A Delay Nonlocal Quasilinear Chafee–Infante Problem: An Approach via Semigroup Theory

A DELAY NONLOCAL QUASILINEAR CHAFEE-INFANTE PROBLEM: AN APPROACH VIA SEMIGROUP THEORY TOM´ AS CARABALLO, A. N. CARVALHO AND YESSICA JULIO Abstract. In this work we study a dissipative one dimensional scalar parabolic problem with non-local nonlinear diffusion with delay. We consider the general situation in which the functions involved are only continuous and solutions may not be unique. We establish conditions for global existence and prove the existence of global attractors. All results are presented only in the autonomous since the non-autonomous case follows in the same way, including the existence of pullback attractors. A particularly interesting feature is that there is a semilinear problem (nonlocal in space and in time) from which one can obtain all solutions of the associated quasilinear problem and that for this semilinear problem the delay depends on the initial function making its study more involved. 1. Introduction Reaction-diffusion equations with non-local terms have attracted great attention during the last twenty years. A few representative references are [11, 19, 10, 12, 17, 3, 7, 1]). In order to explain the problems we wish to consider, let us start with an example of the type of models we have in mind, that is, consider the following quasilinear initial value problem with delay: (1)            ∂w ∂τ −a(l(w))∂w2 ∂x2=λf(w) + γw(τ−ρ) + h(τ), τ > 0, x ∈Ω, w(τ, 0) = w(τ, 1) = 0, w(τ) = φ(τ), τ ∈[−ρ, 0], where Ω = (0,1), γ, λ, ρ > 0, la continuous operator from H1 0(Ω) into R+,a∈C(R) with a(R)⊂[m, M]⊂(0,∞), f∈C(R,R), h∈C(R, L2(0,1)) is bounded and φ∈C([−ρ, 0], H1 0(Ω)). The local problem, i.e. a≡1, without delay, has first shown to have very interesting properties in 1974 in the seminal works of N. Chafee and E. Infante (see [9, 8]). Through the work of many authors, this has become the best understood infinite dimensional dynamical system (see, for example, [14, 13] for the autonomous case and [2] for the non-autonomous case). 1991 Mathematics Subject Classification. 35Q30, 35B41, 35K58, 76D05. Key words and phrases. non-local quasilinear parabolic problems with delay without uniqueness, existence and regularity of solutions, comparison results, multivalued processes, global attractors, uniform bounds. [TC] Partially supported by the Spanish Ministerio de Ciencia e Innovaci´on (MCI), Agencia Estatal de Investigaci´on (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) under the project PID2021-122991NB-C21. [ANC] Partially supported by FAPESP Grant # 20/14075-6 and by CNPq Grant # 308902/2023-8, Brazil. [YJ] Partially supported by CAPES Grant # 88887.695331/2022-00 and by the Colombian Ministerio de Ciencia, Tecnolog´ıa e Innovaci´on (Minciencias). 