Pullback permanence for non-autonomous partial differential equations

Pullback permanence for non-autonomous partial differential equations J.A. Langa1and A. Suarez1 1Departamento de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain. e-mails: [email protected] ; [email protected] January 7, 2002 Abstract A system of differential equations is permanent if there exists a fixed bounded set of positive states strictly bounded away from zero to which, from a time on, any positive initial data enter and remain. However, this fact does not happen for a differential equation with general non-autonomous terms. In this work we introduce the concept of pullback permanence, defined as the existence of a time dependent set of positive states to which all solutions enter and remain for suitable initial time. We show this behaviour in the non-autonomous logistic equation ut−∆u=λu −b(t)u3, with b(t)>0,for all t∈R,limt→∞ b(t) = 0.Moreover, a bifurcation scenario for the asymptotic behaviour of the equation is described in a neighbourhood of the first eigenvalue of the Laplacian. We claim that pullback permanence can be a suitable tool for the study of the asymptotic dynamics for general non-autonomous partial differential equations. Mathematics Subject Classification 2000: 35B05, 35B22, 35B41, 37L05. Key words and phrases: Non-autonomous differential equations, pullback attractors, comparision techniques, permanence. 1 Introduction One of the main questions for a mathematical model from a natural phenomena is that on the long-time behaviour of its solutions. In particular, of special interest for an ecological 1 model is to predict the existence of strictly positive solutions for all future times. In this sense, we can look for strictly positive globally attracting equilibria (or stationary solutions) for the corresponding partial differential equation associated to the model. But only in a restricted set of systems we can assure the existence of stationary solutions. However, the concept of global attractor comes to put some light in the understanding of the asymptotic behaviour of many dissipative systems (Hale [14], Temam [25]). Indeed, we can infer uniform persistence or permanence (Cantrell et al. [4], [5]) of solutions from the presence of a globally attracting positive set rather that a single attracting equilibrium. A system is said to have uniform persistence (Butler et al. [3] ) if there exists a positive set which is bounded away from zero and globally attracting for all positive solutions. Note that this allows the systems to have a more complex dynamics, so that a wider set of (more natural) situations can be considered. On the other hand, we lose some information on the location and size of these new sets, so that any study on the structure of them are really helpful. There exists a substantial literature on this subject for autonomous differential equations (Hale and Waltman [15], Hess [17], Hutson and Schmitt [18]). The system is said to be permanent if it is also dissipative, i.e., the orbits enter into a compact set in a finite time. Afterwards, Cao and Gard [6] introduce the concept of practical persistence, defined as uniform persistence together with some information on the location of the positive attractor. In this work we study problems on permanence of positive solutions for the following non-autonomous logistic equation ut−∆u=λu −b(t)u3, with b(t)>0,the interesting case being when we impose limt→+∞b(t) = 0.Previous works for non-autonomous equations focus on the periodic or bounded by periodic functions in time cases (Cantrell and Cosner [4], Burton and Hutson [2], see also Nkashama [21] for the finite dimensional case with bounded and strictly positive non-autonomous terms). We treat a more general case, so that we allow the equation a very weak dissipation effect as time goes to infinite, and so previous works in the literature are not valid for our purposes. The situation can be summarized as follows: when the parameter λ < λ1,with λ1the first eigenvalue of the negative Laplacian, we get the existence of the zero solution as a globally attracting set. However, a drastic