Extremal bounded complete trajectories for nonautonomous reaction-diffusion equations with discontinuous forcing term

Noname manuscript No. (will be inserted by the editor) Extremal bounded complete trajectories for nonautonomous reaction-diffusion equations with discontinuous forcing term Tom´as Caraballo ·Jos´e A. Langa ·Jos´e Valero Abstract In this paper we establish a strong comparison principle for a nonautonomous differential inclusion with a forcing term of Heaviside type. Using this principle, we study the structure of the global attractor in both the autonomous and nonautonomous cases. In particular, in the last case we prove that the pullback attractor is confined between two special bounded complete trajectories, which play the role of nonautonomous equilibria. Keywords: differential inclusions, reaction-diffusion equations, pullback attractors, nonautonomous dynamical systems, multivalued dynamical systems, structure, comparison of solutions. AMS Subject Classification (2010): 35B40, 35B41, 35B51, 35K55, 35K57 1 Introduction Comparison of solutions for reaction-diffusion equations is a powerful tool in order to study the structure of global attractors. In particular, in the autonomous case it allows us to establish that the global attractor is confined between two stationary solutions, which are the maximal and minimal elements of the attractor. For a class of autonomous reaction-diffusion equations, such result was proved in [5], [31]. It is worth noticing that a general theory of monotone random dynamical systems was This work has been partially supported by Spanish Miniterio de Econom´ıa y Competitividad and FEDER, projects MTM2015-63723-P and MTM2016-74921-P, and by Junta de Andaluc´ıa under Proyecto de Excelencia FQM-1492. T. Caraballo and J.A. Langa Dpto. Ecuaciones Diferenciales y An´alisis Num´erico Facultad de Matem´aticas, Universidad de Sevilla, C/ Tarfia s/n, 41012-Sevilla, Spain E-mail: [email protected], [email protected] J. Valero Centro de Investigaci´on Operativa, Universidad Miguel Hern´andez de Elche, Avda. de la Universidad, s/n, 03202-Elche, Spain E-mail: jv[email protected] 2 Tom´as Caraballo et al. first studied in [1], [2], [17]. These results were extended to multivalued autonomous dynamical systems in [11]. In nonautonomous problems the situation is more complicated because stationary solutions do not exist in general but only in rather particular cases, at least not in the classical sense. For this reason, we need to replace them by a special type of bounded complete trajectories, which play the role of “nonautonomous equilibria”. The general theory of order-preserving nonautonomous dynamical systems was studied in [16], [25]. In this sense, a result proved in [30] (see also [13], [25] and [29]) is remarkable because a complete bounded positive non-degenerate solution was constructed for a nonautonomous reaction-diffusion equation. Using this solution, a nonautonomous interval containing the pullback attractor is provided. In the multivalued nonautonomous framework, similar results were established in [12], where an ordinary nonautonomus differential inclusion was studied. We aim to study the structure of attractors for the following nonautonomous differential inclusion      ∂u ∂t −∂2u ∂x2∈b(t)H0(u) + ω(t)u, on (0,1) ×(τ, ∞), u(0, t) = u(1, t) = 0, u(x, τ) = uτ(x), (1) where H0is a Heaviside function. Problems of this type appear when we have a differential equation driven by a nonlinear function having a discontinuity, which can be rewritten as a differential inclusion by means of a Heaviside function. Well known applications like combustion in porous media [21], the conduction of electrical impulses in nerve axons (see [33], [34]) or the surface temperature on Earth (see [10], [20]) are modeled by inclusions of similar type. The structure of the global attractor for problem (1) in the autonomous case has been studied in detail in [4]. Nevertheless, several challenging problems still remain open. For models concerning the climate on Earth, some results about bifurcations of steady states were proved in [8], [9]. In the multivalued framework, that is, when more than one solution can exist for the Cauchy problem of a differential equation, it is not possible to compare solutions with ordered initial data in the same way as in the single-valued case. Instead, we need to establish some sort of order relationship between the set of solutions corresponding to the ordered initial conditions. In this sense, different definitions have been given in the literature. A strong comparison principle was defined and applied to ordinary differential equations with delays in [11]. A weak comparison principle was established in [38] for reaction-diffusion equations without uniqueness. Also, an intermediate comparison principle was given in [14] for differential inclusions governed by subdifferential maps. In this paper we firstly prove in the second section a strong comparison principle for the solutions of problem (1). Moreover, we obtain also strong comparison between positive solutions of (1) and its corresponding autonomous equation (that is, for b(t), ω (t)identically equal to constants). After that, in the third section, we use this comparison principle and the abstract results from [11] in order to establish that the global attractor of the autonomous problem (1) is confined between maximal and minimal stationary points. Moreover, the expressions of these fixed points are known from [4]. Extremal trajectories for equations with discontinuous forcing term 3 The main results of this paper, concerning the structure of the pullback attractor for the nonautonomous inclusion (1), are given in the fourth section. We prove the existence of two bounded complete trajectories that generate a timedepending interval containing the pullback attractor. These solutions are strictly positive (negative) for any time and any x∈(0,1), that is, they are non-degenerate. Moreover, they are the unique non-degenerate bounded complete trajectories of the problem and play the role of nonautonomous positive equilibria. 2 Comparison of solutions In this section we establish a strong comparison principle for the strong solutions of a nonautonomous differential inclusion in a bounded n-dimensional domain. Let Ω⊂Rnbe a bounded open subset with smooth boundary. We consider the problem      ∂u ∂t −∆u ∈b(t)H0(u) + ω(t)u, on Ω×(τ, ∞), u|∂Ω = 0, u(τ, x) = uτ(x), (2) where b:R→R+, ω :R→R+are continuous functions such that 0< b0≤b(t)≤b1,0≤ω0≤ω(t)≤ω1, and H0(u) =    −1,if u < 0, [−1,1],if u= 0, 1,if u > 0, is the Heaviside function. We rewrite (2) in the abstract form (∂u ∂t +∂ψ(u)−R(t, u(t)) 30, u(τ) = uτ, where ∂ψ is the subdifferential of the proper, convex, lower semicontinuous function ψ:L2(Ω)→(−∞,+∞] given by ψ(u)=1 2RΩ|∇u|2dx, if u∈H1 0(Ω), +∞, otherwise, ∂ψ(u) = ny∈L2(Ω) : y(x) = −∆u(x), a.e. on Ωo, D(∂ψ)=H2(Ω)∩H1 0(Ω)and for any t∈R, R(t, u) = ny∈L2(Ω):y(x)∈b(t)H0(u(x)) +ω(t)u(x), a.e. on Ωo. We note that in our particular case the operator ∂ψ :H2(Ω)∩H1 0(Ω)→L2(Ω) is single-valued and linear. In the sequel, as ∂ψ is also the generator of a C0- semigroup, for convenience we shall use the notation A=∂ψ. Also, we observe that D(ψ) = L2(Ω). 4 Tom´as Caraballo et al. Let us introduce some notation. Throughout this paper we denote by k·kX the norm in the Banach space X, whereas k·k,(·,·)will be used for the norm and scalar product in the space L2(Ω) (and with some abuse of notation also in (L2(Ω))d,d∈N). Also, P(X) will be the set of all non-empty subsets of Xand 2X=P(X)∪∅. The Hausdorff semidistance from the set Cto the set Bis given by dist(C, B) = sup y∈C inf z∈Bky−zk, whereas the Hausdorff distance is defined by distH(C, B) = max{dist(C, B), dist(B, C)}. For C⊂Xan ε-neighborhood is the set Oε(C) = {z∈X:dist(z, C)< ε}. For a multivalued map G:X→2Xwe denote D(G) = {u∈X:G(u)∈ P(X)}. The map Gis called upper semicontinuous if for any u∈D(G) and any neighborhood Oof G(u) there exists δ > 0 such that G(v)⊂Oas soon as ku−vk< δ. it is said to be w-upper semicontinuous if for all  > 0, u ∈D(G),there is δ > 0 such that G(v)⊂O(G(u)) if ku−vk< δ. Any upper semicontinuous map is w-upper semicontinous, the converse being true if Ghas compact values [3, p.45]. We recall the concept of strong solution for problem (2). Definition 1 We say that the function u∈C([τ, +∞), L2(Ω)) is a strong solution of (2) if: 1. u(τ) = uτ; 2. For any δ > 0, τ+δ < T,u(·) is absolutely continuous on [τ+δ, T] and u(t)∈D(A)for a.a. t∈(τ, T); 3. There exists a function r: [τ, +∞)→L2(Ω) such that r(t)∈R(t, u(t)), r∈ L2τ, T;L2(Ω)for any T > τ, and du dt +Au(t) = r(t),for a.a. t∈(τ, +∞),(3) where the equality is understood in the sense of the space L2(Ω). Lemma 1 For every strong solution of (2) the function r(·)belongs to L∞τ, T;L2(Ω) for any T > τ. Proof The statement follows readily from u∈C([τ, +∞), L2(Ω)), r∈L2τ, T;L2(Ω) and the inequality |r(t, x)| ≤ b(t)+ω(t)|u(t, x)|for a.a. (t, x). We denote by f:R×R→P(R) the multivalued function given by f(t, u) = b(t)H0(u).Then fpossesses nonempty, closed, bounded and convex values, and for all t∈Rthe map f(t, ·) : R→P(R) is upper semicontinuous. Moreover, for any t, s ∈R+,u∈R, distH(f(t, u), f(s, u)) = |b(t)−b(s)|, and sup y∈f(t,u)|y|=b(t). Extremal trajectories for equations with discontinuous forcing