Existence of pullback attractor for a reaction-diffusion equation in some unbounded domains with non-autonomous forcing term in H-1

EXISTENCE OF PULLBACK ATTRACTOR FOR A REACTION-DIFFUSION EQUATION IN SOME UNBOUNDED DOMAINS WITH NON-AUTONOMOUS FORCING TERM IN H−1 MAR´ IA ANGUIANO, TOM´ AS CARABALLO, & JOS´ E REAL Dpto. Ecuaciones Diferenciales y An´alisis Num´erico Universidad de Sevilla Apdo. de Correos 1160 41080-Sevilla (Spain) E-mails: [email protected], carabal[email protected], jre[email protected] To the Memory of Professor Valery S. Melnik The existence of a pullback attractor in L2(Ω) for the following non-autonomous reactiondiffusion equation      ∂u ∂t − 4u=f(u) + h(t),in Ω ×(τ, +∞), u= 0,on ∂Ω×(τ, +∞), u(x, τ) = uτ(x), x ∈Ω, (1) is proved in this paper, when the domain Ω is not necessarily bounded but satisfying the Poincar´e inequality, and h∈L2 loc(R;H−1(Ω)). The main concept used in the proof is the asymptotic compactness of the process generated by the problem. Keywords: pullback attractor, asymptotic compactness, evolution process, non-autonomous reaction-diffusion equation. Mathematics Subject Classifications (2000): 35B41, 35Q35, 35Q30, 35K90, 37L30 1. Introduction and setting of the problem Let Ω ⊂RNbe an open set, not necessarily bounded and suppose that Ω satisfies the Poincar´e inequality, i.e., there exists a constant λ1>0 such that ZΩ |u(x)|2dx ≤λ−1 1ZΩ |∇u(x)|2dx, ∀u∈H1 0(Ω) . (2) Let us consider the following problem for a nonautonomous reaction-diffusion equation with zero Dirichlet boundary condition in Ω,      ∂u ∂t − 4u=f(u) + h(t),in Ω ×(τ, +∞), u= 0,on ∂Ω×(τ, +∞), u(x, τ) = uτ(x), x ∈Ω, (3) where τ∈R,uτ∈L2(Ω), h∈L2 loc(R;H−1(Ω)) and f∈C(R) satisfies that there exist constants α1>0, α2>0, l≥0, and p > 2 such that −α1|s|p≤f(s)s≤ −α2|s|p, (4) (f(s)−f(r))(s−r)≤l(s−r)2∀r, s ∈R. (5) 1 2M. Anguiano, T. Caraballo & J. Real Using (4), it follows that |f(s)| ≤ α1|s|p−1∀s∈R. (6) The aim of this paper is to show the existence of a pullback attractor in the phase space L2(Ω) for the problem (3) in the case of open domains not necessarily bounded but satisfying the Poincar´e inequality. This, and the fact that the non-autonomous hbelongs to the space L2 loc(R;H−1(Ω)),are the main novelties of our problem. The lack of compactness of the injection H1 0(Ω) ⊂L2(Ω) (in the case of unbounded domains) implies that the standard techniques previously used, particularly the one involving the so-called flatenning property (see [Kloeden & Langa, 2007], [Li & Zhong, 2007], [Song & Wu, 2007], [Wang & Zhong, 2008], amongst others), which have been successfully used when Ω is bounded and h∈L2 loc(R;L2(Ω)), do not work in our case. Instead, we will use the asymptotic