Well--posedness and asymptotic behaviour for a non-classical and non-autonomous diffusion equation with delay

WELL–POSEDNESS AND ASYMPTOTIC BEHAVIOUR FOR A NON-CLASSICAL AND NON-AUTONOMOUS DIFFUSION EQUATION WITH DELAY TOM´ AS CARABALLO Departamento de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla Spain E-mail: carabal[email protected] ANTONIO M. M´ ARQUEZ-DUR´ AN Departamento de Econom´ıa, M´etodos Cuantitativos e Historia Econo´omica, Universidad Pablo de Olavide, Ctra. de Utrera, Km. 1, 41013Sevilla, Spain E-mail: [email protected] FELIPE RIVERO Departamento de An´alise, Instituto de Matem´atica e Estat´ıstica, Universidade Federal Fluminense, Rua M´ario Santos Braga S/N Valonguinho, 24.020-140 Niter´o, RJ, Brazil E-mail: lf river[email protected] February 27, 2015 In this paper, it is analyzed a non-classical non-autonomous diffusion equation with delay. First, the well-posedness and the existence of a local solution is proved by using a fixed point theorem. Then, the existence of solutions defined globally in future is ensured. The asymptotic behaviour of solutions is analyzed within the framework of pullback attractors as it has revealed a powerful theory to describe the dynamics of non-autonomous dynamical systems. One difficulty in the case of delays concerns the phase space that one needs to consider to construct the evolution process. This yields to the necessity of using a version of the Ascoli-Arzel`a theorem to prove the compactness. Keywords: Delay equations, pullback attractors, non-autonomous problems, evolution processes, non-classical diffusion equations. Mathematics Subject Classifications (2010): 35B40, 35B41, 35K55, 37B55 1 In this work we are interested in the analysis of a non-classical diffusion equation with delays, written in the following abstract functional formulation,              ∂u ∂t −γ(t)∆∂u ∂t −∆u=g(u) +f(t, ut) in (s, +∞)×Ω, u= 0 on (s, +∞)×∂Ω u(t, x) = φ(t−s, x), t ∈[s−h, s], x ∈Ω, (1) where s∈Ris the initial time, Ω ⊂Rnis a smooth bounded domain, ∆ represents the Laplacian operator with respect to the spatial variables, i.e. ∆ = Pn i=1 ∂/∂xi, g∈C1(R),lim sup |a|→+∞ g(a) a≤0 (2) |g(a)−g(b)| ≤ c|a−b|(1 + |a|ρ−1+|b|ρ−1),(3) with 1 < ρ < n+2 n−2. The time-dependent delay term f(t, ut) represents, for instance, the influence of an external force with some kind of delay, memory or hereditary characteristics, although can also model some kind of feedback controls. Here, utdenotes a segment of the solution, in other words, given h > 0 and a function u: [s−h, +∞)×Ω→R, for each t≥swe define the mapping ut: [−h, 0] ×Ω→R by ut(θ, x) = u(t+θ, x),for θ∈[−h, 0], x ∈Ω. In this way, this abstract formulation includes several types of delay terms in a unified way. For instance, terms like F1(t, u(t−h)), F2(u(t−τ(t))), Z0 −h F3(t, θ, u(t+θ)) dθ, (4) where Fi(i= 1,2,3) are suitable functions, and τ:R→[0, h], can