Multivalued dynamics of non-autonomous reaction-diffusion equation with nonlinear advection term

Multivalued dynamics of non-autonomous reaction-diffusion equation with nonlinear advection term Willian Cintraa,∗, Mirelson M. Freitasb, To Fu Maa, Pedro Marín-Rubioc aDepartment of Mathematics, University of Brasília, Brasília-DF, 70910-900, Brazil bFaculty of Mathematics, Federal University of Pará, Augusto Corrêa Street 01, Belém-PA, 66075-110, Brazil cDpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, 41012 C/Tarfia s/n, Seville, Spain Abstract In this paper, we investigate a reaction-diffusion population model with a nonlinear advection term and a time-dependent force given by the equation ut−∆u+~α · ∇up=f(u) + h(t)in (τ, ∞)×Ω, subject to the boundary condition u= 0 on (τ, ∞)×∂Ω. Here, Ω⊂RNwith N≥1is a bounded domain with smooth boundary, τ∈R,~α = (α1, ..., αN)is a given advective direction and p > 1. The presence of the nonlinear advection term ~α · ∇upintroduces technical difficulties in the analysis, leading to a scenario where the uniqueness of weak solutions cannot be guaranteed. Consequently, the equation generates a multi-valued nonautonomous dynamical system. In this context, we establish the existence of minimal pullback attractors, considering universes of bounded and tempered sets. Moreover, we explore the relationships between these pullback attractors. Finally, we prove the upper semicontinuity of pullback attractors with respect to the advective vector ~α. Keywords: Reaction-diffusion-advection, pullback attractor, multi-valued dynamical systems, upper semicontinuity. 2010 MSC: 35B41, 35B40, 74H40, 35K57. 1. Introduction In this paper, we prove the existence and upper semicontinuity of pullback attractors for a reactiondiffusion-advection problem            ut−∆u+~α · ∇up=f(u) + h(t)in (τ, ∞)×Ω, u= 0 on (τ, ∞)×∂Ω, u(τ, x) = uτ(x)in Ω, (1) ∗Corresponding author Email addresses: [email protected] (Willian Cintra ), [email protected] (Mirelson M. Freitas), [email protected] (To Fu Ma), [email protected] (Pedro Marín-Rubio) Preprint submitted to Elsevier December 12, 2023 where Ω⊂RN(N≥1) is a bounded domain with smooth boundary, τ∈R,~α ∈RN,p > 1, and f∈C(R) satisfies the condition −κ−β1|s|q≤f(s)s≤κ−β2|s|q,(2) where β1, β2and κare positive constants and q≥2p. We note that (2) implies that there exists β > 0such that |f(s)| ≤ β(1 + |s|q−1).(3) Reaction-advection-difusion equations have been widely used to model phenomena in population dynamics. In this context, Ωrepresents the habitat of species whose population density at time t>τ and position x∈Ωis given by u(t, x). Such models take into account the movement of organisms, which can occur randomly (diffusion) or in a privileged director vector (advection). The advection can arise either from the behavior of individuals or from physical transport processes, such as the winds influence, river currents, blood flow, etcetera (cf. [28,3,12]). In [26], the author refer to such transport processes as unconditional dispersal. For instance, in the model (1), the advective term ~α · ∇up can be interpreted as a movement in the direction of vector ~α, with a velocity proportional to |~α|and the population density u. The global attractor is a useful object for understanding the dynamic behavior of a dynamical system. For non-autonomous dynamical systems, there are some important concepts of attractors: the uniform attractor [10], the random attractor [13] and the pullback attractor [9,20]. The main goal of this work is to study the existence of minimal pullback attractors for a class of time-dependent families (referred to as the universe) satisfying a tempered condition [8,5,14]. It is important to note that due to the general conditions (2) and the technical difficulties in the analysis of the nonlinear advection term ~α · ∇up, especially because it has no definite sign, the uniqueness of the solution cannot be guaranteed. For this reason, the existence of a minimal pullback attractor is proved using the theory of multi-valued dynamical systems [27,7,19,1,2,22,25]. Additionally, by using the theory in [25], we study the relations of these attractors with those of fixed bounded sets. A natural and fundamental question in Ecology is to understand how movement strategies influence species survival. In particular, we aim to analyze the influence of the advective term in (1) on the set of solutions to this problem and how it affects their dynamics. Specifically, we established the upper semicontinuity of attractors with respect to ~α, i.e., asymptotic dynamics are robust under small perturbations of this parameter. Now, we recall the known-results related with (1). In [4] the authors analyze a model where the nonlinear