compactness already used in the case of non-autonomous 2D-Navier-Stokes (see [Caraballo et al., 2006] and [Caraballo et al., 2006b]), and which was previously used in [Rosa, 1998] for the autonomous case. We would like to emphasize that this technique seems to be the only one which allows to prove the main result of this paper (namely Theorem 4.4) concerning the existence of pullback attractor for our problem. It is also worth mentioning that our problem has received much attention over the last years in the case of a bounded domain or for a less general term h, as we will recall now. In [Caraballo et al., 2003] it is proved the existence of pullback attractor in the space L2(Ω) (and that it possesses finite Hausdorff dimension) when the domain is bounded and his unbounded but with polynomial growth, i.e kh(t)kL2(Ω) ≤k1|t|α+k2 where k1, k2and αare nonnegative constants. When Ω is bounded and h∈L2 loc(R;L2(Ω)) and is translation bounded, i.e. sup m∈RZm+1 m kh(s)kL2(Ω) ds < ∞,(7) the existence of a pullback attractor in the space H1 0(Ω) is proved in [Song & Wu, 2007], while in [Li & Zhong, 2007] the translation bounded condition (7) is weakened to kh(s)k2 L2(Ω) ≤Meα|s|, where 0 ≤α≤λ1, and λ1denotes the first eigenvalue of the Laplacian. In [Wang & Zhong, 2008], the existence of pullback attractor in H1 0(Ω) is shown for a bounded domain and for a h∈L2 loc(R;L2(Ω)) such that Zt −∞ eσs kh(s)k2 L2(Ω) +kh0(s)k2 L2(Ω)ds < +∞ for all t∈Rand certain σ≥0. For a bounded domain Ω, and a translation bounded function h∈L2 loc(R;L2(Ω)), the existence of a uniform attractor in Lp(Ω) is demonstrated in [Song & Zhong, 2008]. Finally, the reader can find similar results for several variants of our model in the references [Wang et al., 2007], [Prizzi, 2003], [Morillas & Valero, 2005], [Sun & Zhong, 2005], among others. We will provide in this paper a sufficient condition ensuring the existence of pullback attractor in L2(Ω) when the domain is not necessarily bounded and h∈L2 loc(R;H−1(Ω)). A case that has not been considered in the literature yet, as far as we know. 2. Existence and uniqueness of solution We state in this section a result on the existence and uniqueness of solution of problem (3). Instead of working directly with our equation, we will establish a general result which, in particular, can be applied to handle our problem. 