2 TOM´ AS CARABALLO, A. N. CARVALHO AND YESSICA JULIO For the non-local problem (anon-constant), without delay, many interesting new features have been discovered relative to what was known for the local case making this problem a very interesting one from the point of view of dynamics (see for example [1, 7, 17, 3]). Of course, nonlocal problems are quite challenging from the analytical point of view making any new discovery even more interesting. In [5] we have dealt with a prototype of non-autonomous scalar one dimensional parabolic problem with the non-local nonlinear diffusion being only continuous and through the semigroup theory. The introduction of the time variable makes the problem quite challenging and interesting already. In this paper we go one step further considering models similar to those treated in [5] with delay. As we will see next, this brings up a new and interesting feature to the problem, that is, the initial value problem has to be considered with a delay depending on the initial function. To obtain the local existence and regularity of mild solutions (continuous in time functions taking value in a suitable phase space and satisfying the variation of constants formula) for (1) we will use the results of [5] but, to that end, we will need to deal with the very interesting new feature of problems with delay depending on the initial function φ. To obtain regularity, some additional assumption is needed on f,a◦l(as in [5]) but also on the initial function φ. To ensure that solutions are globally defined and to be able to apply the method of steps we impose the structural condition (S) Assume that there exist C0, C1∈Rsuch that uf(u)⩽−νC0u2+|u|C1 for all u∈Rand for both ν=m λand ν=M λ. Finally, to obtain the existence of a global attractor we assume, the dissipativity condition (D) Assume that (S) holds for some C0such that the first eigenvalue ωof A+νC0Iis positive and satisfies e−ωρ/m +γ ωm <1. The nonlinear nonlocal diffusion a(l(·)) : H1 0(0,1) →[m, M] makes the above problem a nonlocal (in space) quasilinear problem. Our aim will be to establish a general local existence and regularity result for solutions to (1), prove that if condition (S) is satisfied, solutions are globally defined and, if condition (D) is satisfied, the multivalued semiflow associated to (1) has a global attractor. We also use comparison results to obtain uniform bounds for the solutions in the global attractor. We could work with the nonlinearity as in (1) (being time dependent) but all results would have identical proofs and therefore we have decided to consider only the case h≡0. It will be clear from the proofs that adding a bounded continuous function h:R→L2(0,1),or even more general non-autonomous nonlinearities, will not change the proofs so we choose to omit it for the sake of simplicity in the notation. A DELAY NONLOCAL QUASILINEAR CHAFEE-INFANTE PROBLEM 3 Before we proceed, let us work a little more with the model (1) in order to understand the interesting new feature it brings. Given a solution w: [−ρ, ∞)→H1 0(0,1) of the problem (1), making t:= α−1 (τ) = Zτ 0 