all be described by the following corresponding fidefined as f1(t, φ) = F1(t, φ(−h)), f2(t, φ) = F2(φ(−τ(t))), f3(t, φ) = Z0 −h F3(t, θ, φ(θ)) dθ, (5) where φ: [−h, 0] →X(Xdenotes certain Banach or Hilbert space concerning the spatial variable). Then, when we replace φby utin (5), we obtain (4). Non-classical parabolic equations arise as models to describe physical phenomena such as non-Newtonian flow, soil mechanics, heat conduction, etc (see [Aifantis, 1980], [Kuttlerand & Aifantis, 1987], [Kuttlerand & Aifantis, 1988], [Anh & Bao, 2010], [Camassa & Holm, 1993], [Ma et al., 2012], [Peter & Gurtin], [Sun et al., 2007], [Sun et al., 2008] and references therein). In the work of Aifantis et al., [Aifantis, 1980], [Kuttlerand & Aifantis, 1987], [Kuttlerand & Aifantis, 1988] we can find a quite general approach to deduce these equations in the autonomous case without delay. In the aforementioned papers, it is pointed out that the classical reaction-diffusion equation ut−∆u=g(u) does not contain each aspect of the reactiondiffusion problem, and it neglects viscidity, elasticity, and pressure of medium in the process of solid diffusion. The authors obtained a diffusion theory similar to Fick’s classical model for solute in an undisturbed solid matrix, obtaining a hyperbolic equation ut+D1utt =D2∆u, where D1and D2are positive constants. Assigning viscosity to the diffusing substance, they arrived to que following equation ut+D1utt =D2∆u+D3∆ut, and neglecting the inertia term, finally obtained the non-classical parabolic equation ut=D2∆u+D3∆ut, where D3is also a positive constant. The asymptotic behaviour of the model without delay terms and with constant coefficients ut−µ∆ut−∆u+g(u) = f(x), µ ∈[0,1] is studied in [Wang et al., 2006], where, in particular, it is shown the well-posedness of the problem and the existence of the global attractor either in H1 0(Ω) or in H2(Ω), depending on the regularity of the initial data. They also showed the continuity of 2 Non-classical non-autonomous diffusion equation with delay 3 the global attractor in Housdorff semidistance when µ→0 in H1 0(Ω). The introduction of a time dependence in coefficient γ(t) represents the variability of viscosity in time due to, for example, external environment temperatures. This time dependence provides the system with a non-autonomous nature. Although there exist several possibilities to analyze non-autonomous dynamical systems (e.g. skew-product flows, uniform attractors, kernel section, etc), we have preferred to consider the theory of evolution processes and pullback attractors. This is a general theory which studies the asymptotic behaviour of non-autonomous problems in the pullback sense, that is, when we fix the final time and study the behaviour when we start earlier and earlier in the past. Therefore, we have two different kinds of dynamics in a non-autonomous problem: the forward dynamics, when final time goes to infinity, and the pullback dynamics, when the initial times go to minus infinity. This theory has been widely developed over the last thirty years (see [Caraballo et al., 2010], [Carvalho et al., 2013], [Caraballo et al., 2006a], [Chepyzhov & Vishik, 2002], [Mar´ın-Rubio & Real, 2009]) and it can be seen as a natural generalization of the classical theory of semigroups and global attractors. In [Rivero, 2013], Rivero studied the existence of the pullback attractor and its continuity under non-autonomous perturbations without delay, showing the existence of a concrete structure under some assumptions on the non-linearity. The study of a non-autonomous case with delay appeared in [Caraballo & M´arquez-Dur´an, 2013] for the first time, where it was established the well-posedness of the problem when γ(t)≡γis constant. Also, it was proved in [Caraballo & M´arquez-Dur´an, 2013] the stability of the stationary solutions under some appropriate hypotheses on the delay term. In [Hu & Wang, 2012], Hu and Wang studied this equation with a specific variable delay term with bounded derivative, showing the existence of the pullback attractor in H1 0and H2without neither non-linearity nor variable coefficients. The presence of delays in the models of certain phenomena implies the necessity of working in a quite different phase space. Indeed, our initial data needs now to be a function defined in an interval of certain length h > 0, prior the initial time, in other words, we need to know the values of a mapping defined in [s−h, s] if sis such an initial time. In particular, this fact implies that the compactness property of our non-autonomous dynamical system has to be proved in more complicated functional spaces, for instance, in a Banach space of continuous functions. This requires the use of Ascoli-Arzel`a’s arguments. The content of this paper is as follows. In order to show the existence and uniqueness of solution for (1) and the continuous dependence on the initial data, in Sec. 1 we establish the notation for delay equations, stating the conditions and results ensuring the well-posedness of (1). Sec. 2 is devoted to the study of the global existence of solutions and the existence of a pullback absorbing family within the universe of global bounded families. In Sec. 3 we recall the basic definitions and results for pullback attractors, setting the framework for our analysis. Finally, in Sec. 4 the existence of the pullback attractor for our model is proved. 