term 5 The last equality implies that, for all y∈L2(Ω) and for a.a. t∈R+, sup ξ∈R(t,y)kξk ≤ |Ω|1/2b(t). The following result follows from Lemma 6.28 in [24]. Lemma 2 The map Rsatisfies the following properties: 1. R:R×L2(Ω)→2L2(Ω)has nonempty, closed, bounded and convex values; 2. For any t∈R, the map R(t, ·) : L2(Ω)→P(L2(Ω)) is w-upper semicontinuous; 3. For all y∈L2(Ω),τ∈R, the map R(·, y):[τ, +∞)→P(L2(Ω)) possesses a measurable selection, that is, there exists a measurable function h: [τ, +∞)→ L2(Ω)such that h(t)∈R(t, y)for a.a. t > τ. Theorem 1 For any uτ∈L2(Ω), problem (2) has at least one strong solution. Proof If we fix an interval [τ, T], the existence of a strong solution follows from Theorem 6.11 and Lemma 6.16 in [24]. Also, adapting Lemma 6.31 in [24] to the nonautonomous case we obtain that the concatenation of two strong solutions is again a strong solution, so every solution in an interval [τ, T] can be extended to a global one, that is, defined for t∈[τ, +∞). Let us consider the auxiliary problem (du dt +Au(t) = g(t), t ∈(τ, T), u(τ) = uτ, (4) where g∈L1(τ, T;L2(Ω)). The continuous function u: [τ, T]→L2(Ω)is said to be a strong solution of (4) on [τ, T], if u(·) is absolutely continuous on any compact subset of (τ, T), u(t)∈D(A) for a.a. t∈(τ, T)and du dt +Au(t) = g(t)for a.a. t∈(τ, T). The continuous function u: [τ, +∞)→L2(Ω)is called in general a strong solution if it is a strong solution on every interval [τ, T]. Proposition 1 ([7, Theorem 3.6] or [6, p.189]) For any g(·)∈L2(τ, T;L2(Ω)),uτ∈ L2(Ω), there exists a unique strong solution of inclusion (2) on [τ, T]satisfying √tdu dt ∈L2(τ, T;L2(Ω)), ψ(u(·)) ∈L1(τ, T ).(5) Also, the map t7→ ψ(u(t)) is absolutely continuous on [τ+δ, T],for all 0< δ < T −τ. If, moreover, uτ∈ D(ψ),then du dt ∈L2τ, T;L2(Ω)and t7→ ψ(u(t)) is absolutely continuous on [τ, T]. 6 Tom´as Caraballo et al. Corollary 1 Every strong solution u(·)to problem (2) satisfies u∈L2τ, T;H1 0(Ω),(6) √tdu dt ∈L2(τ, T;L2(Ω)), du dt ∈L2(τ, T;H−1(Ω)), for all T > τ. Moreover: 1. The map t7→ ku(t)k2is absolutely continuous on every interval [τ, T]and d dt ku(t)k2= 2 du dt , u (t)for a.a. t∈(τ, T).(7) 2. The map t7→ k∇u(t)k2is absolutely continuous on every interval [τ+δ, T]with 0< δ < T −τ, d dt k∇u(t)k2= 2 du dt ,−∆u (t),for a.a. t∈(τ, T),(8) and u∈C([τ+δ, +∞), H1 0(Ω)).(9) 3. If uτ∈H1 0(Ω),then du dt ∈L2τ, T;L2(Ω)and t7→ k∇u(t)k2is absolutely continuous on every interval [τ, T].Also, u∈C([τ, +∞), H1 0(Ω)).(10) Proof Equality (7) follows from [15, p.285] and the rest of properties, except (8)- (10), are a consequence of Proposition 1. For the equality (8) see [6, p.189]. If uτ∈ H1 0(Ω), as t7→ u(t)is weakly continuous with respect to H1 0(Ω), t7→ ku(t)kH1 0(Ω) is continuous and H1 0(Ω)is a Hilbert space, property (10) follows. The proof for (9) is analogous. Our aim is to prove the following comparison principle. Definition 2 The strong solutions of problem (2) satisfy a strong comparison principle if for any initial data uτ≤vτthere exist strong solutions u(·), v (·)such that u(τ)=uτ, v (τ)=vτand u(t)≤v(t), u(t)≤v(t),∀t≥τ, where u(·), v (·)are arbitrary strong solutions of problem (2) such that u(τ)= uτ, v (τ)=vτ. Extremal trajectories for equations with discontinuous forcing term 7 In particular, this definition implies that for every initial data there exist a maximal and a minimal strong solution. Let us consider the following parabolic problems      ∂u ∂t −∆u =b(t)fε(u) + ω(t)u, on Ω×(τ, +∞), u|∂Ω = 0, u(τ, x) = uτ(x), (11)      ∂u ∂t −∆u =b(t)fε(u) + ω(t)u, on Ω×(τ, +∞), u|∂Ω = 0, u(τ, x) = uτ(x), (12) where fε, fε∈C1(R), f0 ε(u), f0 ε≥0, f0 ε(u),f0 ε(u)≤Cεfor all u, and fε(u) =    −1,if u≤ −ε, −1≤fε(u)≤1,if −ε < u < 0, 1,if u≥0, fε(u) =    −1,if u≤0, −1≤fε(u)≤1,if 0 < u < ε, 1,if u≥ε. It is straightforward that fε(u)≥sup y∈H0(v) yif u≥v, (13) fε(u)≤inf y∈H0(v)yif u≤v. With obvious little changes, we can extend the definition of strong solutions to problems (11)-(12). Let us show that these problems have a unique strong solution. Let us just consider problem (11). From now on, for v∈L2(Ω), we denote by fε(v) (fε(v)) the element y∈ L2(Ω))such that y(x)=fε(v(x)) (= fε(v(x))) for a.a. x. In the same way, for h∈L1(τ, T;L2(Ω)) we denote h(t):= h(t,·)∈L2(Ω)). We know from [15, p.283] that for any uτ∈L2(Ω) there exists a unique weak solution of problem (11), which means that u∈C([0,+∞), L2(Ω)), u∈ L2(τ, T;H1 0(Ω)), for all T > τ, and d dt (u(t), v)−h∆u, vi=(fε(u(t)), v),∀v∈H1 0(Ω), where the equality is understood in the sense of distributions on every interval (τ, T)and h·,·i is pairing between H−1(Ω) and H1 0(Ω). Lemma 3 For any uτ∈L2(Ω)there exists a unique strong solution uε(·)of problem (11), which coincides with the unique weak solution of (11). 