change in the asymptotic behaviour happens as the paramenter λcrosses the value λ1.We describe in some detail this bifurcation scenario (Sections 3 and 4). Indeed, we firstly show that the equation leads to an order-preserving system and the method of sub and super solutions (Pao [22], Hess [17]) can be adapted to this case. Afterwards, we find a non bounded order interval depending on time in which 2 all the asymptotic behaviour forward in time takes place (Section 4). That is, it does not exist any bounded absorbing set for the problem, and so any result on permanence in the sense of Cantrell and Cosner [4] can be expected. However, very recently the theory of attractors for general non-autonomous differential equations has been introduced (Cheban et al. [8], Kloeden and Schamalfuss [19]; see also Crauel and Flandoli [12], Crauel et al. [13], for the same concept in a stochastic framework). In this case, the semigroup becomes a process, that is, a two-time dependent operator (Sell [24]), as the dependence of the initial times is so important as that on the final times. When the non-autonomous terms are periodic or quasi-periodic, the same concept of attractor in Temam [25] or Hale [14] can be used for these situations (Sell [24], Chepyzhov and Vishik [10]). But important changes in the concept must be introduced when we deal with general non-autonomous terms. Chepyzhov and Vishik [10] define kernel and kernel sections. This last concept is similar to that defined in Cheban et al. [8] as cocycle or pullback attractor. In our opinion, this is one of the right concepts to define the attractor for a general non-autonomous differential equation, as some results on the upper-semicontinuity of pullback attractors to the (autonomous) global attractor show (Caraballo and Langa [7]). The attractor in this situation is a time-dependent family of compact sets, invariant with respect to the cocycle and attracting from ‘−∞’ (see Definition 4). We apply the theory of pullback attractors to our non-autonomous logistic equation. We also apply a result on the upper semicontinuity of this pullback attractor to the global attractor for the autonomous equation. This reinforces the choice of working with the pullback attractor to study the asymptotic behaviour of non-autonomous equations. While forward in time we have not information on the stability of the equation when λ > λ1,we describe a bifurcation scenario at the parameter value λ=λ1from the pullback procedure: the zero solution becomes unstable for λ > λ1and there exists a transfer of stability to the pullback attractor, which is a set strictly bigger than the zero solution, so that a result on permanence follows. We think this is the sensible concept for permanence for general non-autonomous partial differential equations (Definition 8). In Section 4.4 we are able to give more information on the structure of this pullback attractor and so on the bifurcation phenomena. Indeed, by introducing the concepts of sub-trajectories, super-trajectories and complete trajectories for non-autonomous systems in Section 3, as generalization of the theory of sub and super-equilibria in the sense of Hess [17], Arnold and Chueshov [1] and Chueshov [11], we describe the existence of a maximal complete trajectory on the attractor with some stability properties. We give a general theorem 3 which can be applied to more general situations. Finally, some conclusions and possible generalizations are given in the final Section. 2 Non-automous attractors In this section, we introduce the general framework in which the theory of attractors for non-autonomous systems is going to be studied (see Cheban et al. [8] and Schmalfuss [23]). In a first step, we define processes as two-time dependent operators related with the solutions of non-autonomous differential equations. In this way, we are able to treat these equations as dynamical systems. Secondly, we write the general definitions of invariance, absorption and attraction and we finish with a general theorem on the existence of global attractors for these kind of equations. Finally, we give the definition of permanence for non-autonomous