1. Existence of solution We will first analyze the existence and uniqueness of solution to our model. Let us denote CL=C([−h, 0], L2(Ω)) with norm kψkCL= sups∈[−h,0]kψ(s)kL2. In this way, given ψ∈ C([−h, T], L2(Ω)), for any t∈[0, T] we can define ψt: [−h, 0] →L2(Ω) as ψt(θ) = ψ(t+θ) for θ∈[−h, 0]. Obviously ψt∈CL. Analogously, it can be defined CH=C([−h, 0], H1 0(Ω)) with norm kψkCH= sups∈[−h,0] kψ(s)kH1 0. We will pick up an initial value φ∈CLsuch that φ(0) ∈H1 0(Ω). Assume also that f: (s, +∞)× CH→L2(Ω) is continuous in tand locally Lipschitz in CHuniformly in time, that is, for all R > 0 there exists a constant C(R)>0 such that, for any t∈R, kf(t, ξ)−f(t, η)kL2(Ω) ≤C(R)kξ−ηkCH,(6) for all ξ, η ∈CHwith kξkCH,kηkCH≤R. Proceeding as in [Rivero, 2013], we can define operators B(t)=(I+γ(t)A)−1and ˜ A(t) = AB(t), where A=−∆ with Dirichlet boundary conditions and the functions ˜g(t, u) = B(t)g(u) and ˜ f(t, φ) = B(t)f(t, φ), ∀t≥s, ∀φ∈CH. Then, we can write problem (1) as du dt =h(t, ut),(7) 4T. Caraballo et al. with h: (s, +∞)×CH→L2(Ω) defined as h(t, φ) = ˜ A(t)φ(0) + ˜g(t, φ(0)) + ˜ f(t, φ), ∀t≥s, ∀φ∈CH. The domain of the operator ˜ A(t) does not depend on time. In fact, if we define our problem in H1 0(Ω), then D(˜ A(t)) = H1 0(Ω). This operator is uniformly bounded in time and ˜ A(t) = 1 γ(t)I−(1 + γ(t)A)−1,(8) for any t∈R. Also, for any α > 0 and x∈D(Aα), Aα˜ A(t)x=˜ A(t)Aαx. Thanks to the continuity of the function R3 t7→ B(t)∈ L(H1 0(Ω)), we obtain the following estimate (see [Rivero, 2013]) k˜ A(t)−˜ A(s)kL(H1 0(Ω)) ≤C|γ(t)−γ(s)|, for a constant C∈R. We can now state and prove the existence of solution to our problem. Theorem 1.1. For each φ∈CHand under the assumptions (2),(3) and (6), there exists δ > 0such that in the interval [s−h, s +δ)there is a unique solution of problem (1). In other words, there exists a function u∈C([s−h, s +δ), H1 0(Ω)) with u(t, s;φ) = φ(t−s)for all t∈[s−h, s]which satisfies u(t, s;φ) = φ(0) + Zt s h(r, ur)dr, for all t∈[s, s+δ), Moreover, if fis continuous in time, then u∈C([s−h, s +δ), H1 0(Ω)) ∩C1((s, s + δ), H1 0(Ω)), i.e. uis a strong solution. Proof. The proof is based on the contraction mapping theorem. To this end, for the given initial datum φ∈CH, and for a positive Tto be determined later on, we define the following space XT φ=u∈C[s−h, T), H1 0(Ω):u(t) = φ(t−s) for all t∈[s−h, s], and kukXT φ≤2kφkCHo, (9) where kukXT φ= supσ∈[s−h,T)ku(σ)kH1 0. This space XT φis a complete metric space (since it is a closed subset of a Banach space). Now we consider the operator Φ : XT φ→XT φ given by Φ(u)(t) =      φ(t−s), t ∈[s−h, s] φ(0) + Zt s h(r, ur)dr, t ∈(s, T). Let us first check that Φ is well defined, i.e. Φ(u)∈ XT φfor all u∈XT φ. Given u∈XT φ, the mapping Φ(u)(·) : [s−h, T)→L2(Ω) is continuous thanks to the continuity of φand the mapping t∈(s, T)→h(t, ut). Moreover, for all t≥swe have that kΦ(u)(t)kH1 0=   φ(0) + Zt s h(r, ur)dr   H1 0 ≤ kφ(0)kH1 0+Zt s kh(r, ur)kH1 0dr ≤ kφ(0)kH1 0+Zt s k˜ A(r)kL(H1 0)ku(r)kH1 0dr +Zt s kB(r)g(u(r))kH1 0dr +Zt s kB(r)f(r, ur)kH1 0dr. (10) But, k˜ A(r)kL(H1 0)≤a, for all r≥s, and proceeding as in [Rivero, 2013], we can prove that B(t)◦gis locally lipschitz in H1 0(Ω) uniformly in t. Indeed, by (3), kg(u)−g(v)kL2n n+2 ≤cZΩ|u−v|(1 + |u|ρ−1+|v|ρ−1))2n n+2 n+2 2n ≤˜cku−vkL2n n−21 + kukρ−1 Ln(ρ−1) 2 +kvkρ−1 Ln(ρ−1) 2 ). (11) Since 2n(ρ−1) 4≤2n n−2, we have that H1 0(Ω) ⊂Ln(ρ−1) 2. Due to H1 0(Ω) g −−−→ H−1(Ω) B(t) −−−−→ H1 0(Ω), Non-classical non-autonomous diffusion equation with delay 5 function B(t)◦gis locally Lipschitz in H1 0(Ω). Therefore, for R= 2kφkCHthere exists Lg(R) such that kB(r)g(u(r))kH1 0 ≤ kB(r) (g(u(r)) −g(0)) + B(r)g(0)kH1 0 ≤Lg(R)ku(r)kH1 0+b0|g(0)|, (12) where b0>0 is a constant such that kB(t)kL(H1 0)≤ b0,for all t. On the other hand, as fis locally Lipschitz, kB(r)f(r, ur)kH1 0 ≤ kB(r) (f(r, ur)−f(r, 0)) + B(r)f(r, 0)kH1 0 ≤b0C(R)kurkCH+b0kf(r, 0)kL2(Ω). (13) Then, for all t≥swe have that kΦ(u)(t)kH1 0≤ kφ(0)kH1 0+aZt s ku(r)kH1 0dr +Lg(R)Zt s ku(r)kH1 0dr +b0|g(0)|(t−s) +b0C(R)Zt s kurkCHdr +b0Zt s kf(r, 0)kL2(Ω)dr. Therefore, kΦ(u)(t)kH1 0≤ kφ(0)kH1 0 + (a+Lg(R) + b0C(R)) Zt s kurkCHdr +b0|g(0)|(t−s) + b0Zt s kf(r, 0)kL2(Ω)dr. Taking into account that kurkCH= sup θ∈[−h,0] ku(r+θ)kH1 0 ≤sup σ∈[s−h,T) ku(σ)kH1 0 =kukXT φ ≤R, for all s≤r≤t<T. Then, we obtain that kΦ(u)(t)kH1 0≤ kφkCH +R(t−s) (a+Lg(R) + b0C(R)) +b0|g(0)|(t−s) +b0Zt s kf(r, 0)kL2(Ω)dr, for all t∈(s, T). If we write T=s+δ, we then have kΦ(u)(t)kH1 0≤ kφkCH+Rδ (a+Lg(R) + b0C(R)) +b0|g(0)|δ+b0Zs+δ s kf(r, 0)kL2(Ω)dr, for all t∈(s, T), and considering δ > 0 small enough, we can ensure that kΦ(u)(t)kH1 0≤2kφkCH,∀t∈(s, T). We also have the same conclusion for t∈[s−h, s], in fact kΦ(u)(t)kH1 0≤ kφkCH,∀t∈[s−h, s], and therefore, we can conclude that the operator Φ is well defined. Now, by using the contraction mapping theorem, we prove the existence of a fixed point for Φ(·), which will be the solution of our problem. To this end, we need to prove that Φ is a contracting mapping. Let us take u, v ∈XT φ. Then ku(t)kH1 0,kv(t)kH1 0≤Rfor all t∈[s, T) and kΦ(u)(t)−Φ(v)(t)kH1 0 ≤Zt s kh(r, ur)−h(r, vr)kdr ≤Zt s k˜ A(r)kL(H1 0)ku(r)−v(r)kH1 0dr +Zt s kB(r)(g(u(r))−g(v(r))+f(r, ur)−f(r, vr))kH1 0dr. Using the uniform bound in time for ˜ A(t) and B(t), and (6) kΦ(u)(t)−Φ(v)(t)kH1 0 ≤K1Zt s ku(r)−v(r)kH1 0dr +K2Zt s kf(r, ur)−f(r, vr)kL2dr ≤K1Zt s ku(r)−v(r)kH1 0dr +K(R)Zt s sup θ∈[−h,0] ku(r+θ)−v(r+θ)kH1 0dr. Taking supremum in [s, T) with T=s+δ kΦ(u)−Φ(v)kXT φ ≤K1δku−vkkXT φ +K(R)δ sup r∈[s,T) sup θ∈[−h,0] ku(r+θ)−v(r+θ)kH1 0!, 6T. Caraballo et al. but, if u, v ∈XT φ, sup r∈[s,T) sup θ∈[−h,0] ku(r+θ)−v(r+θ)kH1 0 = sup r∈[s−h,T) ku(r)−v(r)kH1 0 ≤ ku−vkXT φ. Therefore, for δ > 0 small enough, Φ is well defined and is a contraction in Xφ. The proof is therefore complete. 