∈ Rn For F (x, s) = s f (x, r)dr, we assume γ1 |u|p − φ3 (x) ≤ F (x, u) ≤ γ2 |u|p + φ4 (x), where γ1 , γ2 > 0, φ3 , φ4 ∈ L1 (Rn ) are nonnegative functions This implies there exist positive constants ζ1 , ζ2 , C3 , C4 such that ζ1 u p Lp (Rn ) F (x, u)dx ≤ ζ2 u − C3 ≤ Rn p Lp (Rn ) + C4 (1.5) < +∞, ∀t ∈ R, (1.6) 1,2 (R; L2 (Rn )) satisfies (H3) The external force g ∈ Wloc t eσs −∞ g(s) L2 (Rn ) and + g (s) L2 (Rn ) t eσs |g(s)|2 = 0, lim sup k→+∞ −∞ (1.7) |x|≥k where σ < min{λ, 2/ε} Definition 1.1 A function u(t, x) is called a weak solution of (1.1) on (τ, T ) iff u ∈ L∞ (τ, T ; H (Rn )) ∩ Lp (τ, T ; Lp (Rn )), ∂u ∈ L2 (τ, T ; H (Rn )), ∂t u|t=τ = uτ a.e in Rn , and T τ Rn ∂u v + ε∇ut ∇v + ∇u∇v + f (x, u)v + λuv = ∂t T gv, τ Rn for all test functions v ∈ C0∞ ([0, T ] × Rn ) 1.2 Existence and uniqueness of solutions In this Section, we prove that under assumptions (H1) - (H3), the problem (1.1) has a unique weak solution Denote by · , (·, ·) the norm and scalar product of L2 (Rn ) and C an arbitrary constant, which may be different from line to line (and even in the same line) In order to prove the existence of a weak solution, we consider the Dirichlet problem in a bounded domain   ut − ε∆ut − ∆u + f (x, u) + λu = g(x, t), x ∈ ΩR , t > τ, (1.8) u|∂ΩR = 0,   u(τ, x) = uτ,R (x), x ∈ ΩR , where ΩR is the open ball of radius R ≥ centered at 0, u0,R = u0 ψR (|x|), and ψR is a smooth function verifying  if ≤ r ≤ R − 1,  1, ψR (r) = ≤ ψR (r) ≤ 1, if R − ≤ r ≤ R,   0, if r > R By the Galerkin method, one can easily show that problem (1.8) has a unique weak solution uR for any initial datum uτ,R ∈ H01 (ΩR ) ∩ Lp (ΩR ) Theorem 1.1 Let conditions (H2) − (H3) hold Then, problem (1.1) has a unique weak solution for any uτ ∈ H (Rn ) ∩ Lp (Rn ) Proof Let urj , rj → +∞, be a sequence of solutions of (1.8) We easily conclude from Dirichlet problem that {uτ,rj } is bounded in L2 (Rn )∩H (Rn ) We have ∂t urj − ε∆(∂t urj ) − ∆urj + f (x, urj ) + λurj = g(t) (1.9) Multiplying (1.9) with urj in L2 (Rn ), we get 1d dt |urj |2 + ε Ω rj + |∇urj |2 Ω rj |∇urj |2 + Ω rj |urj |2 = f (x, urj )urj + λ Rn Ωr j (1.10) g(t)urj Ωr j large enough, we get |u|p ≤ C Ω(u≥2M ) |ut ||u − M | + ε Ω1 |∇ut ||∇u| Ω1 |g(t)||u − M | + Cmes(Ω(u ≥ 2M )) + Ω1 1/2 ≤ ut |u − M | 1/2 + ε ∇ut Ω1 |∇u| Ω1 1/2 |u − M |2 + g(t) + Cmes(Ω(u ≥ 2M )) Ω1 (2.42) By Lemmas 2.9-2.10, there exist τ0 and M such that (|u(t) − M |2 + |∇u(t)|2 ) < η and mes(Ω(u(t) ≥ 2M )) < η Ω(|u(t)|≥M ) for all τ ≤ τ0 , uτ ∈ D Hence, from (2.42), using Lemma 2.6 we have |u(t)|p ≤ Cη, Ω(u(t)≥2M ) provided τ ≤ τ0 Similarly, replace (u − M )+ by (u + M )− , we can deduce that |u(t)|p ≤ Cη Ω(u(t)≤−2M ) The proof