partial differential equations. Let (X, d) be a complete metric space (with the metric d) with an order relation ‘≤’ and {S(t, s)}t≥s,t, s ∈Rbe a family of mappings satisfying: i) S(t, s)S(s, τ)u=S(t, τ)u, for all τ≤s≤t, u ∈X, ii) u7→ S(t, τ)uis continuous in X. This map is called a process. In general, we have to consider S(t, τ)uas the solution of a non-autonomous equation at time twith initial condition uat time τ. Let Dbe a non-empty set of parameterized families of non-empty bounded sets {D(t)}t∈R. In particular, D(t)≡B∈ D, where B⊂Xis a bounded set. In what follows, we will consider fixed this base of attraction D,so that the concepts of absorption and attraction in our analysis are always referred to it. For A, B ⊂Xdefine the Hausdorff semidistances as, dist(A, B) = sup a∈A inf b∈Bd(a, b)Dist(A, B) = inf a∈Ainf b∈Bd(a, b). Definition 1 Given t0∈R, we say that K(t0)⊂Xis attracting at time t0if for every {D(t)} ∈ D we have that lim τ→−∞ dist(S(t0, τ)D(τ), K(t0)) = 0. A family {K(t)}t∈Ris attracting if K(t0)is attracting at time t0,for all t0∈R. The previous concept considers a fixed final time and moves the initial time to −∞. Note that this does not mean that we are going backwards in time, but we consider the 4 state of the system at time t0starting at τ→ −∞.This is called pullback attraction in the literature (cf. [19], [23]). Definition 2 Given t0∈R, we say that B(t0)⊂Xis absorbing at time t0if for every {D(t)} ∈ D there exists T=T(t, D)∈Rsuch that S(t0, τ)D(τ)⊂B(t0),for all τ≤T. A family {B(t)}t∈Ris absorbing if B(t0)is absorbing at time t0,for all t0∈R. Note that every absorbing set at time t0is attracting. Definition 3 Let {B(t)}t∈Rbe a family of subsets of X. This family is said to be invariant with respect to the process Sif S(t, τ)B(τ) = B(t),for all (τ, t)∈R2, τ ≤t. Note that this property is a generalization of the classical property of invariance for semigroups. However, in this case we have to define the invariance with respect to a family of sets depending on a parameter. We define the omega-limit set at time t0of D≡ {D(t)} ∈ D as Λ(D, t0) = ∩s≤t0∪τ≤sS(t0, τ)D(τ). From now on, we assume that there exists a family {K(t)}t∈Rof compact absorbing sets, that is, K(t)⊂Xis non-empty, compact and absorbing for each t∈R. Note that, in this case, Λ(D, t0)⊂K(t0),for all {D(t)} ∈ D,t0∈R.As in the autonomous case, it is not difficult to prove that under these conditions Λ(D, t0) is non-empty, compact and attracts {D(t)} ∈ D at time t0. The proof is similar to that in Crauel et al. [13], where the set D consists only of bounded sets. Definition 4 The family of compact sets {A(t)}t∈Ris said to be the global non-autonomous (or pullback) attractor associated to the process Sif it is invariant, attracting every {D(t)} ∈ D (for all t0∈R) and minimal in the sense that if {C(t)}t∈Ris another family of closed attracting sets, then A(t)⊂C(t)for all t∈R. Remark 5 Chepyzhov and Vishik [9] define the concept of kernel sections for non-autonomous dynamical systems which corresponds to our definition of global non-autonomous attractor with {D(t)} ≡ B⊂Xbounded. 5 The general result on the existence of non-autonomous attractors is a generalization of the abstract theory for autonomous dynamical systems (Temam [25], Hale [14]): Theorem 6 (Crauel et al. [13], Schmalfuss [23]) Assume that there exists a family of compact absorbing sets. Then, the family {A(t)}t∈Rdefined by A(t) = ∪D∈DΛ(D, t) is the global non-autonomous attractor. Remark 7 All the general theory of non-autonomous attractors can be written in the framework of cocycles (cf., among others, Cheban et al. [8], Crauel and Flandoli [12], Kloeden and Schmalfuss [19], Schmalfuss [23]). We could have also followed this notation here, but we think that, in this case, it is more clear to keep the explicit dependence on time of the attractor, which in addition, allows us to compare more straightforward with the concept of attractor in an autonomous framework. From the concept of non-autonomous attractor, we can now give the following definition of permanence, which will be suitable for non-autonomous partial differential equations. Definition 8 We say that a system has the property of pullback permanence (or that it is permanent in the pullback sense) if there exists a time-dependent family of sets U:R7−→ X, satisfying 1. U(t)is absorbing every bounded set D⊂X(cf. Definition 2). 