2. Global solution and pullback absorbing family In this section we will prove that the local solution, whose existence has been proved in Theorem 1.1, is in fact a global one, i.e. it is defined in the whole future and not only in a small time interval. However, we will deduce this result after obtaining some a priori estimates which will be also useful to deduce the existence of absorbing sets for the process generated by our model. For any ϕ∈H1 0(Ω), taking into account (2) and arguing as in [Hale, 1989], for each δ > 0 there is a constant Kδ>0 such that ZΩ g(u)u≤δkuk2 L2(Ω) +Kδ, ZΩ G(u)≤δkuk2 L2(Ω) +Kδ (14) for all u∈L2(Ω), where G(r) = Rr 0g(θ)dθ. Let Lb(ϕ) be the following energy functional Lb(ϕ) = 1 2kϕk2 L2+bkϕk2 H1 0−bZΩ G(ϕ),(15) with b≥0. It is easy to prove that for δ=λ1 6, Lb(ϕ)≥b 3kϕk2 H1 0−bKλ1 6 (16) and for any δ > 0, Lb(ϕ)≤1 + b(λ1+ 2δ) 2λ1 kϕk2 H1 0+bKδ,(17) with λ1the first eingenvalue of A. Taking a solution u(t, s;φ) of (1) and for b > 0, d dtLb(u)=(u, du dt )L2+b(u, du dt )H1 0−bZΩ g(u)du dt ≤ − 1−γ1ε1 2−2δ+ε2 2λ1kuk2 H1 0 +ε2+ 1 2ε2 kf(t, ut)k2 L2 +γ(t)1 2ε1 −bkdu dt k2 H1 0+Kδ, for ε1, ε2, δ > 0. Taking ε1=1 2γ1,ε2=λ1 4,δ=λ1 8 and b≥1 2ε1= 2γ1, we obtain d dtLb(u)≤ −1 2kuk2 H1 0+λ1+ 4 2λ1kf(t, ut)k2 L2+Kλ1 8 ≤ − λ1 1 + b(λ1+ 2˜ δ)Lb(u) +λ1+ 4 2λ1kf(t, ut)k2 L2 +λ1 1 + b(λ1+ 2˜ δ)K˜ δ+Kλ1 8 . Denoting Cb=λ1 1+b(λ1+2˜ δ),Cλ1=λ1+4 2λ1 and ˜ Kb=CbK˜ δ+Kλ1 8 , d dt eCbtLb(u)=CbeCbtLb(u) + eCbtd dtLb(u) ≤eCbt(Cλ1kf(t, ut)k2 L2+˜ Kb). Integrating between sand tand using (6), eCbtLb(u(t)) ≤eCbsLb(φ(0)) + Cλ1Zt s eCbrkf(r, ur)k2 L2dr +˜ Kb CbeCbt−eCbs ≤eCbsLb(φ(0)) + Cλ1C(R)Zt s eCbrkurk2 CHdr +Cλ1Zt s eCbrkf(r, 0)k2 L2dr +˜ Kb CbeCbt−eCbs. Replacing tby t+θwith θ∈[−h, 0], by (16) Non-classical non-autonomous diffusion equation with delay 7 and (17), b 3eCbtku(t+θ)k2 H1 0 ≤eCb(s+h)(˜ Cbkφ(0)k2 H1 0+bKδ2) +Cλ1C(R)eCbhZt+θ s eCbrkurk2 CHdr +Cλ1eCbhZt+θ s eCbrkf(r, 0)k2 L2dr +eCbt ˜ Kb Cb +bKλ1 6!, where δ2is chosen such that ˜ Cb=1 + b(λ1+ 2δ2) 2λ1 . If t > s +h, then eCbtku(t+θ)k2 H1 0 ≤max (sup θ∈[−h,s−t] eCbtkφ(t+θ−s)k2 H1 0, sup θ∈[s−t,0] 3 beCb(s+h)(˜ Cbkφ(0)k2 H1 0+bKδ2) +3 bCλ1C(R)eCbhZt+θ s eCbrkurk2 CHdr +3 bCλ1eCbhZt+θ s eCbrkf(r, 0)k2 L2dr +3 beCbt ˜ Kb Cb +bKλ1 6!), but, sup θ∈[−h,s−t] eCbtkφ(t+θ−s)k2 H1 0 ≤sup θ∈[−h,0] eCb(s+h)kφ(θ)k2 H1 0 =eCb(s+h)kφk2 CH, and sup θ∈[s−t,0] Zt+θ s eCbrkurk2 CHdr+Zt+θ s eCbrkf(r, 0)k2 L2dr ≤Zt s eCbrkurk2 CHdr+Zt s eCbrkf(r, 0)k2 L2dr. Therefore, b 3eCbtkutk2 CH≤eCb(s+h)˜ Cbkφk2 CH+bKδ2 +Cλ1eCbhZt s eCbrkf(r, 0)k2 L2dr +eCbt ˜ Kb Cb +bKλ1 6! +Cλ1C(R)eCbh Zt s eCbrkurk2 CHdr. Assuming that C(R)<bCb 3Cλ1eCbh(18) and calling β=3 bCλ1C(R)eCbh(it means β < Cb) and α(t) = 3 beCb(s+h)˜ Cbkφk2 CH+bKδ2 +3 bCλ1eCbhZt s eCbrkf(r, 0)k2 L2dr +3 beCbt ˜ Kb Cb +bKλ1 6!, by the Gronwall Lemma we obtain that eCbtkutk2 CH≤α(t) + βZt s α(r)eβ(t−r)dr. Now, βZt s α(r)eβ(t−r)dr ≤3 beCb(s+h)eβ(t−s)˜ Cbkφk2 CH+bKδ2 +3 b β Cb−β ˜ Kb Cb +bKλ1 6!eCbt +3 bβCλ1e(Cbh+βt) Zt s e(Cb−β)rkf(r, 0)k2 L2dr. Then, kutk2 CH ≤e−Cbtα(t) +3 beCbhe(Cb−β)(s−t)˜ Cbkφk2 CH+bKδ2 +3 b β Cb−β ˜ Kb Cb +bKλ1 6! +3 bβCλ1e(Cbh−(Cb−β)t) Zt s e(Cb−β)rkf(r, 0)k2 L2dr. 8T. Caraballo et al. Assuming that there exists a η0≥0 such that for any η∈[0, η0], Zt −∞ eηrkf(r, 0)k2 L2dr < +∞,(19) we have kutk2 CH s→−∞ −−−−→ l(t),(20) where l(t) = 3 b ˜ Kb Cb +bKλ1 6!1 + β Cb−β +3 bβCλ1eCbhZt −∞ eCbrkf(r, 0)k2 L2dr +e−(Cb−β)tZt −∞ e(Cb−β)rkf(r, 0)k2 L2dr. Then, we have the global existence of any solution u(t, s;φ) of (1), i.e. for each φ∈CH,u(·, s;φ)∈ C([s−h, +∞), H1 0(Ω)) in Theorem 1.1, and, once we justify in the next section that the solutions of our problem generates a non-autonomous dynamical system, this also ensures the existence of a family of closed subsets BCH(0, l1/2(t)) : t∈Rwhich pullback attracts bounded subsets of CH. Also we need a result on the continuous dependence on the initial data. Proposition 2.1. Under the assumptions of Theorem 1.1, any solution u(t, s;φ)of (1) is continuous with respect to the initial condition φ∈CH.More precisely, if ui,for i= 1,2,are the corresponding solutions to the initial data φi∈CH, i = 1,2,the following estimate holds: ku1 t−u2 tkCH≤ kφ1−φ2kCHe(a+Lg(R)+C(R))t,(21) for all t∈[s, T),where R≥0is given by R= max(2kφ1kCH,2kφ2kCH). Proof. Let ui,for i= 1,2,be the corresponding solutions to the initial data φi∈CH, i = 1,2,in the interval [s−h, T),for a fixed T > s. Then we have that u1(t)−u2(t) = φ1(0) −φ2(0) +Zt s (h(r, u1 r)−h(r, u2 r))dr, for t∈(s, T). Now, taking into account (6), (9), that k˜ A(r)kL(H1 0)≤a, and function B(t)◦gis locally Lipschitz in H1 0(Ω), for R= max(2kφ1kCH,2kφ2kCH) it is not difficult deduce that ku1(t)−u2(t)kH1 0≤ kφ1(0) −φ2(0)kH1 0 + (a+Lg(R) + C(R)) Zt s ku1 r−u2 rkCHdr, for t∈(s−h, T). Thus, replacing now tby t+θwith θ∈[−h, 0] and taking supremum in θ, ku1 t−u2 tkCH≤ kφ1−φ2kCH + (a+Lg(R) + C(R)) Zt s ku1 r−u2 rkCHdr, por t∈(s, T), and therefore, thanks to the Gronwall lemma, we deduce (21). 3. Abstract results on the theory of attractors. Existence of pullback attractors In this section we recall some abstract results on the theory of pullback attractors. We present a summary of some results on the existence of minimal pullback attractors obtained in [Carvalho et al., 2013] (see also [Caraballo et al., 2010, Caraballo et al., 2011, Caraballo et al., 2006a, Caraballo et al., 2006b, Garc´ıa-Luengo et al., 2012, Mar´ın-Rubio & Real, 2009]). Consider given a metric space (X, dX). For A, Bsubsets of Xlet dist(A, B) denote the Hausdorff semidistance between Aand B, i.e. dist(A, B) = sup a∈A inf b∈BdX(a, b). Definition 3.1. An evolution process in a metric space (X, dX) is a family of continuous maps {S(t, s) : t≥s}from Xinto itself with the following properties (1) S(t, t) = I, for all t∈R, (2) S(t, s) = S(t, τ)S(τ, s), for all t≥τ≥s, (3) {(t, s)∈R2:t≥s} × X 3 (t, s, x)7→ S(t, s)x∈ X is continuous. Let us denote P(X) the family of all nonempty subsets of X,and consider a family of nonempty Non-classical non-autonomous diffusion equation with delay 9 sets b D0={D0(t) : t∈R} ⊂ P(X). Observe that we do not require any additional