a(l(w(r)))−1dr,τ∈[−ρ, ∞), we have that the function udefined by u(t) = w(τ) will be a solution of the problem (2)                    ut=uxx +λf(u) + γu(α(t)−ρ) a(l(u)) , t > 0, x ∈Ω, u(t, 0) = u(t, 1) = 0, u(t) = φ(α0(t)), t ∈[α−1(−ρ),0], where α−1(−ρ)=Z−ρ 0 a(l(φ(r)))−1 dr<0 and, since a(R)⊂[m, M]⊂(0,∞), ρ M⩽−α−1(−ρ)⩽ρ m. We note that, if τ⩾0 α(t) = Zt 0 a(l(u(r))dr and α0: [α−1(−ρ),0] →[−ρ, 0] is the only solution of the integral equation α(s) = Zs 0 a(l(φ◦α(r)))dr, s ∈(α−1(−ρ),0]. On the other hand, given φ∈C(−ρ, 0], H1 0(Ω)) we define α0and α−1(ρ) as above. If u: [α−1(−ρ),∞)→H1 0(0,1) is a solution of the semilinear delay differential problem (2), making τ=α(t) = Zt 0 a(l(u(r)))dr, the function w(τ) := u(t) will be a solution of problem (1). Problem (2) is a non-local (in time and in space) non-autonomous semilinear problem. Even though, for a fixed initial condition φ∈C([−ρ, 0], H1 0(Ω)), the solution of the problem (2) may not be unique, the delay is always the same, determined by the initial function φ. In fact, it is a striking feature of this model that α0and α−1(ρ) are uniquely determined by φonly and that, to solve the (1) with initial function φ, corresponds to solve the nonlocal non-autonomous semilinear delay differential problem (2) with delay determined by φ. With this, we will prove the existence of solutions to problem (2) and those will give us solutions for (1). We will use the method of steps which consists of solving the problem iteratively, in intervals of time of length α−1(ρ) = Z0 −ρ a(l(φ(r)))−1 dr. In each step we will apply the results obtained in [5] for the semilinear non-local (in time and in space) non-autonomous problem (2). As in [5], we will proceed as abstract as possible in order that the theory can be applied to other similar models with little effort. Let us now recall the results of [5]. For a Banach space X, let C(X) (L(X)) denote the space of continuous (linear continuous) transformations from Xinto itself. For u∈C([0, T ], Xα), let 4 TOM´ AS CARABALLO, A. N. CARVALHO AND YESSICA JULIO ut(·) = u(·)[0,t],t∈[0, T]. Consider the Cauchy problem (3)          du(t) dt +Au(t) = g(t, ut(·)) t > 0, u(0) = w0∈Xα, where −Ahas compact resolvent and is the infinitesimal generator of an exponentially decaying analytic semigroup {e−At :t⩾0},Xα, α ⩾0, is the fractional power spaces associated to A [14, 16], and g: [0, T]×C([0, T], Xα)→Xis a continuous function. With this preliminaries, we introduce the definitions of strong and mild solution for (3). Definition 1.1. [20] A function u: [0, T )→Xis a strong solution of (3) on [0, T),if uis continuous on [0, T )and continuously differentiable on (0, T ), u(t)∈D(A)for 0< t < T, u(0) = w0and (3) is satisfied on [0, T ). Definition 1.2. [20] Let Abe the infinitesimal generator of an analytic semigroup S(t),w0∈Xα and g: [0, T)×C([0, T], Xα)→X. A function u∈C([0, T ]; Xα)such that (4) u(t) = S(t)w0+Zt 0 S(t−s)g(s, us(·))ds, 0⩽t⩽T, is called a mild