2.1. An abstract result Let Hbe a separable Hilbert space with scalar product (·,·) and norm |·|. Let Vi,i= 1, ..., m, be m≥1 reflexive and separable Banach spaces such that m [ i=1 Vi⊂H, m \ i=1 Viis dense in H, and Vi, i= 1, ..., m is included in Hwith continuous injection. Pullback Attractors for reaction-diffusion equations 3 By k·kiwe denote the norm in Vi, by k·k∗ithe norm in V0 i,i= 1, ..., m and, by Vthe space V= m \ i=1 Vi, with the norm kvk= m X i=1 kvki,∀v∈V. It is easy to see that Vis a separable Banach space. We will use h·,·i to denote the duality product between V0 iand Vi, for each i= 1, ..., m. We identify Hwith its dual H0using the Riesz Theorem, but if Viis a Hilbert space, we do not identify Viwith V0 i. Let u∈H, we identify uwith Tu∈ ∩m i=1V0 isuch that hTu, vi= (u, v),∀v∈Vi,∀i= 1, ..., m. Let τ∈Rbe an initial time, and let Ai: (τ, ∞)× Vi→V0 i,i= 1, ..., m, be moperators, in general nonlinear, such that A1) For each v∈Vi, the function t∈ (τ, ∞)7−→ Ai(t, v)∈V0 iis Lebesgue measurable. A2) Each operator Aiis hemicontinuous, i.e, for all t∈(τ, ∞) and u, v, w ∈Vi, the function θ∈R7−→ hAi(t, u +θv), wi ∈ Ris continuous. Suppose that there exist 2 ≤pi<+∞,i= 1, ..., m, and there exist constants c > 0, α > 0 and λ≥0, and a nonnegative function C(t)∈L1(τ, T ), for all T > τ, such that for each i= 1, ..., m, A3) (Boundedness) kAi(t, v)k∗i≤c(1 + kvkpi−1 i), for all t∈(τ, ∞), v ∈Vi. A4) (Monotonicity) hAi(t, v)−Ai(t, w), v −wi+λ|v−w|2≥0, for all t∈(τ, ∞) and for all v, w ∈Vi. A5) (Coercivity) For each ithere exists a seminorm [·]iin Vi, such that there exists λi≥0 for which [v]i+λi|v| is another norm in Vi. Moreover, [v]i+λi|v|and k·kiare equivalent, and hAi(t, v), vi+λ|v|2+C(t)≥α[v]pi i, for all t∈(τ, ∞) and for all v∈Vi. Consider mfunctions hi(t)∈Lp0 i(τ, T ;V0 i),∀T > τ, i = 1, ..., m, (8) and the initial condition uτ∈H. (9) If we set A(t, v) = m X i=1 Ai(t, v), h(t) = m X i=1 hi(t), we can consider the following problem          u∈ m \ i=1 Lpi(τ, T ;Vi)∀T > τ, u0(t) + A(t, u(t)) = h(t),in D0(τ, ∞;V0), u(τ) = uτ. (10) The proof of the following result is similar to that of Theorem 1.4, Chapter 2 in [Lions, 1969]. Theorem 2.1. Assume A1)-A5), (8) and (9). Then, there exists a unique solution uof (10), such that u∈C([τ, ∞); H),u0∈ m X i=1 Lp0 i(τ, T ;V0 i),(11) for all T > τ. 2.2. Existence and uniqueness of solution of problem (3) We use Theorem 2.1 to show the existence and uniqueness of solution of (3). Consider m= 2, H =L2(Ω), V1=H1 0(Ω) and V2=Lp(Ω) ∩L2(Ω) with p > 2, and denote V0 1= H−1(Ω), V0 2=Lp0(Ω) + L2(Ω). Recall that |·| denotes the norm in H, by k·k1= |∇·| we will denote the norm in V1, and by k·k2= k·kLp(Ω) +|·| the norm in V2. If we set A1(t, u) = −∆u, A2(t, u) = −f(u), and h1(t) = h(t), h2(t) = 0, then, it is not difficult to apply Theorem 2.1 with p1= 2 and p2=p, and we obtain 4M. Anguiano, T. Caraballo & J. Real Theorem 2.2. Assume that f∈C(R)satisfies (4) and (5), and h∈L2 loc(R;H−1(Ω)). Then, for all τ∈R,uτ∈L2(Ω), there exists a unique solution u(t) = u(t;τ, uτ)of (3) such that u∈L2(τ, T ;H1 0(Ω)) ∩Lp(τ, T ;Lp(Ω)) ∀T > τ, d dt(u(t), v)− h∆u(t), vi=hf(u(t)), vi +hh(t), vi,in D0(τ, ∞),∀v∈H1 0(Ω) ∩Lp(Ω) , u(τ) = uτ. Moreover, u∈C([τ, ∞); L2(Ω)), and usatisfies the energy equation, 1 2 d dt|u(t)|2+|∇u(t)|2=hf(u(t)), u(t)i +hh(t), u(t)iin D0(τ, ∞).