is complete Lemma 2.13 Assume that ≤ q < ∞ and {U (t, τ )} has an (H (Rn ) ∩ Lp (Rn ), Lq (Rn )) - pullback attractor Then, for any η > 0, any t ∈ R and any D ∈ B(H (Rn ) ∩ Lp (Rn )), there exist τ0 ≤ t and m0 ∈ N such that |(I − Pm )U (t, τ )uτ |q ≤ Cη, for any τ ≤ τ0 , uτ ∈ D, m ≥ m0 , Rn where Pm is the canonical projection of Lq (Rn ) onto an m− dimensional subspace 27 Proof Let η > Assume that Aˆ = {A(t) : t ∈ R} is the (H (Rn ) ∩ Lp (Rn ), Lq (Rn )) - pullback attractor of {U (t, τ )}, then for any t ∈ R and any D ∈ B(H (Rn ) ∩ Lp (Rn )), there exists τ0 ≤ t such that U (t, τ )D ⊂ N (A(t), η, Lq ), τ ≤τ0 where N (A(t), η, Lq ) is the η - neighborhood of A(t) in Lq (Rn ) Since A(t) is compact in Lq (Rn ), there exist k ∈ N and vi ∈ Lq (Rn ), i = 1, k such that k q τ ≤τ0 U (t, τ )D ⊂ i=1 N (vi , η, L ) For each vi there is an mi such that |(I − Pmi )vi |q ≤ η Rn Taking m0 = max{m1 , , mn } Denote Qm0 = I − Pm0 , for any τ ≤ τ0 , any uτ ∈ D, there exists some vi such that |Qm0 U (t, τ )uτ |q = Rn |Qm0 U (t, τ )uτ − Qm0 vi + Qm0 vi |q Rn ≤ 2q |Qm0 U (t, τ )uτ − Qm0 vi |q + 2q Rn |Qm0 vi |q Rn ≤ 2q Cq |U (t, τ )uτ − vi |q + 2q Rn |Qm0 vi |q Rn q ≤ (Cq + 1)η, where Cq depends only on q This completes the proof Theorem 2.14 The process {U (t, τ )} corresponding to (1.1) satisfies Condition (PDC) in H (Rn ), thus {U (t, τ )} possesses an (H (Rn )∩Lp (Rn ), H (Rn )∩ Lp (Rn )) - pullback attractor Aˆε = {Aε (t) : t ∈ R} Proof Since H (Rn ) is separable, we can choose {ω1 , ω2 , } which forms an orthogonal basis in L2 (Rn ) and H (Rn ) Let Hm = span{ω1 , ω2 , , ωm }, Pm be the canonical projector on Hm and I be the indentity Then for any u ∈ H (Rn ), u has a unique decomposition: u = u1 + u2 , where u1 = Pm u ∈ Hm and u2 = (I − Pm )u Let η > be arbitrary Taking u2 as a test function in (1.1), we obtain 1d u2 + ε ∇u2 + ∇u2 + (f (x, u), u2 ) + λ u2 = (g(t), u2 ), dt (2.43) 28 thus, d u2 dt 2 + ε ∇u2 + ∇u2 + 2λ u2 ≤ f (x, u) Lq (Rn ) (2.44) u2 + g(t) u2 Lp (Rn ) Some standard computations give us eσt ( u2 (t) + ε ∇u2 (t) ) t στ ≤ e ( uτ eσs f (x, u(s)) + ε ∇uτ ) + Lq (Rn ) u2 (s) Lp (Rn ) τ t eσs g(s) u2 (s) +2 τ (2.45) From Lemma 2.13 and Theorem 2.12, there exist m1 and k > such that u2 (s) Lp (Rn ) ≤ η, u2 (s) ≤ η, ∀τ ≤ s − k, ∀m ≥ m1 (2.46) We have t eσs f (x, u(s)) Lq (Rn ) u(s) Lp (Rn ) τ τ +k eσs f (x, u(s)) = u(s) Lq (Rn ) Lp (Rn ) τ t eσs f (x, u(s)) + u(s) Lq (Rn ) (2.47) Lp (Rn ) τ +k 1/q τ +k ≤ e σs f (x, u(s)) τ q Lq (Rn ) +η e e σs f (x, u(s)) τ +k σs p Lp (Rn ) u(s) τ 1/q t 1/p τ +k q Lq (Rn ) 1/p t σs e τ +k By (1.3), we have f (x, u(s)) q Lq (Rn ) p Lp (Rn ) + φ2 q Lq (Rn ) ), eσs u(s) p Lp (Rn ) + C φ2 ≤ C( u(s) (2.48) thus τ +k σs e τ f (x, u(s)) q Lq (Rn ) τ +k ≤C τ q τ +k Lq (Rn ) e (2.49) 29 Moreover, we conclude from (2.11) that τ +k eσs u(s) τ p Lp (Rn ) τ +k στ ≤ C(e ( uτ 2 τ +k + ε ∇uτ )) + Ce s eσr g(r) +C −∞ τ τ +k ≤ C(eστ ( uτ + ε ∇uτ )) + Ceτ +k + Ck eσr g(r) −∞ → 0, as τ → −∞, t σs −∞ e |g(s)|2 since exists τ2 such that (2.50) is finite Now combining (2.47)-(2.50), we see that there t eσs f (x, u(s)) Lq (Rn ) u(s) Lp (Rn ) ≤ Cη, ∀τ ≤ τ2 , ∀m ≥ m0 , (2.51) τ where C is independent of τ and m By the same technique, we can get τ3 such that t eσs g(s) u2 (s) ≤ Cη, ∀τ ≤ τ3 , ∀m ≥ m0 (2.52) τ From (2.45), (2.51) and (2.52), we can get τ4 ≤ min{τ1 , τ2 , τ3 } such that u2 (t) + ε ∇u2 (t) ≤ Cη, ∀τ ≤ τ4 , ∀m ≥ m0 , (2.53) where C is independent of τ , m, and η This show that {U (t, τ )} satisfies condition (ii) in Definition 2.2 The condition (i) is obvious since τ ≤τ0 U (t, τ )D is bounded when τ0 → −∞ and Pm is a bounded projector for any m Hence, {U (t, τ )} satisfies Condition (PDC) in H (Rn ), thus {U (t, τ )} has an (H (Rn ) ∩ Lp (Rn ), H (Rn ) ∩ Lp (Rn )) - pullback attractor Aˆε = {Aε (t) : t ∈ R} 2.2 The upper semicontinuity of pullback attractors at ε = In the case ε = 0, under conditions (H1) − (H3), it is proved in [20] the existence of a pullback attractor Aˆ0 = {A0 (t) : t ∈ R} in the space 30 H (Rn ) for the problem (1.1) The aim of this section is to prove the upper semicontinuity of pullback attractors Aˆ at ε = in L2 (Rn ) To prove the upper semicontinuity of the pullback attractors, we assume the exponent σ in condition (H3) satisfies σ < λ, α 2λ , 2λ + ζ2 (2.54) Proposition 2.15 Under assumptions (H1)-(H3), there exists a family of pullback absorbing sets for {U (t, τ )} which is uniform with respect to ε ∈ (0, 1] Proof Multiplying (1.1) by u + ut and integrating over Rn , we get 1d (2λ + 1) u dt + ∇u 2 + (ε + 1) ∇u +2 F (x, u) Rn + ut + ε ∇ut + f (x, u)u + λ u (2.55) Rn = (g(t), u + ut ) ≤ 1 + 41 42 From (2.54), we can choose that ∇u + (1 − ) ut 1, 2 g(t) + ut + u small enough and a constant C > such + ε ∇ut f (x, u)u + (λ − ) u + Rn ≥ σ (2λ + 1) u + (ε + 1) ∇u F (x, u) − C +2 Rn Thus, we deduce from (2.55) that d y(t) + σy(t) ≤ C(1 + g(t) ), dt (2.56) where C is independent of ε and y(t) = (2λ + 1) u + (ε + 1) ∇u +2 F (u) Rn 31 By Gronwall’s lemma, we get t −σ(t−τ ) y(t) ≤ e −σt eσs g(s) y(τ ) + C + e (2.57) τ Obviously, we have y(t) ≥ ∇u (t) y(τ ) = (2λ + 1) uτ 2 + 2ζ1 u + (ε + 1) ∇uτ p Lp (Rn ) − 2C3 , and F (x, uτ ) +2 Rn ≤ C + uτ + ∇uτ + uτ pLp (Rn ) (2.58) Since u(τ ) ∈ D is bounded, combining (2.57)-(2.58), we obtain τ0 ≤ t such that ∇u(t) + u(t) p Lp (Rn ) t ≤C 1+e −σt eσs g(s) , (2.59) −∞ for all