8 Tom´as Caraballo et al. Proof Let uε(·)be the unique weak solution of problem (11). If we put gε(t) = fε(uε(t)) + ω(t)uε(t), then gε∈L2(τ, T;L2(Ω)), for all T > τ, and we can consider the linear problem      ∂z ∂t −∆z =gε(t),on Ω×(τ, +∞), z|∂Ω = 0, z(τ, x) = zτ(x). (14) On the one hand, uε(·)is the unique weak solution of problem (14). On the other hand, it follows from Proposition 1 that problem (14) possesses a unique strong solution z(·). If we were able to show that z(·)is also a weak solution of (14), then we would obtain that z=uε, so uε(·)would be a strong solution of (11). Indeed. In view of Proposition 1 we obtain that z∈L2(τ, T;H1 0(Ω)) for all T > τ. From the equality in (14) we infer then that dz dt ∈L2(τ, T;H−1(Ω)), which implies by [28, Lemma 7.4] that dz dt , v−h∆z, vi=(gε(t), v),∀v∈H1 0(Ω). Hence, by [32, p. 250] we have d dt (z, v)−h∆z, vi=(gε(t), v), so z(·)is a weak solution of (14). Thus, z=uεis a strong solution of problem (11). It remains to check uniqueness. Let v(·)be an arbitrary strong solution of (11). Then, it is a strong solution of problem (14) with gε(t) = fε(v(t)) + ω(t)v(t). By the previous argument v(·)is a weak solution of (14) and then a weak solution of (11) as well. Therefore, vis equal to uε,the unique weak solution of (11). Thus, v=z. Remark 1 The function hε(t) = b(t)fε(uε(t))+ω(t)uε(t)belongs to L∞τ, T;L2(Ω) for any T > τ. Corollary 2 For any uτ∈L2(Ω)the function uε∈C([τ, +∞), L2(Ω))is a strong solution to problem (11) if and only if it is a weak solution. It is well known [27, Chapter 7] that operator −A=∆u :D(A) = H2(Ω)∩ H1 0(Ω)→L2(Ω)is the generator of a strongly continuous semigroup of bounded linear operators S(t):L2(Ω)→L2(Ω),t≥0, which will be denoted in the sequel by S(t)=e−At. Moreover, it is a semigroup of contractions, that is,   e−At  ≤1. For every x∈D(A) the function u(t)=e−Atxis the unique classical solution (see the definition below) to the problem (du dt +Au(t) = 0, t > 0, u(0)=x. (15) Also, the semigroup e−At is positive for all t≥0 [13, Chapter 12]. Let us defined the concept of mild solution for the inhomogeneous problem (4). Extremal trajectories for equations with discontinuous forcing term 9 Definition 3 Let uτ∈L2(Ω)and g∈L1 loc(τ, +∞;L2(Ω)). Then the function u∈C([τ, +∞), L2(Ω)) is called a mild solution to problem (4) if u(t)=e−A(t−τ)uτ+Zt τ e−A(t−s)g(s)ds,τ≤t < ∞.(16) It is called a classical solution if u(·)is continuously differentiable on (τ, +∞), u(t)∈D(A) for any t∈(τ, +∞),u(τ)=xand the equality in (4) is satisfied for every t∈(τ, ∞). For every uτ∈L2(Ω)and g∈L1 loc(τ, +∞;L2(Ω)) there exists a unique mild solution to problem (4). Moreover, if uτ∈D(A)and gis continuously differentiable on [τ, +∞), then the mild solution is the unique classical solution [27, p.107]. We can also define mild solutions for problems (11) and (2). Definition 4 Let uτ∈L2(Ω). Then the function uε∈C([τ, +∞), L2(Ω)) is called a mild solution to problem (11) if uε(t)=e−A(t−τ)uτ+Zt τ e−A(t−s)(b(t)fε(uε(t)) + ω(t)uε(t))ds,τ≤t < ∞. We note that for any uε∈C([τ, ∞), L2(Ω)) the map fε(uε(t, ·)) belongs to L∞ loc 0,+∞;L2(Ω)⊂L2 loc 0,+∞;L2(Ω). Definition 5 Let uτ∈L2(Ω). Then the function u∈C([τ, +∞), L2(Ω)) is called a mild solution to problem (2) if there exists hsuch that h∈L2 loc 0,+∞;L2(Ω), for any T > τ, h (t, x)∈H0(u(t, x)) ,for a.a. (t, x), and u(t)=e−A(t−τ)uτ+Zt τ e−A(t−s)(b(t)h(t) + ω(t)u(t))ds,τ≤t < ∞. Lemma 4 For any uτ∈L2(Ω)and g∈L2 loc 0,+∞;L2(Ω)the function u∈ C([τ, +∞), L2(Ω))is a strong solution to problem (4) if and only if it is a mild solution. Proof The proof follows the same lines of [39, Lemma 2], but we provide it in detail for the sake of completeness. Let u(·)be a strong solution. We take sequences un τ∈D(A),gn(·)∈C1[τ, +∞), L2(Ω) such that un τ→uτin L2(Ω), gn→gin L2τ, T;L2(Ω)∀T > τ. Denote by un(·)the unique classical solution to the problem (dun dt =Aun(t) + gn(t), t > τ, un(τ)=un τ. Let T > τ and 0 < ε < T −τbe arbitrary. We note that wnis the unique strong solution to problem (4) with eg(t)=gn(t)−g(t)and w(τ)=un τ−uτ, so by (7) we have 2d dt (un−u), un−u=d dt  un(t)−u(t) 2for a.a. t∈(τ+ε, T). 