solution of the initial value problem (3) on [0, T]. We summarize the results of [5] next. Theorem 1.3 ([5]).Let Xbe a Banach space, Aa sectorial operator such that (λ+A)−1is compact for all λ∈ρ(−A), and let {S(t) : t⩾0}be the analytic semigroup generated by −A. If g: [0, T]×C([0, T], Xα)→X, 0⩽α⩽1is a continuous map, then for each w0∈Xα there exists a T1=T1(w0)∈(0, T ]such that the initial value problem (3) has a mild solution u∈C([0, T1]; Xα). Furthermore, T1may be chosen uniformly for w0in bounded subsets of Xα. In addition, if g: [0, T ]×C([0, T ], Xα)→X, 0⩽α⩽1, is such that, given u∈C([0, T ], Xα) (5) kg(t, ut(·)) −g(s, us(·))kX⩽w(ku(t)−u(s)kα) + w(|t−s|β),0< β < 1−α, where w: [0,∞)→[0,∞)is an increasing continuous function such that w(0) = 0 and (6) Zt 0 u−1w(uβ)du < ∞. A continuous function u: [0, T1]→Xαsatisfying u(t) = S(t)w0+Zt 0 S(t−s)g(s, us(·))ds, 0⩽t⩽T1, is a strong solution of (3). Next we will show how to apply the results in [5] to establish existence and regularity of solutions for (2). Of course, since f:R→R,a:R+→[m, M]⊂(0,∞), l:H1 0(0,1) →Rand φ∈C([−ρ, 0], H1 0(0,1)) we have that, for T∈[0, α−1(ρ)), g(t, ut(·))(x) = λf(u(t)(x)) + γφ(α(t)−ρ) a(l(ut(t))) A DELAY NONLOCAL QUASILINEAR CHAFEE-INFANTE PROBLEM 5 is a continuous map from [0, T ]×C([0, T ], H1 0(0,1) into X=L2(0,1). The only point that has to be analyzed more carefully is the continuity of [0, α−1(ρ)] 3t7→ φ(α(t)−ρ)∈L2(0,1), but that follows from the continuity of φand from the fact that |α(t)−α(t0)|=|Zt t0 a(l(u(r)))dr|t→t0 −→ 0. It follows that the following theorem holds. Theorem 1.4. Let X=L2(0,1),D(A) = H2(0,1) ∩H1 0(0,1) and A:D(A)⊂X→Xbe the operator defined by Au =uxx. Then −Ais positive and self-adjoint (hence sectorial), with fractional power spaces Xα:= D(−Aα)with the graph norm, α⩾0,X1 2=H1 0(0,1),(λ+A)−1 has compact resolvent, λ∈ρ(−A)and the semigroup {S(t) : t⩾0}generated by Ais a compact and exponentially decaying analytic semigroup. If f:R→Ris a continuous function satisfying (S), given φ∈C([−ρ, 0], H1 0(0,1)) the initial value problem (2) has a mild solution u∈C([0,∞); H1 0(0,1)), furthermore u∈C([0,∞); Xα), for all α∈(0,1). For regularity, we assume also that l:H1 0(0,1) →Ris continuous and that f:R→R, a:R+→[m, M], l:H1 0(0,1) →Rand φ: (−ρ, 0] →H1 0(0,1) satisfy |f(s)−f(s0)|⩽w(|s−s0|) |a(l(u)) −a(l(v))|⩽w(ku−vkH1 0(0,1)), |h(s)−h(s0)|⩽w(|s−s0|β), kφ(t)−φ(t0)kX⩽w(|t−t0|β), (7) where w: [0,∞)→[0,∞) is a continuous increasing function, w(0) = 0 and Z1 0 u−1w(uβ)du < ∞,for some β∈(0,1−α). We only need to pay attention to the function [0, α−1(ρ)] 3t7→ φ(α(t)−ρ)∈L2(0,1) and check that, under the above conditions, this function satisfies the conditions of Theorem 1.3 for the existence of a mild solution. That follows from kφ(α(t)−ρ)−φ(α(t0)−ρ)kX⩽w(|α(t)−α(t0)|) =w(|Zt t0 a(l(u(r)))dr|)⩽w(M|t−t0|). Theorem 1.5. Under the assumptions of Theorem 1.4 and that (7) is satisfied, the initial value problem (1) has a strong solution w∈C([0,∞); H2(0,1) ∩H1 0(0,1)). 