τ ≤ τ0 and uτ ∈ D Hence, {U (t, τ )} has a family of pullback absorbing B = {B(t) : t ∈ R} in H (Rn ) ∩ Lp (Rn ), where B is independent of ε ∈ [0, 1] The following lemma is the key of this section Lemma 2.16 For each t ∈ R, for each compact subset K of H (Rn ) ∩ Lp (Rn ) and each T > 0, we have √ Uε (t, τ )uτ − U0 (t, τ )uτ ≤ C ε, ∀τ ∈ [t − T, t], ∀uτ ∈ K, (2.60) where the constant C is independent of τ and uτ Proof Denote Uε (t, τ )uτ by u(t), and U0 (t, τ )uτ by v(t) Let w(t) = u(t) − v(t), we claim that w(t) satisfies wt − ε∆ut − ∆w + λw + f (x, u) − f (x, v) = (2.61) Multiplying this equation by w, then integrating over Rn we get 1d w +ε(∇ut , ∇w)+ ∇w +λ w +(f (x, u)−f (x, v), w) = (2.62) dt 32 We have (f (x, u) − f (x, v))(u − v) ≥ −γ u − v , (2.63) (f (x, u) − f (x, v), w) = Rn and −ε(∇ut , ∇w) ≤ ε ∇ut ∇w (2.64) Applying (2.63) and (2.64) in (2.62), we have d w dt ≤ 2γ w + 2ε ∇ut ∇w (2.65) Hence t w(t) e2γ(t−s) ∇ut (s) ∇w(s) ≤ 2ε τ t ≤ 2e2γT ε ∇ut (s) ∇w(s) τ (2.66) t τ Now, we estimate the term on the right-hand side of (2.66) Multiplying the first equation in (1.1) by ut , then integrating over Rn , we obtain ut +ε ∇ut + d 1d ( ∇u +λ u )+ dt dt F (x, u) ≤ Rn g(t)ut (2.67) Rn Using Cauchy’s inequality, we conclude that d dt ∇u(t) + λ u(t) F (x, u(t)) + 2ε ∇ut (t) +2 ≤ Rn g(t) (2.68) Integrating from τ to t, τ ∈ [t − T, t], we find that t ∇ut (s) 2ε ≤ ∇uτ + λ uτ τ ≤ C( ∇uτ +2 F (x, uτ ) + Rn + λ uτ 33 + uτ pLp (Rn ) ) t g(s) τ +C + t g(s) , τ (2.69) thus t ∇ut (s) τ C + ∇uτ ≤ ε C(K, T, g) ≤ , ε because of uτ ∈ K and g ∈ one get w(t) 2 + uτ uτ pLp (Rn ) + L2loc (R; L2 (Rn )) t + t−T (2.70) Now, using (2.70) in (2.66), t √ ≤ C(K, T, g) ε g(s) ∇w(s) (2.71) τ Using (2.59) and noting that τ ∈ [t − T, t] we have ∇w(t) ≤ ∇u(t) ≤ 2C e−σ(t−τ ) ( uτ + u(τ ) ≤ 2C uτ + ∇v(t) 2 + ∇uτ t p Lp (Rn ) −σt eσs g(s) + 1) + + e τ + ∇uτ + u(τ ) p Lp (Rn ) t −σt eσs g(s) +2+e τ t ≤ C(K) + g(s) τ t ≤ C(K) + g(s) t−T (2.72) Thus, t t ∇w(s) s ≤ C(K) t − τ + τ g(r) τ t ≤ C(K) T + T s−T g(r) (2.73) t−T ≤ C(K, T, g) Combining (2.71) and (2.73) we get w(t) √ ≤ C(K, T, g) ε 34 (2.74) The proof is complete Theorem 2.17 If g satisfies an addition condition t e−σ(t−s) g(s) lim sup t→−∞ < +∞, (2.75) −∞ then, for any bounded interval I ∈ R, the family of pullback attractors {Aε (.) : ε ∈ [0, 1]} is upper semicontinuous in L2 (Rn ) at for any t ∈ I, that is, (2.76) lim sup distL2 (Rn ) (Aε (t), A0 (t)) = ε→0 t∈I Proof We will verify the conditions (i) - (iii) in Theorem 2.3 First, condition (i) follows directly from Lemma 2.16 Let Bε (.) = B(r0 (.)) be the corresponding family of pullback absorbing sets of (1.1) which is uniform with ε By the definition