condition on these sets as compactness or boundedness. Let Dbe a nonempty class of families parameterized in time b D={D(t) : t∈R}⊂P(X).The class Dwill be called a universe in P(X). Definition 3.2. It is said that b D0={D0(t) : t∈ R}⊂P(X) is pullback D−absorbing for the process {S(t, s) : t≥s}on Xif for any t∈Rand any b D∈ D,there exists a s0(t, b D)≤tsuch that S(t, s)D(s)⊂D0(t) for all s≤s0(t, b D). Definition 3.3. The family AD={AD(t) : t∈R} is the D-pullback attractor for the process {S(t, s) : t≥s}in Xif: (a) for any t∈R,the set AD(t) is a nonempty compact subset of X. (b) ADis pullback D−attracting, i.e., lim τ→−∞ dist(S(t, s)D(s),AD(t)) = 0 for all b D∈ D,for t∈R, (c) ADis invariant, i.e., S(t, s)AD(s) = AD(t) for all s≤t. The family ADis minimal in the sense that if b C= {C(t) : t∈R} ⊂ P(X) is a family of closed sets such that for any b D={D(t) : t∈R} ∈ D, lim τ→−∞ distX(S(t, s)D(s), C(t)) = 0, then AD(t)⊂C(t). Definition 3.4. Given a family parameterized in time, b D={D(t) : t∈R} ⊂ P(X),it is said that a process {S(t, s) : t≥s}on Xis b D−asymptotically compact if for any t∈Rand any sequences {sn} ⊂ (−∞, t] and {xn} ⊂ X bounded satisfying sn→ −∞ and xn∈D(sn) for all n, the sequence {S(t, sn)xn}is relatively compact in X. Definition 3.5. A process {S(t, s) : t≥s}on X is said to be pullback D−asymptotically compact if it is b D-asymptotically compact for any b D∈ D. Denoting by ω(b D0, t) := \ s≤t[ τ≤s S(t, τ)D0(τ) X for all t∈R, the natural generalization of the ω-limit set in the pullback sense, we have the following result on existence of pullback attractors. Theorem 3.6. Consider a process {S(t, s) : t≥ s}in X, a universe Din P(X),and a family b D0={D0(t) : t∈R} ⊂ P(X)which is pullback D−absorbing and assume also that the process is pullback b D0−asymptotically compact. Then, the family AD={AD(t) : t∈R}defined by AD(t) = [ b D∈D ω(b D, t) X t∈R, is the D-pullback attractor. 4. Existence of pullback attractors for the process associated to (1) On account of the previous results, we will be able to construct a non-autonomous dynamical system. More precisely, we will be able to construct a process S:CH→CHassociated to (1), and prove the existence of a pullback attractor for such a process. Assume hypotheses (2), (3), (6), with f(t, 0) = 0 for all t∈R(this assumption is not a real restriction as we can subtract such term to f(t, ·) and add it to the term g), and suppose that C(R) satisfies (18). For each φ∈CHand s∈R, Theorem 1.1 and the a priori estimates in Section 2 ensure the existence of a unique global solution u(·;s, φ) of (1). Then we can define the family of maps S(t, s) in CHas S(t, s)φ=ut(·;s, φ)∀t≥s. According to the previous results, it is not difficult to prove that S(·,·) is a process and we can also write (S(t, s)φ)(θ) = ut(θ;s, φ) =u(t+θ;s, φ) =T(t+θ, s)φ(0) +Zt+θ s T(t+θ, τ)˜ f(τ, uτ)dτ +Zt+θ s T(t+θ, τ)˜g(τ, u)dτ, (22)