2. Existence of Solutions Proof of Theorem 1.5. We are going to use the method of steps. Let us review the reasoning and the needed details to apply Theorem 1.3. We know that w(τ) = φ(τ) for τ∈(−ρ, 0]. Then, for τ∈(0, ρ] , τ−ρ∈(−ρ, 0] and then we have w(τ−ρ) = φ(τ−ρ), so that equation (1) 6 TOM´ AS CARABALLO, A. N. CARVALHO AND YESSICA JULIO becomes the following non-autonomous quasilinear non-local scalar one-dimensional parabolic partial differential equation (8)            ∂w ∂τ =a(l(w))∂w2 ∂x2+λf(w) + γφ(τ−ρ), τ > 0, x ∈Ω, w(τ, 0) = w(τ, 1) = 0, w(0) = φ(0). One can perform a change in the time scale in order to obtain the semilinear problem (9)            ut=uxx +λf(u) + γφ(α(t)−ρ) a(l(u)) , t > 0, x ∈Ω, u(t, 0) = u(t, 1) = 0, u(0) = φ(0), where τ=Zt 0 a(l((u(r)))dr =: α(t). It is clear, from the discussion in the Introduction, that this initial value problem has at least a mild solution that we will call g1(see [5]). Under Assumption (S), since [0, α−1(ρ)] 3t7→ φ(α(t)−ρ)∈H1 0(0,1) is continuous, this solution exists in the interval [0, α−1(ρ)]. Thus, we just need to check regularity. To that end, set f1: [0, α−1(ρ)] ×C([0, α−1(ρ)], Xα)→Xdefined by f1(t, ut(·)) = λf(u(t)) + γφ(α(t)−ρ) a(l(u(t))) , α(t) = Zt 0 a(l(u(θ)))dθ. Then, for g1(t), g1(s)∈V, for Va neighborhood of φ(0),we have that kf1(t,gt 1(·))−f1(s, gt 1(·))k ⩽    λf(g1(t))+γφ(α(t)−ρ) a(l(g1(t))) −λf(g1(s))+γφ(α(s)−ρ) a(l(g1(s))) ±λf(g1(s))+γφ(α(s)−ρ) a(l(g1(t)))     ⩽m−1kλ(f(g1(t)) −f(g1(s)))k+m−1kγ(φ(α(t)−ρ)−φ(α(s)−ρ))k +k(λf(g1(s)) + γφ(α(s)−ρ))k a(l(g1(s))) −a(l(g1(t))) a(l(g1(t)))a(l(g1(s)))  . (10) Since for t>swe have |α(t)−α(s)|=Zt s a(l(g1(θ)))dθ ⩽M|t−s|. (11) It follows that (12) kf1(t, gt 1(·)) −f1(s, gs 1(·))k⩽K1w(kg1(t)−g1(s)kα) + K2w(c|t−s|β), for some positive numbers K1, K2and c, and since Zt 0 u−1w(uβ)du < ∞, for 0< β < 1−α, we can apply Theorem 1.3 and conclude that g1is a strong solution to equation (9) in the interval [0, α−1(ρ)]. Also w(τ) = g1(t) is a solution of (8) in the interval [0, ρ]. A DELAY NONLOCAL QUASILINEAR CHAFEE-INFANTE PROBLEM 7 Now, for τ∈[ρ, 2ρ], τ −ρ∈[0, ρ] and we have that w(τ) = u(α−1(τ)) = g1(τ), so equation (1) becomes (13)                    ∂w ∂τ =a(l(w))∂w2 ∂x2+λf(w) + γg1(τ−ρ), τ > ρ, w(τ, 0) = w(τ, 1) = 0, w(ρ) = g1(ρ). Again, by making a change in the time scale we have the initial value problem (14)                      ∂u ∂t =∂u2 ∂x2+λf(u) + γg1(α(t)−ρ) a(l(u)) , t ∈(α−1(ρ), α−1(2ρ)], u(t, 0) = u(t, 1) = 0, u(α−1(ρ)) = g1(α−1(ρ)). Since g1∈C((0, α−1(ρ)], Xα), we have that the application f1(t, u(t), u(·)) = λf(u(t)) + γg1((α(t)−ρ)) a(l(u(t)) is continuous as long as α(t)∈[ρ, 2ρ], which means that t∈[α−1(ρ), α−1(2ρ)]. Thus, again we just need to check the regularity of the solution. Hence, we need to see that kf1(t, ut(·)) −f1(s, us(·))k⩽w(ku(t)−u(s)kα) + w(|t−s|β), but that follows exactly as before. To proceed, we make g2(τ) = u(α−1(τ)) for τ∈[ρ, 2ρ]. Continuing in this