16 Tom´as Caraballo et al. We now study a comparison principle between the solutions in the nonautonomus and autonomous cases. Let us consider the problem      ∂u ∂t −∆u ∈bH0(u) + ωu, on Ω×(τ, ∞), u|∂Ω = 0, u(τ, x) = uτ(x), (29) where 0 < b, 0 ≤ω. Let us denote a strong solution to problem (29) by ub,ω (·) (and we do not take into account in this notation the initial time τas the solution is the same whatever the value of τ). Theorem 3 For any initial datum uτ≥0there exists a non-negative solution ub1,ω1(·) to problem (29) with u(τ)=uτ, b =b1, ω =ω1such that v(t)≤ub1,ω1(t),∀t≥τ, (30) where v(·)is an arbitrary strong non-negative solution to (2) with u(τ)=uτ. On the other hand, there exist a non-negative solution u(·)to (2) with u(τ)=uτ such that u(t)≥ub0,ω0(t),∀t≥τ, (31) where ub0,ω0(·)is an arbitrary strong non-negative solution to (29) with u(τ)= uτ, b =b0, ω =ω0. Proof Let F:V→C([τ, τ +t0], L2(Ω)) be defined by F(u)(t) = e−A(t−τ)uτ+Zt τ e−A(t−s)(b1fε(u(s)) + ω1u(s))ds, (32) where V={u∈C([τ, τ+t0], L2(Ω)) : u(τ) = uτ, u (t)≥0,ku(t)−uτk ≤ 1,∀t∈[τ, τ+t0]}, and t0>0 satisfies (17). Let z0∈L2(Ω)be such that z0(x)= 1 for a.a. x. Since fε(u(s)) = z0, for any u∈V, and e−At ≥0, we have that F(u)(t)≥0.Then, arguing in the same way as in Theorem 2 we obtain that F:V→Vis a contracting map, so Fpossesses a unique fixed point u∈V, which is a mild solution and, by Lemma 4, coincides with the unique strong solution uεto problem (11) with b(t)=b1, ω (t)=ω1. Let v(·)be a non-negative strong solution of (2) such that v(τ)=uτ. Then there exists hsuch that h∈L∞τ, T;L2(Ω), for any T > τ,h(s, x)∈H0(v(s, x)) for a.a. (s, x), and v(·)is the unique strong solution to problem (19). Moreover, v(·) satisfies (20) and it is the unique fixed point of the contractive map F1:V1→V1 given by F1(u)(t) = e−A(t−τ)uτ+Zt τ e−A(t−s)(b(s)h(s) + ω(s)u(s))ds, where V1={u∈C([τ, τ +t0], L2(Ω)) : u(τ) = uτ,ku(t)−uτk ≤ 1,∀t∈[τ, τ +t0]}. Extremal trajectories for equations with discontinuous forcing term 17 Let us define the set b V={u∈V:u(t)≥v(t)}, which is obviously non-empty as v∈b V. For any u∈b V, since fε(u(·, s)) = z0and u(s)≥v(s)≥0, we have b1fε(u(s)) −b(s)h(s) = b1z0−b(s)h(s)≥(b1−b(s)) z0≥0, ω1u(s)−ω(s)v(s)≥0. Then, by e−At ≥0, we obtain F(u)(t)−v(t)=Zt τ e−A(t−s)(b1fε(u(s)) −b(s)h(s) + ω1u(s)−ω(s)v(s))ds ≥0, for any t∈[τ, τ +t0]. Therefore, F(b V)⊂b V. Since Fis a contraction in V, it is a contraction in b Vas well. We deduce that Fpossesses a unique fixed point u∈b V, which is equal to the solution uεto problem (11) with b(t)=b1, ω (t)=ω1. As before, by a standard continuation argument, it follows that uε(t)≥v(t)for any t≥τ. Passing to the limit in exactly the same way as in Theorem 2 we obtain the existence of a solution ub1,ω1(·)to problem (29) such that (30) holds. Let F2:V→C([τ, τ +t0], L2(Ω)) be defined by F2(u)(t) = e−A(t−τ)uτ+Zt τ e−A(t−s)(b(s)fε(u(s)) + ω(s)u(s))ds. (33) Let z0∈L2(Ω)be such that z0(x)= 1 for a.a. x. Since fε(u(s)) = z0, for any u∈V, and e−At ≥0, we have F(u)(t)≥0.Then, arguing in the same way as in Theorem 2 we obtain that F2:V→Vis a contracting map, so F2possesses a unique fixed point u∈V, which is a mild solution and, by Lemma 4, coincides with the unique strong solution uεto problem (11). Let ub0,ω0(·)be a non-negative strong solution of (29) such that ub0,ω0(τ)=uτ, b=b0, ω =ω0. Then there exists hsuch that h∈L∞τ, T;L2(Ω), for any T > τ, h(s, x)∈H0(ub0,ω0(s, x)) for a.a. (s, x), and ub0,ω0(·)is the unique strong solution to problem (19) with b(t)=b0, ω (t)=ω0. Moreover, ub0,ω0(·)satisfies (20) and it is the unique fixed point of the contractive map F3:V1→V1given by F3(u)(t) = e−A(t−τ)uτ+Zt τ e−A(t−s)(b0h(s) + ω0u(s))ds. We define the set b V1={u∈V:u(t)≥ub0,ω0(t)}, which is non-empty as ub0,ω0∈b V. As before, for any u∈b V1we have b(s)fε(u(s)) −b0h(s) = b(s)z0−b0h(s)≥(b(s)−b0)z0≥0, ω(s)u(s)−ω0ub0,ω0(s)≥0, 18 Tom´as Caraballo et al. and F2(u)(t)−ub0,ω0(t)=Zt τ e−A(t−s)b(s)fε(u(s)) −b0h(s) + ω(s)u(s)−ω0ub0,ω0(s)ds ≥0, for any t∈[τ, τ +t0]. Thus, F2(b V)⊂b V. Since F2is a contraction in V, it is a contraction in b Vas well, so F2possesses a unique fixed point u∗∈b V, which is equal to the solution uεto problem (11). As before, by a standard continuation argument, it follows that uε(t)≥ub0,ω0(t)for any t≥τ. Again, passing to the limit we obtain the existence of a solution u(·)to problem (29) such that (31) holds. 3 Characterization of the global attractor in the autonomous case In this section we will study the autonomous differential inclusion (2) in the scalar case and will deduce from the strong comparison principle some properties concerning the structure of the global attractor. Hence, we consider the autonomous problem      ∂u ∂t −∂2u ∂x2∈bH0(u) + ωu, on (0,1) ×(τ, ∞), u|∂Ω = 0, u(0, x) = u0(x), (34) where 0 < b, 0 ≤ω. We assume also throughout this section that 0≤ω < π2, where π2is the first eigenvalue of the operator −∂2 ∂x2in H1 0(0,1).This restriction is necessary in order to guarantee the existence of a global attractor. Let D(u0) denote the set of all strong solutions of (34) for u0∈L2(Ω) and τ= 0. We define the multivalued family of operators G:R+×L2(Ω)→P(L2(Ω)), where