(12) 3. Preliminaries on the theory of pullback attractors Now, we will recall the main points from the theory of pullback attractors which will be needed in order to prove our objective (see [Caraballo et al., 2006] and [Caraballo et al., 2006b] for more details). Let us consider a process (also called a twoparameter semigroup) Uon a metric space X, i.e., a family {U(t, τ); −∞ < τ ≤t < +∞} of continuous mappings U(t, τ) : X→X, such that U(τ, τ)x=x, and U(t, τ) = U(t, r)U(r, τ) for all τ≤r≤t. (13) Suppose Dis a nonempty class of parameterized sets b D={D(t); t∈R} ⊂ P(X),where P(X) denotes the family of all nonempty subsets of X. Definition 3.1. The process U(·,·) is said to be pullback D-asymptotically compact if for any t∈ R, any b D∈ D,any sequence τn→ −∞,and any sequence xn∈D(τn), the sequence {U(t, τn)xn}is relatively compact (i.e. pre-compact) in X. Definition 3.2. It is said that b B∈ D is pullback D-absorbing for the process U(·,·) if for any t∈R and any b D∈ D, there exists a τ0(t, b D)≤tsuch that U(t, τ)D(τ)⊂B(t)for all τ≤τ0(t, b D). Definition 3.3. The family b A={A(t); t∈R} ⊂ P(X) is said to be a pullback D-attractor for U(·,·) if 1. A(t) is compact for all t∈R, 2. b Ais pullback D-attracting, i.e., lim τ→−∞ dist(U(t, τ)D(τ), A(t)) = 0, for all b D∈ D, and all t∈R, 3. b Ais invariant, i.e., U(t, τ)A(τ) = A(t),for −∞ < τ ≤t < +∞. We have the following result. Theorem 3.4. Suppose that the process U(·,·)is pullback D-asymptotically compact and that b B∈ D is a family of pullback D-absorbing sets for U(·,·). Then, the family b A={A(t); t∈R}⊂P(X) defined by A(t) = Λ( b B, t), t ∈R,where for each b D∈ D Λ( b D, t) = \ s≤t [ τ≤s U(t, τ)D(τ) , is a pullback D-attractor for U(·,·)which satisfies in addition that A(t) = Sb D∈D Λ( b D, t),for t∈R. Furthemore, b Ais minimal in the sense that if b C={C(t); t∈R} ⊂ P(X)is a family of closed sets such that limτ→−∞ dist(U(t, τ)B(τ), C(t)) = 0, then A(t)⊂C(t). 4. Existence of the pullback attractor Now, we can prove our main aim in this paper. First, we need a continuity result which is established in the next subsection. 