2. Dist(U(t),{0})>0for all t∈R. 3 Order-preserving non-autonomous differential equations We now introduce order-preserving systems and the concepts of sub, super and complete trajectories as a generalization of sub, super and equilibria in Hess [17], and of sub, super and equilibria in Arnold and Chueshov [1] in the stochastic case and Chueshov [11] in the non-autonomous case under stronger conditions. Definition 9 We say that the process {S(t, s) : X→X}t≥sis order-preserving if there exists an order relation ‘≤’ in Xsuch that, if u0≤v0,then S(t, s)u0≤S(t, s)v0,for all t≥s. 6 Definition 10 Let Sbe an order-preserving process. We call u(u) : R→Xa subtrajectory (super-trajectory) of Sif it satisfies S(t, s)u(s)≥u(t),for all t≥s(S(t, s)u(s)≤u(t),for all t≥s). Definition 11 We call the continuous map v:R→Xa complete trajectory if, for all s∈R,we have S(t, s)v(s) = v(t),for t≥s. From a sub and super-trajectory (u, u) of a process such that u(t)≤u(t),for all t∈R,we can define the “interval” Iu u(t) = {u∈X:u(t)≤u≤u(t)}. Clearly, it is a closed forward invariant set, i.e. S(t, s)Iu u(s)⊂Iu u(t),for all t≥s. The following result gives sufficient conditions for the existence of upper and lower asymptotically stable complete trajectories, giving some information on the structure of the non-autonomous attractor, adapting to our case the main results in Arnold and Chueshov [1] and Chueshov [11]. Note that we slightly generalize the results in [11] as we do not impose the set of parameters to be a compact set. Suppose the pullback attractor attracts time-dependent families of sets in a base of attraction D. Theorem 12 Let Sbe an order-preserving process and A(t)its associated pullback attractor. Let u, u be sub and super-trajectories such that u(t)≤u(t),for all t∈R,and Iu u(t)the corresponding associated interval, such that A(t)⊂Iu u(t),for all t∈Rand u, u∈ D.Suppose that there exists τ > 0such that S(τ+s, s)Iu u(s)is relatively compact, for all s∈R.Then, there exist complete trajectories u∗(t), u∗(t)∈ A(t)such that i) u(t)≤u∗(t)≤u∗(t)≤u(t),and A(t)⊂Iu∗ u∗(t),for all t∈R. ii) u∗(u∗) is minimal (maximal) in the sense that it does not exist any complete trajectory in the interval Iu∗ u(Iu u∗). iii) u∗(t) is globally asymptotically stable from below, that is, for all v∈ D with u(t)≤ v(t)≤u∗(t),for all t∈R, we have that lim s→+∞d(S(t, −s)v(−s), u∗(t)) = 0. 7 u∗(t) is globally asymptotically stable from above, that is, for all v∈ D with u∗(t)≤ v(t)≤u(t),for all t∈R, we have that lim s→+∞d(S(t, −s)v(−s), u∗(t)) = 0. Proof. Write an(t) = S(t, −nt0)u(−nt0), bn(t) = S(t, −nt0)u(−nt0).Then, we have that u(t)≤an(t)≤am(t)≤bm(t)≤bn(t)≤u(t),for all m > n. (1) Indeed, an(t) = S(t, −nt0)u(−nt0)≥u(t),since uis a sub-trajectory. Moreover, for s=σ+r, r > 0, as(t) = S(t, −st0)u(−st0) = S(t, −(σ+r)t0)u(−(σ+r)t0) =S(t, −σt0)S(−σt0,−(σ+r)t0)u(−(σ+r)t0)≥S(t, −σt0)u(−σt0) = aσ(t). On the other hand, we have an+1(t) = S(t, −(n+ 1)t0)u(−(n+ 1)t0) =S(t, t −t0)S(t−t0,−(n+ 1)t0)u(−(n+ 1)t0) = S(t, t −t0)an+1(t−t0), and so an+1(t)∈S(t, t −t0)Iu u(t−t0),for all n∈N.Thus, from (1) and the relative compactness of S(t, t −t0)Iu u(t−t0), there exists the following limit lim n→+∞an(t)$u∗(t). Clearly, u∗:R→Xis a complete trajectory, as, by the continuity of the process S(t, s), S(t, s)u∗(s) = S(t, s) lim n→+∞S(s, −nt0)u(−nt0) = lim n→+∞S(t, s)S(s, −nt0)u(−nt0) = lim n→+∞S(t, −nt0)u(−nt0) = u∗(t). We now prove that u∗(t), u∗(t)∈ A(t).Indeed, dist(S(t, s)u∗(s),A(t)) ≤d(S(t, s)u∗(s), S(t, s)u(s)) + dist(S(t, s)u(s),A(t)), and this right hand of the inequality tends to zero when s→ −∞.As S(t, s)u∗(s) = u∗(t), for all s∈R, u∗(t)∈ A(t). Is is also straightforward that, for all u(t)∈ A(t) u(t)≤u∗(t)≤u(t)≤u∗(t)≤u(t), by the definition of u∗and u∗,the invariance of A(t) and the order in X. 8 On the other hand, for any complete trajectory v(·) such that u(t)≤v(t)≤u∗(t),for all t∈R,and by the order in the process, u∗(t) = lim