P(X) stands for the set of all non-empty subsets of the space X, by G(t, u0) = {u(t) : u(·)∈ D(u0)}. It is well-known [36] that Gis a strict multivalued semiflow, i.e., G(t+s, u0) = G(t, G(s, u0)), for all t, s ≥0, u0∈L2(Ω), possessing a global compact invariant attractor A. This means that: –A=G(t, A) for all t≥0 (strict invariance); –dist(G(t, B),A)→0 as t→+∞for any bounded set B(attracting property). Moreover, the attractor Ais connected [37] and bounded in H1 0(Ω) [36]. Also, Ghas compact values and the operator u07→ G(t, u0) is upper semicontinuous for any t≥0 (see [36] again). We recall the concept of order-preserving multivalued semiflow, which was introduced in [11]. Extremal trajectories for equations with discontinuous forcing term 19 Definition 6 The multivalued semiflow Gis called order-preserving if for any u0≤ v0and t≥0 we have: 1. There exists y∈G(t, u0) such that y≤yfor all y∈G(t, v0). 2. There exists y∈G(t, v0) such that y≤yfor all y∈G(t, u0). Theorem 2 implies that the semiflow Ggenerated by the solutions of (2) is order-preserving. We recall that xis called a fixed point (or equilibrium) of the multivalued semiflow Gif x∈G(t, x) for all t≥0. The set of strong solutions R=∪u0∈L2(Ω)D(u0) satisfies the following properties [18]: (H1)For any x∈L2(Ω) there exists ϕ∈ R such that ϕ(0)=x. (H2)ϕτ(·)=ϕ(·+τ)∈ R for any τ≥0, ϕ(·)∈ R (translation property). (H3)Let ϕ1, ϕ2∈ R be such that ϕ2(0) = ϕ1(s), where s > 0. Then the function ϕ(·),defined by ϕ(t) = ϕ1(t)if 0 ≤t≤s, ϕ2(t−s)if s≤t, belongs to R(concatenation property). (H4)For any sequence ϕn(·)∈ R such that ϕn(0)→ϕ0in L2(Ω), there exists a subsequence ϕnkand ϕ∈ R such that ϕnk(t)→ϕ(t),∀t≥0. The element xis called a fixed point (or equilibrium) of Rif ϕ(t)≡x∈ R. Since (H1)−(H4)hold, it is well-known that xis a fixed point of Gif and only if it is a fixed point of R[23, Lemma 7]. Applying Theorem 2 in [11] we obtain the following result. Theorem 4 There exist two equilibria x∗,y∗∈ A such that: 1. x∗≤z≤y∗for all z∈ A. 2. If the solutions corresponding to the initial conditions x∗,y∗are unique, then dist(G(t, u0), x∗)→0, as t→+∞, for any u0≤x∗,(35) dist(G(t, u0), y∗)→0, as t→+∞, for any u0≥y∗.(36) We observe that in [11] the following additional assumption concerning the order relation ’≤’ was assumed: for any bounded set Bthere exists a, d such that a≤y≤dfor all y∈B, which means that Bis contained in an interval [a, d]. Though this assumption is not true in the space L2(Ω), in the proof of Theorem 2 in [11] it is only necessary to use this property for the global attractor A, and not for an arbitrary bounded set B. Since Ais bounded in H1 0(Ω), which is continuously embedded in C([0,1]), 20 Tom´as Caraballo et al. the global attactor is in fact contained in an interval [a, d], so Theorem 2 in [11] is applicable. The fixed points of Gwere described explicitly in [4]. There exists an infinite but countable number of fixed points, denoted by v0= 0, v+ 1(x), v− 1(x), ..., v+ n(x), v− n(x), ..., where v± j(x) possess exactly j+ 1 zeroes in [0,1] and v+ j(x) = −v− j(x). The exact expression for the point v+ 1is given by v+ 1(x) = b0 ω0cos (√ω0x)+b0(1−cos (√ω0)) ω0sin (√ω0)sin (√ω0x)−b0 ω0 , which is the unique solution of the boundary-value problem u00 +ω0u=−b0, u (0)=u(1)= 0. From the analysis in [4] we infer that v− 1(x)≤v± j(x)≤v+ 1(x) for all j≥1, and then x∗=v− 1,y∗=v+ 1so Theorem 4 implies that v− 1≤z≤v+ 1for all z∈ A. Also, since the points v± 1are stable [4, Theorem 6.3], the solutions corresponding to the initial conditions v± 1are unique. Hence, the convergences (35), (36) hold true. Finally, we remark that in [4, Theorem 6.3] the structure of the global attractor was studied. We recall that a map φ:R→L2(Ω) is a complete trajectory if φ(·+h)|[0,∞)∈ R,for all h∈R. The global attractor Aconsists of all bounded complete trajectories and it consists if fact of the fixed points and all complete bounded trajectories ψ(·)connecting two fixed points, that is, ψ(t)→z1as t→+∞,(37) ψ(t)→z2as t→ −∞, where zjare fixed points. Partial results related to how the fixed points are connected were given in [4, Theorem 6.3]: if v zmeans that there is a connection from vto z, then: 1. v0 v± j,∀j≥1; 2. v+ j v± j−1,v− j v± j−1,∀j≥2; 3. v+ j v± 1, v− j e± 1,∀j≥2; 4. If v± k v± j(k, j 6= 0), then v± kn v± jn,∀n≥1; 5. If 1 ≤k≤j, then v± k v± j,v± k v0are forbidden. Now we have completed this description by showing that all these bounded complete trajectories lie inside the interval [v− 1, v+ 1]. Extremal trajectories for equations with discontinuous forcing term 21 4 Characterization of the pullback attractor in the nonautonomous case In this section we will treat the