way, if gn−1(τ) = u(α−1(τ)) for τ∈[(n−2)ρ, (n−1)ρ] we have, for n⩾3, (15)            ∂w ∂τ −a(l(w))∂w2 ∂x2=λf(w) + γgn−1(τ−ρ), τ ∈((n−1)ρ, nρ], w(τ, 0) = w(τ, 1) = 0, w((n−1)ρ) = gn−1((n−1)ρ). has a mild solution gn: [(n−1)ρ, nρ]→H1 0(0,1). Now if w:[−ρ, ∞)→H1 0(0,1) is given by (16) w(τ) =    φ(τ), τ ∈[−ρ, 0]; gn(τ), τ ∈((n−1)ρ, nρ], n ⩾1, it is a strong solution of (1). Using the variation of constants formula we have g1(τ) = u(t) = S(t)φ(0) + Zt 0 S(t−s)λf(g1(α(s))) + γφ(α(s)−ρ) a(l(g1(α(s)))) ds, 0⩽t⩽α−1(ρ) g2(τ) = u(t) = S(t−α−1(ρ))g1(ρ) + Zt α−1(ρ) S(t−s)(λf(g2(α(s))) + γg1(α(s)−ρ)) a(l(g2(α(s)))) ds, for t∈[α−1(ρ), α−1(2ρ)]. We know that g1(ρ) = S(α−1(ρ))φ(0) + Zα−1(ρ) 0 S(α−1(ρ)−s)λf(g1(α(s))) + γφ(α(s)−ρ) a(l(g1(α(s)))) ds, 8 TOM´ AS CARABALLO, A. N. CARVALHO AND YESSICA JULIO then, we obtain that S(t−α−1(ρ))g1(ρ) = S(t)φ(0) + Zα−1(ρ) 0 S(t−s)λf(g1(α(s))) + γφ(α(s)−ρ) a(l(g1(α(s)))) ds and g2(τ) = u(t) = S(t)φ(0) + Zα−1(ρ) 0 S(t−s)λf(g1(α(s))) + γφ(α(s)−ρ) a(l(g1(α(s)))) ds +Zt α−1(ρ) S(t−s)(λf(g2(α(s))) + γg1(α(s)−ρ)) a(l(g2(α(s)))) ds. Similarly, for n > 1 we obtain that gn(τ) = u(t) = S(t)φ(0) + n−1 X m=1 Zα−1(mρ) α−1((m−1)ρ) S(t−s)(λf(gm(α(s))) + γgm−1(α(s)−ρ)) a(l(gm(α(s)))) ds +Zt α−1((n−1)ρ) S(t−s)(λf(gn(α(s))) + γgn−1(α(s)−ρ)) a(l(gn(α(s)))) ds.  3. Comparison results and global existence Throughout this section, φis the initial function of problem (1), K > 0 is such that −K⩽ φ(t)⩽Kand “ ⩽” is a partial ordering in H1 0(Ω), that is: u⩽vin H1 0(Ω) ⇔u(x)⩽v(x) a.e. for xin Ω. We consider the initial value problems : ∂u ∂t =Au+λf(u)−K M, t>0, x∈Ω, u(0) = −K, u|∂Ω= 0, (17) ∂u ∂t =Au+λf(u)+K m, t>0, x∈Ω, u(0) = K, u|∂Ω= 0, (18) and we write f−(u) = λf(u)−γK M, uf−(u)⩽νC0u2+γK M+C1|u|, f+(u) = λf(u) + γK m, uf+(u)⩽νC0u2+γK m+C1|u|, g(t, ut(·)) = λf(u(t)) + γu(Rt 0a(l(u(θ))) −ρ)dθ a(l(u(t))) , where A:D(A)→L2(Ω) is the linear operator defined in the following way, D(A) = H2(Ω) ∩ H1 0(Ω) and Au =uxx, u ∈D(A); f:R→Rand g: [0, T ]×C([0, T ], Xα)→Xare continuous functions. For u(·)∈C([0, T ], Xα), ut(·) = u(·)|[0,t], u(t) = ut(t). We shall prove that, under some structural condition on fand assuming that for each r > 0 there is a κ=κ(r) such that u7→ κu +f(u) is an increasing function in [−r, r], then for each n∈N, there is a constant Kn such that the solutions u(t, φ) of (2) are globally defined, and there are u(t, Kn), u(t, −Kn), A DELAY NONLOCAL QUASILINEAR CHAFEE-INFANTE PROBLEM 9 solutions of (18) and (17) with Kreplaced by Kn, such that u(t, −Kn)⩽u(t, φ)⩽u(t, Kn), t∈[α−1((n−1)ρ), α−1(nρ)]. That is, thanks to the fact that the solutions of (17) and (18) are globally defined, we can guarantee that the solutions of (2) are defined in the interval [0, α−1(ρ)] and there is a positive constant K1such that −K1⩽u(t, φ)⩽K1, 0 ⩽t⩽α−1(ρ). We can therefore iterate this procedure to obtain that the solutions of (2) are globally defined. To prove results described above, we need to impose the structural condition (S) stated in the introduction to the non-linear forcing term. This condition ensures that the solutions of (18) and (17) are global and consequently, using the procedure described above we obtain that the solutions of (1) are global. Observe that condition (D) imposes a restriction on the constant C0that appears in condition (S). It was used in [5] to ensure the existence of pullback attractor. Here some special care needs to be taken due to the procedure described above with changing constants Kat intervals of length ρ. In fact, the following result holds Theorem 3.1. Assume that (S) holds for a continuous function f:R→Rsuch that, for every r > 0, there exists a constant κ=κ(r)>0such that s7→ κs +f(s)is increasing in [−r, r]. Let φ: [−ρ, 0] →H1 0(0,1) and K > 0such that −K⩽φ(τ)(x) = φ(α(t))(x)⩽Kfor all τ∈[−ρ, 0] and x∈[0,1]. If u(t, φ)is a solution of (19)          ut=uxx +g(t, ut(·)), t > 0, x ∈Ω, u(t, 0) = u(t, 1) = 0, u(r) = φ(α(r)), r ∈[α−1(−ρ),0], there are u(t, −K),u(t, K)solutions of (17) and (18) such that u(t, −K)⩽u(t, φ)⩽u(t, K), for t∈[0, α−1(ρ)]. As an immediate consequence of this, u(·, φ)is defined for all t⩾0. Proof. We will use the iterative step method. For that, first we consider t∈(0, α−1(ρ)],then, α(t)−ρ∈(−ρ, 0],and w(τ−ρ) = φ(α(t)−ρ), so that the equation (1) becomes the initial value problem (20)            ut=uxx +λf(u) + γφ(α(t)−ρ) a(l(u)) , t > 0, x ∈Ω, u(t, 0) = u(t, 1) = 0, u(0) = φ(0). Using [5, Corollary 4.4], since to −K⩽φ(0) ⩽K, we obtain the existence of u+(t, K),and u−(t, −K), solutions (18) and (17), defined in [0, α−1(ρ)], such that u−(t, −K)⩽u(t, φ)⩽ u+(t, K) for t∈[0, α−1(ρ)]. Now, taking K1= supt∈[0,α−1(ρ)],x∈[0,1] |u(t, φ)(x)|, we may repeat this procedure to ensure that u(t, φ) is defined in [α−1(ρ), α−1(2ρ)] and, by induction, for all t⩾0. To that, in each step, we use comparison in the interval [α−1(iρ), α−1((i+ 1)ρ)] to ensure 16 TOM´ AS CARABALLO, A. N. CARVALHO AND YESSICA JULIO [17] Li, Y., Carvalho, A. N., Luna, T. L. M., and Moreira, E. M. A non-autonomous bifurcation problem for a non-local scalar one-dimensional parabolic equation. Commun. Pure Appl. Anal. 19, 11 (2020), 5181–5196. [18] Melnik, V. S., and Valero, J. On attractors of multivalued semi-flows and differential inclusions. SetValued Analysis 6, 1 (1998), 83–111. [19] Michel, C., and Lue, M. Asymptotic behaviour of some nonlocal diffusion problems. Applicable Analysis 80, 3-4 (2001), 279–315. [20] Pazy, A. Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied mathematical sciences. Springer, 1983. (TC) Depto. Ecuaciones Diferenciales y Anal. Num., Facultad de Matem´ aticas, Universidad de Sevilla, C/ Tarfia s/n, 41012-Sevilla (Spain) Email address, TC: [email protected] (ANC and YJ) Instituto de Ciˆ encias Matem´ aticas e de Computac¸˜ ao Universidade de S˜ ao Paulo, Campus de S˜ ao Carlos, Caixa Postal 668, S˜ ao Carlos SP, Brazil. Email address, ANC: [email protected] Email address, YJ: [email protected]