n→+∞S(t, −nt0)u(−nt0)≤lim n→+∞S(t, −nt0)v(−nt0) = v(t)≤u∗(t), so that v(t) = u∗(t),for all t∈R.Note that this implies that u∗and u∗are uniquely defined by the order in X. Finally, for iii), let be v∈ D with u(t)≤v(t)≤u∗(t),for all t∈R.Then, by the attraction property of A(t) u∗(t) = lim s→+∞S(t, −s)u(−s)≤lim s→+∞S(t, −s)v(−s)≤lim s→+∞S(t, −s)u∗(−s) = u∗(t). All these arguments also hold for u∗. Note that the same conclusions can be got under weaker hypotheses: Corollary 13 Let Sbe an order-preserving process and A(t)its associated pullback attractor. Let u, u ∈ D be such that u(t)≤u(t),for all t∈R,and assume that A(t)⊂Iu u(t),∀t∈R. Then there exists two trajectories u∗(t), u∗(t)∈ A(t)such that i) u∗(t)≤u≤u∗(t),∀t∈Rand ∀u∈ A(t). ii) u∗(u∗)is minimal (maximal) in the sense that it does not exist any complete trajectory in the interval Iu∗ u(Iu u∗). iii) u∗(t) is globally asymptotically stable from below and u∗(t) is globally asymptotically stable from above. Proof. Since Iu u(t)⊂ D, the attractivity property of A(t) implies that dist(S(t, −s)Iu u(−s),A(t)) →0,as s→+∞. Now, the compactness of A(t) and the order relation in Ximply that there exist u∗(t), u∗(t)∈ A(t) with lim s→+∞S(t, −s)u(−s) = u∗(t) and lim s→+∞S(t, −s)u(−s) = u∗(t) and the argument follows as in the previous theorem. 9 Remark 20 From (14) we conclude that the non-autonomous attractor A(t)attracts not only the “pullback pseudotrajectories” ∪s≤tS(t, s)u0,but we have a stronger attraction property: Consider the base of attraction D={v:R→Xcontinuous, such that, lim s→−∞ e2λs |v(s)|2 0 = 0} Then, we have that, given v∈ D, lim s→−∞ dist(S(t, s)v(s),A(t)) = 0.(15) Indeed, we have that for ssmall enough |u(t, s;v(s))|2 0≤e2λt e2λs |v(s)|2 s + 2 Rt se2λτ b(τ)dτ ≤r2 1(t). Note that every map v, with v(t)≡v0,for all t, is in D. 4.3 Upper semicontinuity of non-autonomous attractors to the global attractor Let bσbe a family of functions satistying (Hb) and Sσ(t, s) be the non-autonomous dynamical system associated to ut−∆u=λu −bσ(t)u3,lim σ&0bσ(t) = α > 0 (16) uniformly on bounded sets of t∈R, λ > λ1,defined as a small perturbation of the given semigroup S0associated to the autonomous equation ut−∆u=λu −αu3.(17a) Remark 21 Note that this holds, for example, for 0< bσ(t) = αe−σ|t|. The asymptotic behaviour of this autonomous logistic equation is very well known (cf., for example, Hale [14]). Indeed, it can be proved that (17) has a global attractor A,that is, a compact invariant set (S0(t)A=A,for all t≥0) attracting every bounded set in Xforward in time, i.e. limt→+∞dist(S(t)D, A)=0,for all D⊂Xbounded. Moreover, we can get some information about the structure of this set. Indeed, at the parameter value λ=λ1we find a pitchfork bifurcation, that is, it bifurcates from the zero solution a globally asymptotically stationary solution u+∈X, i.e. u+satisfies lim t→+∞|u(t;u0)−u+|0= 0,for all u0∈ V+. 16 Note that the asymptotic behaviour of (16) is rather different. However, we can apply a result on the upper semicontinuity of attractors associated to (16) and (17). Indeed, suppose, for all t > s, lim σ&0d(Sσ(t, s)u0, S0(t)u0) = 0 (h1) uniformly on bounded sets of X. On the other hand, suppose that there exist the pullback attractors Aσ(t) and A, associated to Sσand S0respectively, such that Aσ(t)⊂Kσ(t),A ⊂ K, where Kσ(t) and Kare compact absorbing sets associated to the corresponding flows, and satisfying lim σ&0dist(Kσ(t), K) = 0,for every t∈R.(h2) Then we have (Caraballo and Langa [7]) Theorem 22 Under the preceding assumptions (h1), (h2), it follows that, for all t∈R, lim σ&0dist(Aσ(t),A) = 0. It remains to prove that conditions (h1) and (h2) are satisfied in our case. Indeed, (h2) is a consequence of the expression for rσ 1(t),with rσ 1(t) the corresponding radius of the absorbing ball associated to (2) with bσ(t).From (14), lim σ&0rσ 1(t)2= lim σ&0 e2λt Rt −∞ e2λτ bσ(τ)dτ = 2λ α, which is independent of t∈R,so that the same is true for Rσ 1(t) and (h2) holds. On the other hand, (h1) is the content of the following lemma Lemma 23 Given uσ(t, s;u0), u(t;u0)solutions of (16) and (17) respectively with initial data uσ(s) = u(s) = u0,it holds that, for all t > s, lim σ&0|uσ(t, s;u0)−u(t;u0)|0= 0.