nonautonomous differential inclusion (2) in the scalar case, that is, we consider the nonautonomous problem      ∂u ∂t −∂2u ∂x2∈b(t)H0(u) + ω(t)u, on (0,1) ×(τ, ∞), u|∂Ω = 0, u(τ, x) = u0(x), (38) where b(t), ω (t), H0(u) are as in Section 2. Additionally, we assume in the sequel that ω1< π2. We start by defining a multivalued process associated with the strong solutions of problem (38) and showing that it possesses a pullback attractor. After that we will give a characterization of this attractor in a similar manner as in Theorem 4 for the autonomous case. In particular, we will show the existence of a bounded complete non-degenerate trajectory at −∞ which is unique in the class of nondegenerate bounded complete trajectories in the whole line. Let Dτ(uτ) be the set of all strong solutions of (38) with initial condition uτat time τand let Rτ=∪x∈L2(Ω)Dτ(x). In the same way as in the autonomous case, one can prove that the sets Rτsatisfy the following properties: (K1)For any τ∈Rand x∈L2(Ω) there exists ϕ∈ Rτsuch that ϕ(τ)=x. (K2)ϕs=ϕ|[τ+s,∞)∈ Rτ+sfor any s≥0, ϕ∈ Rτ(translation property). (K3)Let ϕ, ψ ∈ R be such that ϕ∈ Rτ,ψ∈ Rrand ϕ(s) = ψ(s) for some s≥r≥τ. Then the function θdefined by θ(t) := ϕ(t), t ∈[τ, s], ψ(t), t ∈[s, ∞), belongs to Rτ(concatenation property). (K4)For any sequence ϕn∈ Rτsuch that ϕn(τ)→ϕ0in L2(Ω), there exists a subsequence ϕnkand ϕ∈ Rτsuch that ϕnk(t)→ϕ(t),∀t≥τ. We define the multivalued family of operators U:R2 ≥×L2(Ω)→P(L2(Ω)), where R2 ≥={(t, s)∈R2:t≥s}, by U(t, s, x) = {u(t) : u(·)∈ Ds(x)}. It easily follows from (K1)−(K3)that Uis a strict multivalued process, that is, U(t, t, ·)=Id is the identity map and U(t, s, x)=U(t, τ, U (τ, s, x)) for all s≤τ≤t, x∈L2(Ω).Moreover, (K4)implies that the graph of the map x7→ U(t, s, x) is closed for all (t, x)∈Rd. We recall that the familiy of sets {K(t)}t∈Ris called pullback attracting for U if it attracts every bounded set Bin the pullback sense, that is, dist(U(t, s, B), K(t)) →0,as s→ −∞.(39) Lemma 5 The process Uhas a pullback attracting family of compact sets {K(t)}t∈R. 22 Tom´as Caraballo et al. Proof For any strong solution multiplying equality (3) by uwe have 1 2 d dt kuk2+   ∂u ∂x(t)    2 =Z1 0 r(t, x)u(t, x)dx ≤b(t)Z1 0|u(t, x)|dx +ω(t)ku(t)k2 ≤b1ku(t)k+ω1ku(t)k2 ≤b2 1 4ε0+(ε0+ω1)ku(t)k2, where ε0is chosen such that ε0+ω1< π2. Then, as π2is the first eigenvalue of the operator −∂2u ∂x2in H1 0(Ω), we obtain d dt kuk2+δku(t)k2≤d dt kuk2+δ π2   ∂u ∂x(t)    2 ≤b2 1 2ε0=C1, where δ= 2(π2−ω1−ε0)>0. By Gronwall’s lemma we get ku(t)k2≤e−δ(t−s)ku(s)k2+C1 δfor all t≥s. (40) Also, integrating over (t−α, t), where 0 < α ≤1,we have Zt t−α   ∂u ∂x    2 dr ≤π2C1 δ+π2 δku(t−α)k2.(41) Further, we multiply (3) by du dt and use Corollary 1 to obtain that    du dt     2 +1 2 d dt    ∂u ∂x    2 ≤b(t)Z1 0∂u ∂x(t, x)dx +ω(t)Z1 0 u(t, x)∂u ∂x(t, x)dx (42) ≤b1   ∂u ∂x (t)   +ω1ku(t)k   ∂u ∂x (t)    ≤b2 1 2+ω2 1 2ku(t)k2+   ∂u ∂x (t)    2 . For s≤t−α≤r≤twe integrate over the interval (r, t). Hence, by (40) and (41) we have    ∂u ∂x(t)    2 ≤   ∂u ∂x(r)    2 +b2 1+ω2 1Zt rku(τ)k2dτ + 2 Zt r   ∂u ∂x (τ)    2 dτ ≤   ∂u ∂x(r)    2 +b2 1+ (ω2 1+2π2 δ)e−δ(t−α−s)ku(s)k2+C1 δ(ω2 1+ 2π2+2π2 δ). Integrating now with respect to the variable rover the interval (t−α, t)and using again (40) and (41) we get α   ∂u ∂x(t)    2 ≤π2C1 δ+π2 δku(t−α)k2 +b2 1+ (ω2 1+2π2 δ)(e−δ(t−α−s)ku(s)k2) + C1 δ(ω2 1+ 2π2+2π2 δ) ≤C2+C3e−δ(t−α−s)ku(s)k2,(43) Extremal trajectories for equations with discontinuous forcing term 23 where C2, C3>0 are some constants. We define the family K(t) by K(t) = {y∈H1 0(Ω) : kyk2 H1 0≤2C2 α}. The compact embedding H1 0(Ω)⊂L2(Ω) implies that K(t) are relatively compact in L2(Ω). Also, as K(t) is weakly closed in H1 0(Ω), it is closed in L2(Ω). Thus, K(t) are compact in L2(Ω). Finally, we obtain readily from (43) that for any bounded set Band any t∈Rthere exists T(B, t) such that U(t, s, B)⊂K(t) for all s≤T(B, t). Thus, (39) follows. We recall the definition of pullback attractor. Definition 7 The family of compact sets {A(t)}t∈Ris called a pullback attractor if: 1. It is pullback attracting. 2. A(t)⊆U(t, s, A(s)), for all t≥s(negative semi-invariance); 3. {A(t)}t∈Ris minimal in the sense that if {K(t)}t∈Ris a pullback attracting family of closed sets, then A(t)⊂K(t) for all t∈R. The pullback attractor is strictly invariant if A(t) = U(t, s, A(s)), for any t≥s. Theorem 5 The process Upossesses a strictly invariant pullback attractor {A(t)}t∈R. Moreover, ∪t∈RA(t)is bounded in H1 0(Ω)and ∪t∈RA(t)is compact in L2(Ω). Proof Since there exists a pullback attracting family of compact sets {K(t)}t∈R and the map x7→ U(t, τ, x) has closed graph for all t≥τ(by (K4)), there exists a compact pullback attractor {A(t)}t∈R, which satisfies A(t)⊂K(t) for all t∈R (see Theorem 1 in [22]). Moreover, since A(t)⊂K(t) and the definition of K(t) we deduce that ∪t∈RA(t) is bounded in H1 0(Ω) and ∪t∈RA(t) is compact in L2(Ω). Using this and the fact that Uis a strict process we also obtain that the pullback attractor is strictly invariant (see Lemma 2.5 in [12] or Proposition 4.3 in [19]). A map γ:R→L2(Ω) is called a complete trajectory if ϕ=γ|[τ,+∞)∈ Rτ,for all τ∈R.