4.1. Weak Continuity Let f∈C(R) be a function, and suppose that f satisfies (4) and (5), and h∈L2 loc(R;H−1(Ω)). Thanks to Theorem 2.2, we can define a process {U(t, τ), τ≤t}in L2(Ω), as U(t, τ)uτ=u(t;τ, uτ)∀uτ∈L2(Ω) ,∀τ≤t. (14) From the uniqueness of solution to problem (3), it follows that (14) defines a process in L2(Ω). In Pullback Attractors for reaction-diffusion equations 5 addition, it can be proved that the process defined by (14) is continuous in L2(Ω). Moreover, Uis weakly continuous, and more exactly the following result holds true. We will denote by “*” the weak convergence in the corresponding indicated space, while “→” will denote the strong convergence, as usual. Proposition 4.1. Let {uτn} ⊂ L2(Ω) be a sequence converging weakly in L2(Ω) to an element uτ∈L2(Ω). Then, for all T > τ, it follows U(t, τ)uτn* U (t, τ)uτin L2(Ω) ∀t≥τ,(15) U(·, τ)uτn* U (·, τ)uτin L2(τ, T;H1 0(Ω)),(16) U(·, τ)uτn* U (·, τ)uτin Lp(τ, T;Lp(Ω)),(17) f(U(·, τ)uτn)* f (U(·, τ)uτ)in Lp0(τ, T;Lp0(Ω)). (18) If Ωis a bounded set, then U(·, τ)uτn−→ U(·, τ)uτin L2(τ, T ;L2(Ω)).(19) Proof. From the assumptions, we deduce that there exists a positive constant Csuch that |uτn| ≤ C∀n≥1.(20) Fix τ∈R, and set un(t) = U(t, τ)uτn, u(t) = U(t, τ)uτ. (21) Using (4), it follows d dt |un(t)|2+ 2 |∇un(t)|2≤ −2α2kun(t)kp Lp(Ω) +2 hh(t), uni. Integrating between τand t, we obtain |un(t)|2+ 2 Zt τ |∇un(s)|2ds + 2α2Zt τ kun(s)kp Lp(Ω) ds ≤ |uτn|2+ 2 Zt τ hh(s), unids ∀t≥τ. (22) On the other hand, Zt τ hh(s), unids ≤Zt τ khkH−1(Ω) |∇un|ds ≤1 2ZT τ khk2 H−1(Ω) ds +1 2Zt τ |∇un|2ds, which, jointly with (20) and (22) imply |un(t)|2+Zt τ |∇un(s)|2ds + 2α2Zt τ kun(s)kp Lp(Ω) ds ≤C2+Zt τ khk2 H−1(Ω) ds ∀t > τ. We deduce that {un}is bounded in L2(τ, T ;H1 0(Ω))∩Lp(τ, T ;Lp(Ω))∩C([τ, T ]; L2(Ω)), (23) for all T > τ. Fix T > τ. In particular, {un(T)}is bounded in L2(Ω). On the other hand, from (6) we have kf(un(t))kp0 Lp0(Ω) ≤αp0 1kun(t)kp Lp(Ω) , whence f(un) is bounded in Lp0(τ, T;Lp0(Ω)). Then, there exists a subsequence {uµ}⊂{un} such that uµ∗ * v weak-star in L∞(τ, T ;L2(Ω)), uµ* v in Lp(τ, T ;Lp(Ω)), (24) uµ(T)* ξ in L2(Ω) , (25) uµ* v in L2(τ, T ;H1 0(Ω)), (26) and f(uµ)* χ in Lp0(τ, T ;Lp0(Ω)). (27) Now (26) imply that ∆uµ*∆vin L2(τ, T ;H−1(Ω)). From (26), (27), and thanks to the equation u0 µ(t) = ∆uµ(t) + f(uµ(t))+h(t), (28) it is a standard matter to prove that we can pick an element in the equivalence class of vsatisfying v(t) = uτ+Zt τ (∆v(s) + χ(s) + h(s))ds, (29) for all t∈[τ, T ]. We are now in position to show that ξ=v(T) and χ(t) = f(v(t)). Let w∈H1 0(Ω) ∩Lp(Ω). Integrating (28) between τand T, we obtain (uµ(T), w) = uτµ, w +ZT τ h∆uµ(s) + f(uµ(s))+h(s), wids, 6M. Anguiano, T. Caraballo & J. Real and thus (ξ, w) = (uτ, w) + ZT τ h∆v(s) + χ(s) + h(s), wids, when µ−→ +∞. By density and using (29), it follows v(T) = ξ. (30) To prove that χ(t) = f(v(t)),we argue similarly to [Rosa, 1998]. Integrating the equality d ds (uµ(s), w) = −(∇uµ(s),∇w) +hf(uµ(s)), wi+hh(s), wi, between tand t+a, with a∈(0, T −τ), t∈ (τ, T −a),and using the H¨older inequality, we obtain (uµ(t+a)−uµ(t), w) ≤Zt+a