(18) Proof. Since limσ&0bσ(t) = α > 0,for σsufficiently small, positive constants are supersolutions of (16), see (12), and so |uσ(t, s;u0)|0≤M(independent of σ.) (19) 17 If we call vσ(t, s;u0) = uσ(t, s;u0)−u(t;u0),and using the variation of constants formula, we have vσ(t, s;u0) = Zt s e−A(t−r)(λvσ(r, s;u0)+(α−bσ(r))u3 σ(r, s;u0) + α(u3 σ(r, s;u0)−u3(r;u0)))dr. Since ° °e−A(t−r)° °op ≤e−δ(t−r)≤1,we get |vσ(t, s;u0)|0≤λZt s |vσ(r, s;u0)|0dr +Zt s |α−bσ(r)|sup σ¯¯u3 σ(r, s;u0)¯¯0dr +3αZt s η2 σ(ξr)|vσ(r, s;u0)|0dr. Using now (19) and Gronwall’s lemma, we get (18). 4.4 Bifurcation scenario for positive solutions In this section we describe the changes in the asymptotic behaviour of the equation as the parameter value crosses λ1.For λ < λ1,note that the zero solution is globally asymptotically stable, in both the forward and the pullback sense. Indeed, from (8) we have that lim t→+∞|u(t, s;u0)|2 0= lim s→−∞ |u(t, s;u0)|2 0= 0. This means that, in this case, the non-autonomous attractor reduces to a fixed (not depending on time) point, i.e. A(t)≡ {0},for all t∈R. On the other hand, a nontrivial attractor exists for values of the parameter bigger than λ1.In particular, we prove that the attractor is bigger than the zero solution, i.e. {0} A(t). Proposition 24 Given u0∈C1 0(¯ Ω) strictly positive, λ > λ1and t∈R,there exists ε > 0 such that, for all s≤t |S(t, s)u0|0> ε. Proof. Since λ > λ1and u0∈C1 0(¯ Ω) is strictly positive, it is not hard to prove that u=εϕ1is a subsolution of (2) provided that εverifies 0< ε ≤min x∈¯ Ω u0(x) ϕ1(x)and ε2B2≤λ−λ1. So, taking εsufficiently small, we get εϕ1≤u(t, s;u0) whence the result follows. 18 Remark 25 Note that this implies that {0} A(t).In fact, there exists a subset of A(t) bounded away from zero “attracting” every u0∈C1 0(¯ Ω) strictly positive. 4.4.1 On the structure of the pullback attractor In this Section we apply the results of Section 3 (Corollary 13) to our equation (2). We take (u(t), u(t)) = (0, r1(t)).Observe that r1(t)∈ D, because lim s→−∞ e2λs r2 1(s)= 0. Moreover, A(t)⊂Ir1 0(t),by (14). So, applying Corollary 13, there exists a complete trajectory u∗(·) in the pullback attractor. Since A(t)6={0},we have that u∗(t)6={0},for all t∈R. Remark 26 Note that u∗(·)∈ A(·)is a very special complete trajectory. On the one hand, it is a maximal trajectory, an upper bound on the pullback attractor in the positive cone, that is, for all u(t)∈ A(t) u(t)≤u∗(t). On the other hand, it is globally asymptotically stable from above (cf. Theorem 12). Finally, we conclude Theorem 27 Assume (Hb) and λ > λ1.Then, (2) is permanent in a pullback sense. Proof. Given a bounded set D⊂Xand u0∈D, by (11) we have that 0< V (x)≤u(t, s;u0) for any s≤T(t, D). Thus, the permanence follows for U(t) = {u∈X:V(x)≤u≤r1(t)}. 19 5 Conclusions We have studied permanence for a general non-autonomous logistic equation. We allow the non-autonomous term to tend to zero, so that we have obtained some information on the transfer of stability at the parameter value λ=λ1from the zero solution to the pullback attractor. On the one hand, we have introduced in a general way the concepts of sub, super and trajectories to get some information on the structure of the pullback attractor of orderpreserving systems, finding u∗(t) as a maximal complete trajectory in the attractor with some properties of stability. We think that all this general framework could be appropiate to study bifurcation phenomena and permanence for other non-autonomous partial differential equations. But note that we do not have that u∗it is globally asymptotically stable, that is, for all u0∈ V+,lims→+∞|S(t, −s)u0−u∗(t)|0= 0.Observe that this being true would lead to a pitchfork bifurcation scenario, just as in the autonomous case for the logistic equation. But, on the other hand, it should be possible to define and apply to particular nonautonomous examples a general theory of stable and unstable manifolds, so that problems on the structure of attractors can be completed as it is known in the autonomous case (Hale [14]). 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