of pullback absorbing sets, for any t ∈ R, there exists τ0 = τ0 (t) ≤ t such that Uε (t, τ )Bε (τ ) ⊂ Bε (t) = B(r0 (t)) (2.77) τ ≤τ0 By Theorem 2.2, we see that Aε (t) = Uε (t, τ )Bε (τ ) (2.78) s≤t τ ≤s From (2.77) and (2.78), we get Aε (t) ⊂ B(r0 (t)) (2.79) Now, for given t0 ∈ R, we can write Aε (t) ⊂ ε∈[0,1] t≤t0 B(r0 (t)) (2.80) t≤t0 We have t −σt t σs r0 (t) = C + e e g(s) −∞ e−σ(t−τ ) g(s) =C 1+ , −∞ (2.81) 35 thus lim sup r0 (t) < +∞, since (2.75) Hence, from (2.80) we have t→−∞ Aε (t) is bounded in L2 (Rn ) for given t0 , ε∈[0,1] t≤t0 i.e., the condition (ii) of Theorem 2.3 is satisfied We remain to show that Aε (t) is compact in L2 (Rn ) for each t ∈ R 0 0, there exist finite balls with radius less than η cover Aε (t) By the invariant of Aˆε , for any u ∈ Aε (t), and 0 0, which is independent of u, such that η |u|2 ≤ , ∀u ∈ Aε (t) (2.83) |x|≥k ε∈(0,1] Thus, there exists K such that u L2 (ΩcK ) η ≤ , ∀u ∈ Aε (t) ε∈(0,1] On the other hand, by (2.80), there is a constant C > such that u H (ΩK ) ≤ C, ∀u ∈ Aε (t) ε∈(0,1] 36 (2.84) Since H1 (ΩK ) → L2 (ΩK ) compactly, ε∈(0,1] Aε (t) is precompact in L2 (ΩK ), hence there a finite covering in L2 (ΩK ) of balls of radius less than η4 Combining this with (2.84), we get ε∈(0,1] Aε (t) is precompact in L2 (Rn ) This completes the proof 37 Conclusion The thesis is concerned with non-autonomous nonclassical diffusion equation in unbounded domains with polynomial type nonlinearity We proved the existence of a pullback attractor Aε for each ε ∈ [0, 1], moreover, we prove the family of pullback attractors {Aε }ε∈[0,1] is upper semicontinuous at ε = The method is combining a technique so-called ”tail estimates” and abstract results on upper semicontinuity of pullback attractors Up to the best of our knowledge, this is the first result on pullback attractors for nonclassical diffusion equations in unbounded domains However, the obtained results are not very satisfied because we are able to prove the existence of pullback attractors in H (Rn ) ∩ Lp (Rn ), the upper semicontinuity of family of pullback attractors is ontained only 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equations, J Math Phys 51 (2010) [25] Y Xiao, Attractors for a nonclassical diffusion equation, Acta Math Appl... hyperbolic equations This brings some difficulty in establishing the existence of pullback attractors for nonclassical diffusion equations On the other hand, notice that the domain Rn for (1.1)... on pullback attractors for nonclassical diffusion equations in unbounded domains However, the obtained results are not very satisfied because we are able to prove the existence of pullback attractors