t |∇uµ(s)| |∇w|ds +Zt+a t kf(uµ(s)kLp0(Ω) kwkLp(Ω) ds +Zt+a t kh(s)kH−1(Ω) |∇w|ds ≤ |∇w|a1/2kuµkL2(τ,T ;H1 0(Ω)) +kwkLp(Ω) a1/p kf(uµ)kLp0(τ,T ;Lp0(Ω)) +|∇w|a1/2khkL2(τ,T ;H−1(Ω)) , and thanks to (23), we deduce that there exists a constant C(1) such that (uµ(t+a)−uµ(t), w) ≤C(1)(a1/2+a1/p)(|∇w|+kwkLp(Ω)). If we take in the last inequality w=uµ(t+a)− uµ(t)∈H1 0(Ω) ∩Lp(Ω) a.e. t∈(τ, T −a),we obtain |uµ(t+a)−uµ(t)|2 ≤C(1)(a1/2+a1/p)|∇uµ(t+a)− ∇uµ(t)| +C(1)(a1/2+a1/p)kuµ(t+a)−uµ(t)kLp(Ω) , a.e. t∈(τ, T −a). Integrating between τand T−a, ZT−a τ |uµ(t+a)−uµ(t)|2dt ≤C(1)(a1/2+a1/p)ZT−a τ |∇uµ(t+a)|dt +C(1)(a1/2+a1/p)ZT−a τ |∇uµ(t)|dt +C(1)(a1/2+a1/p)ZT−a τ kuµ(t+a)kLp(Ω) dt +C(1)(a1/2+a1/p)ZT−a τ kuµ(t)kLp(Ω) dt, it follows ZT−a τ |uµ(t+a)−uµ(t)|2dt ≤2C(1)(a1/2+a1/p)ZT τ |∇uµ(s)|ds + 2C(1)(a1/2+a1/p)ZT τ kuµ(s)kLp(Ω) ds, and using the H¨older inequality, we obtain ZT−a τ |uµ(t+a)−uµ(t)|2dt ≤2C(1)(a1/2+a1/p)(T−τ)1/2kuµkL2(τ,T ;H1 0(Ω)) + 2C(1)(a1/2+a1/p)(T−τ)1/p0kuµkLp(τ,T ;Lp(Ω)) . Thanks to (23) we deduce that there exists a constant e CTsuch that ZT−a τ |uµ(t+a)−uµ(t)|2dt ≤e CT(a1/2+a1/p), for all µ, and all a∈(0, T −τ),and thus lim a→0sup µZT−a τ |uµ(t+a)−uµ(t)|2dt= 0. (31) Now, for all m∈Z,m≥1, we denote Ωm= Ω ∩x∈RN:|x|RN< m, where |·|RNdenotes the Euclidean norm in RN. Let φ∈C1([0,+∞)) be a function such that 0≤φ(s)≤1, φ(s)=1 ∀s∈[0,1], and φ(s) = 0∀s≥2. For each µand m≥1, we define vµ,m(x, t) = φ |x|2 RN m2!uµ(x, t) . Pullback Attractors for reaction-diffusion equations 7 From (23), for all m≥1, we obtain that {vµ,m}µ≥1is bounded in L2(τ, T ;H1 0(Ω2m)) ∩ Lp(τ, T ;Lp(Ω2m)) ∩L∞(τ, T ;L2(Ω2m)). In particular, lim a→0sup µZτ+a τ |vµ,m(t)|2 L2(Ω2m)dt + +ZT T−a |vµ,m(t)|2 L2(Ω2m)dt= 0. (32) On the other hand, from (31) we deduce that for all m≥1, lim a→0sup µZT−a τ |vµ,m(t+a)−vµ,m(t)|2 L2(Ω2m)dt = 0. (33) Moreover, as Ω2mis a bounded set, then H1 0(Ω2m) is included in L2(Ω2m) with compact injection. Then, by the compactness Theorem 13.3 of [Temam, 1983] with X=L2(Ω2m), Y=H1 0(Ω2m), r= 2 and G={vµ,m}µ≥1, we obtain that{vµ,m}µ≥1 is relatively compact in L2τ, T;L2(Ω2m), and thus, taking into account that vµ,m(x, t) = uµ(x, t) for all x∈Ωm,we deduce that, in particular, for all m≥1 nuµ|Ωmoµ≥1is pre-compact in L2τ, T ;L2(Ωm). (34) It is not difficult to conclude from (34), (26) and (2), via a diagonal procedure, the existence of a subsequence {uµ µ}µ≥1⊂ {uµ}µ≥1such that uµ µ→va.e. in Ωm×(τ, T) as µ−→ ∞ ∀m≥1. Then, as fis continuous, f(uµ µ)→f(v) a.e. in Ωm×(τ, T ) , and as {f(uµ µ)}is bounded in Lp0(Ωm×(τ, T)),by Lemma 1.3, Chapter 1 in [Lions, 1969], we obtain f(uµ µ)* f(v) weakly in Lp0τ, T;Lp0(Ωm). From (27) f(uµ)* χ|Ωm×(τ,T )weakly in Lp0τ, T ;Lp0(Ωm). By the uniqueness of the weak