(44) It is obvious that any complete trajectory satisfies γ(t)∈U(t, s, γ (s)) for all s≤t. (45) The complete trajectory γis said to be bounded if ∪r∈Rγ(r)is a bounded set. By the pullback attracting property and (45) it is easy to see that if γ(·)is a bounded complete trajectory, then γ(t)⊂ A(t) for any t∈R, where {A(t)}t∈Ris the pullback attractor. We have the following characterization of the pullback attractor. Lemma 6 A(t)={γ(t):γis a bounded complete trajectory}. 24 Tom´as Caraballo et al. Proof Since (K1)−(K4)are satisfied and ∪t∈RA(t) is bounded, the result follows either from [12, Corollary 2.10] or [12, Corollary 2.12]. We will also prove that the sets A(t) are compact in H1 0(Ω). Lemma 7 Let un τ→uτin L2(Ω)and let un∈ Dτ(un τ). Then there exists a subsequence {unk}and u∈ Dτ(uτ)such that unk→uin C([τ+r, T], H1 0(Ω)) for all 0< r < T −τ,T > τ. (46) Proof We know by (K4)that there exists u∈ Dτ(uτ) such that, up to a subsequence, un(t)→u(t)in L2(Ω) for any t≥τ. We need to prove that (46) holds for this solution u. We fix an interval [τ+r, T]. Taking in (43) s=τ,t−α=τ, we obtain a constant D1=D1(r) such that  un(t) H1 0(Ω)≤D1,∀t∈[τ+r, T].(47) By integrating (42) over (τ+r, T )and using (40) and (47) we have a constant D2=D2(τ, r, T ) satisfying ZT τ+r   dun dt     2 ds ≤D2.(48) Hence, Ascoli-Arzel´a theorem implies, passing to a subsequence, that un→uin C([τ+r, T], L2(Ω)). Also, from (48) and equality (3) we get a constant D3=D3(τ, r, T ) such that ZT τ+r   ∂2un ∂x2    2 ds ≤D3. These inequalities and the Compactness Theorem [26, p.58] imply that un→uweakly star in L∞(τ+r, T;H1 0(Ω)), un→uweakly in L2(τ+r, T;H2(Ω)), dun dt →du dt weakly in L2(τ+r, T;L2(Ω)), un→ustrongly in L2(τ+r, T;H1 0(Ω)), un(t)→u(t) in H1 0(Ω) for a.a. t∈(τ+r, T). In view of (9) we obtain also that un, u ∈C([τ+r, T ], H1 0(Ω)). Now, making use of (42) and (47) we deduce the existence of D4=D4(r) satisfying  un(t) 2 H1 0(Ω)≤ un(s) 2 H1 0(Ω)+D4(t−s),for τ+r≤s≤t≤T, and the same inequality is true for the limit function u. Hence, the functions Jn(t) = kun(t)k2 H1 0(Ω)+D4t,J(t) = ku(t)k2 H1 0(Ω)+D4tare continuous and nonincreasing in the interval [τ+r, T ]. Moreover, Jn(t)→J(t) for a.a. t∈(τ+r, T ). Extremal trajectories for equations with discontinuous forcing term 25 Let tn∈[τ+r, T] and tn→t0∈(τ+r, T]. We choose tm∈(τ+r, t0) such that tm→t0and Jn(tm)→J(tm) for each m. It is important to observe that when we fix mthe elements tnare greater than tmfor nbig enough. By the above properties we have Jn(tn)−J(t0) = Jn(tn)−Jn(tm) + Jn(tm)−J(tm) + J(tm)−J(t0) ≤Jn(tm)−J(tm) + J(tm)−J(t0)≤ε, if n≥N(ε, m(ε)), where ε > 0. We infer that lim sup ku(tn)k2 H1 0(Ω)≤ ku(t0)k2 H1 0(Ω). Since u(tn)→u(t0) weakly in H1 0(Ω), then lim inf ku(tn)k2 H1 0(Ω)≥ ku(t0)k2 H1 0(Ω), so lim ku(tn)k2 H1 0(Ω)=ku(t0)k2 H1 0(Ω)and thus un(tn)→u(t0)in H1 0(Ω). As this argument is valid also in the interval [τ+r 2, T], this convergence holds for tn→τ+ras well. Using a standard diagonal procedure we obtain that (46) is true. Corollary 6 The sets A(t)are compact in H1 0(Ω). Proof Let yn∈ A(t), t∈R. Since A(t) is compact in L2(Ω), up to a subsequence yn→yin L2(Ω). The invariance of A(t) implies the existence of solutions un(·)∈ Rt−1such that un(t) = ynand un(t−1) ∈ A(t−1). Again, passing to a subsequence un(t−1)→uin L2(Ω). Hence, by Lemma 7 we obtain the existence of u(·)∈ Dt−1(u) such that yn=un(t)→u(t)in H1 0(Ω). This proves that the sets A(t) are relatively compact in H1 0(Ω). As they are closed in L2(Ω), so they are in H1 0(Ω). Thus, A(t) are compact in H1 0(Ω). Further we are going to give a deeper characterization of the pullback attractor by showing that any bounded complete trajectory is contained in an interval defined by two special bounded complete trajectories. Let w+ bi,ωi,i= 0,1, denote the positive fixed point v+ 1of problem (29) for the parameters b=bi,ω=ωi. Theorem 6 There exists a bounded complete trajectory ξMsuch that any complete bounded trajectory γsatisfies −ξM(t)≤γ(t)≤ξM(t)for all t∈R.(49) Moreover, w+ b0,ω0≤ξM(t)≤w+ b1,ω1,(50) −ξM(t)≤y≤ξM(t)for all y∈ A(t),(51) −ξM(t)≤lim inf s→−∞ u(t)≤lim sup s→−∞ u(t)≤ξM(t),(52) uniformly for u∈ Ds(uτ),uτ∈B, where Bis bounded.