limit, we have χ=f(v) a.e. in Ωm×(τ, T )∀m≥1, and thus, taking into account that ∞ [ m=1 Ωm= Ω, we obtain χ=f(v) a.e. in Ω ×(τ, T ) . (35) From (29) and (35), and by the uniqueness of solutions we have v(t) = u(t) for all t∈[τ, T ]. And then, if we consider (30) in (26) and (35) in (27), we have uµ* u in L2(τ, T ;H1 0(Ω)), uµ* u in Lp(τ, T ;Lp(Ω)), uµ(T)* u(T) in L2(Ω) , f(uµ)* f(u) in Lp0(τ, T ;Lp0(Ω)). Then, by a contradiction argument we deduce un* u in L2(τ, T ;H1 0(Ω)), un* u in Lp(τ, T ;Lp(Ω)), un(T)* u(T) in L2(Ω) , f(un)* f(u) in Lp0(τ, T ;Lp0(Ω)), and, as T > τ has been taken arbitrarily, the first part of the proof is finished. Now, if Ω is bounded, we deduce from (34) that uµ|Ωµ≥1is pre-compact in L2τ, T ;L2(Ω). (36) Finally, (19) follows from (36). Remark 4.2. From the proof, it is clear that for any m≥1, U(·, τ)uτn→U(·, τ)uτin L2(τ, T ;L2(Ωm)). Moreover, it is possible to prove that U(t, τ)uτn→U(t, τ)uτin L2(Ωm), for all t > τ. Remark 4.3. Notice that all the results obtained in the previous analysis hold true for a general nonempty open subset Ω ⊂RN. 8M. Anguiano, T. Caraballo & J. Real 4.2. The existence of the global pullback attractor Let Rλ1be the set of all functions r:R→(0,+∞) such that lim t→−∞ eλ1tr2(t) = 0, and denote by Dλ1the class of all families b D={D(t) : t∈R}⊂P(L2(Ω) ) such that D(t)⊂ B(0, r b D(t)), for some rb D∈ Rλ1, where B(0, r b D(t)) denotes the closed ball in L2(Ω) centered at zero with radius rb D(t). Now, we can prove the following result. Theorem 4.4. Suppose that Ωsatisfies (2), and suppose that f∈C(R)satisfies (4) and (5) with l= 0. Let h∈L2 loc(R;H−1(Ω)) such that Zt −∞ eλ1skh(s)k2 H−1(Ω) ds < +∞ ∀t∈R. Then, there exists a unique global pullback Dλ1attractor for the process U, which belongs to Dλ1, and is defined by (14). Proof. Let τ∈R, and uτ∈L2(Ω) be fixed, and denote u(t) = u(t;τ, uτ) = U(t, τ)uτ∀t≥τ. Taking into account (4) and the energy equality, d dt eλ1t|u(t)|2+ 2eλ1t|∇u(t)|2 =λ1eλ1t|u(t)|2+ 2eλ1thf(u(t)), u(t)i + 2eλ1thh(t), u(t)i(37) ≤λ1eλ1t|u(t)|2 +eλ1t|∇u(t)|2+eλ1tkh(t)k2 H−1(Ω) , and thus, from (2), we obtain d dt eλ1t|u(t)|2≤eλ1tkh(t)k2 H−1(Ω) . Integrating between τand t, it follows eλ1t|u(t)|2≤Zt τ eλ1skh(s)k2 H−1(Ω) ds +eλ1τ|uτ|2 ≤Zt −∞ eλ1skh(s)k2 H−1(Ω) ds +eλ1τ|uτ|2. Let b D∈ Dλ1be given. Then |U(t, τ)uτ|2≤e−λ1tZt −∞ eλ1skh(s)k2 H−1(Ω) ds +eλ1(τ−t)r2 D(τ), (38) for all uτ∈D(τ) and for all t≥τ. Denote by Rλ1(t) the nonnegative number given for each t∈Rby R2 λ1(t) = e−λ1tZt −∞ eλ1skh(s)k2 H−1(Ω) ds + 1. (39) Observe that lim t→−∞ eλ1tR2 λ1(t) = 0, and, consequently, Rλ1∈ Rλ1. Now, consider the family b Bλ1of closed balls in L2(Ω) b Bλ1={Bλ1(t) : t∈R}, defined by Bλ1(t) = v∈L2(Ω) : |v| ≤ Rλ1(t). It is straightforward to check that b Bλ1∈ Dλ1, and moreover, by (38), the family b Bλ1is pullback Dλ1-absorbing for the process U. According to Theorem 3.4, to finish the proof of the theorem we only have to prove that Uis pullback Dλ1-asymptotically compact. Let us fix b D∈ Dλ1, a sequence τn→ −∞, a sequence uτn∈D(τn), and t∈R. We have to prove that from the sequence {U(t, τn)uτn}we can extract a subsequence that converges in L2(Ω). As the family b Bλ1is pullback Dλ1-absorbing, for each integer k≥0, there exists a τD(k)≤t−k such that U(t−k, τ)D(τ)⊂Bλ1(t−k)∀τ≤τD(k). (40) Again, by a diagonal procedure, it is not difficult to conclude from (40), that there exist a subsequence τn0, uτn0⊂ {(τn, uτn)}, and a sequence {wk;k≥0} ⊂ L2(Ω) such that for all k≥0, and wk∈Bλ1(t−k), U(t−k, τn0)uτn0* wkin L2(Ω) . (41) Pullback Attractors for reaction-diffusion equations 9 Observe that, by Proposition 4.1. w0=weak −lim n0→∞ U(t, τn0)uτn0 =weak −lim n0→∞ U(t, t −k)U(t−k, τn0)uτn0 =U(t, t −k)weak −lim n0→∞ U(t−k, τn0)uτn0. i.e., U(t, t −k)wk=w0∀k≥0. (42) Then, by the lower semi-continuity of the norm, using (41) we obtain |w0| ≤ lim inf n0→∞ U(t, τn0)uτn0. (43) If we now prove that also lim sup n0→∞ U(t, τn0)uτn0≤ |w0|, (44) then we will have lim n0→∞ U(t, τn0)uτn0=|w0|. And this, together with the weak convergence, will imply the strong convergence in L2(Ω) of U(t, τn0)uτn0to w0. In order to prove (44), consider [u] := |∇u|2−λ1 2|u|2− hf(u), ui. (45) From (37), and integrating between τand t, eλ1t|u(t)|2−eλ1τ|uτ|2=−2Zt τ eλ1s[u(s)] ds +2 Zt τ eλ1shh(s), u(s)ids, i.e., |U(t, τ)uτ|2=eλ1(τ−t)|uτ|2(46) + 2 Zt τ eλ1(s−t)(hh(s), U(s, τ)uτi − [U(s, τ)uτ]) ds. From (46) it is immediate that for all k≥0 and all τn0≤t−k, U(t, τn0)uτn02 =U(t, t −k)U(t−k, τn0)uτn02(47) =U(t−k, τn0)uτn02e−λ1k + 2 Zt t−k eλ1(s−t)h(s), U(s, t −k)U(t−k, τn0)uτn0ds −2Zt t−k eλ1(s−t)U(s, t −k)U(t−k, τn0)uτn0ds. As, thanks to (40), U(t−k,τn0)uτn0∈Bλ1(t−k)∀τn0≤τD(k), k≥0, we have lim sup n0→∞ U(t−k, τn0)uτn02e−λ1k ≤R2 λ1(t−k)e−λ1k∀k≥0. (48) On the other hand, from (41) and Proposition 4.1 we deduce that U(·, t −k)U(t−k, τn0)uτn0* U(·, t −k)wk(49) in L2(t−k, t;H1 0(Ω)). Taking into account that, in particular, eλ1(s−t)h(s)∈L2(t−k, t;H−1(Ω)), we obtain from (49), lim n0→∞ Zt t−k eλ1(s−t)h(s), U(s, t −k)U(t−k, τn0)uτn0ds =Zt t−k eλ1(s−t)hh(s), U(s, t −k)wkids. (50) Now we will prove that Zt t−k eλ1(s−t)[U(s, t −k)wk]ds (51) ≤lim inf n0→∞ Zt t−k eλ1(s−t)U(s, t −k)U(t−k, τn0)uτn0ds. Denote Jk(v) = J(1) k(v) + J(2) k(v), where J(1) k(v) = Zt t−k eλ1(s−t)|∇v(s)|2−λ1 2|v(s)|2ds, and J(2) k(v) = −Zt t−k eλ1(s−t)hf(v), vids, for all v∈L2(t−k, t;H1 0(Ω)) ∩Lp(t−k, t;Lp(Ω)). Then, we want to prove Jk(U(·, t −k)wk) ≤lim inf n0→∞ Jk(U(·, t −k)U(t−k, τn0)uτn0),