advection term represents directed movement up the gradient of fitness. They prove the existence and 2 uniqueness of a single positive stationary solution and describe the behavior of equilibria of the problem with respect to certain parameters of the equation. They further show that in certain cases the unique positive steady state is globally asymptotically stable among positive solutions. A study of the influence of the advective term on stability is also carried out by [15], considering a system of two species. The stationary solutions of (1) in the presence of a logistic reaction term are studied in [11] by using bifurcation theory and the method of sub and supersolutions. To the best of our knowledge, models with nonlinear advection terms have not yet been studied in the literature from the perspective of pullback attractors nor their upper semicontinuity with respect to ~α. The organization of the paper, as well as its main contributions, is as follows. In Section 2, we prove the existence of weak solutions using a Faedo-Galerkin scheme and the compactness method. In Section 3, we recall some abstract results on the theory of multi-valued nonautonomous dynamical systems. In Section 4, we adapt the methods from [17,18,23,25,14,6] to prove the existence of minimal pullback attractors. Additionally, we establish some relations between pullback attractors. In Section 5, we obtain the upper semicontinuity of pullback attractors with respect to ~α. More specifically, we prove that, as ~α →~α0, the family of pullback attractors indexed by ~α converges upper semicontinuously to the pullback attractor associated with the problem (1) with ~α =~α0. 2. Existence of a weak solution We denote the inner product in L2(Ω) by (·,·)and the associated norm by |·|. We denote the inner product in H1 0(Ω) by ((·,·)) and the associated norm by k·k. We denote the duality product between H−1(Ω) and H1 0(Ω) by h·,·i and the norm in H−1(Ω) by k · k∗. We identify L2(Ω) with its dual, and therefore, we have the sequence of compact and dense embeddings H1 0(Ω) ,→L2(Ω) ,→H−1(Ω). We note that as a result of the previous identification, we can make an abuse of notation by considering l∈L2(Ω) and denoting (l, u) as l(u). We denote the duality product between Lq(Ω) and Lq0(Ω) (where q0is the conjugate exponent of q) by (·,·), and we denote the norm in Lq(Ω) will be represented by k · kLq(Ω).We also denote by h·,·i the duality product between H−1(Ω) + Lq0(Ω) and H1 0(Ω) ∩Lq(Ω). Finally, we denote the norm in Lr(τ, T;X) by k·kLr(τ,T ;X)where Xis a Banach space. Here on we consider h∈L2 loc(R;H−1(Ω)). Now, let us introduce the precise definition of a weak solution to problem (1). Definition 2.1. Given an initial data uτ∈L2(Ω), a weak solution of (1)is an element usuch that for all T > τ, u ∈L∞(τ, T;L2(Ω)) ∩L2(τ, T;H1 0(Ω)) ∩Lq(τ, T;Lq(Ω)) and it satisfies d dt(u(t), v) + ((u(t), v)) −(up, ~α · ∇v)=(f(u(t)), v) + hh(t), vi(4) for all v∈H1 0(Ω) ∩Lq(Ω) in D0(τ, ∞),and u(τ) = uτ. 3 Remark 2.2. Since a weak solution to (1)satisfies u∈Lq(τ, T ;Lq(Ω))⊂L2p(τ, T;L2p(Ω)) for any T > τ, then ~α · ∇up∈L2(τ, T ;H−1(Ω)).Indeed, for any ϕ∈H1 0(Ω) ˆΩ ~α · ∇upϕdx=ˆΩ up~α · ∇ϕdx≤ |~α|kukp L2p(Ω)kϕk.(5) From above and also by using (3)we conclude that u0∈L2(τ, T, t;H−1(Ω))+Lq0(τ, T ;Lq0(Ω)) for any T > τ (where qand q0are conjugate exponents). Therefore, u∈C([τ, ∞); L2(Ω)) and the initial data u(τ)makes sense. Moreover, integrating by parts (zp, ~α · ∇z)=0holds for any z∈H1 0(Ω) ∩Lq(Ω).Indeed, (zp, ~α · ∇z) = 1 p+ 1 ˆΩ ~α · ∇zp+1dx =−1 p+ 1 ˆΩ zp+1∇ · ~αdx +1 p+ 1 ˆ∂Ω zp+1~α ·ηd∂Ω=0, where η=η(x)the outward unit normal vector at x∈∂Ω. Therefore the following energy identity holds, 1 2 d dt|u(t)|2+ku(t)k2= (f(u(t)), u(t)) + hh(t), u(t)i,a.e. t > τ. (6) Theorem 2.3. Suppose that fsatisfies (2). Then, for any initial data uτ∈L2(Ω), problem (1)has at least a weak solution. Proof. We will use the Faedo-Galerkin scheme and the compactness method. Fix T > τ. We split the proof into five steps. Step 1. Approximate problems. Let us consider initial data uτ∈L2(Ω). It is known from spectral theory that there exists a sequence {ωj}∞ j=1 of eigenfunctions of −∆in H1 0(Ω) (with corresponding eigenvalues 0< λ1≤λ2≤ · · · ), which forms a Hilbert basis for L2(Ω). We seek coefficients ajn(·)such that un(t) = n X j=1 ajn(t)ωj, is a local solution of d dt(un(t), ωj) + ((un(t), ωj)) −(up n, ~α · ∇ωj)=(f(un(t)), ωj) + hh(t), ωji, t ∈(τ, ∞), (un(τ), ωj)=(uτ, ωj), j = 1, . . . , n. (7) A local solution to this ODE system is guaranteed in an interval (τ, tn)for some tn≤T. Step 2. A first estimate. Multiplying equation (7) by ajn(t)and then summing for j= 1, . . . , n, we obtain d dt|un|2−(up n, ~α · ∇un)+2kunk2= 2(f(un), un)+2hh(t), uni, where the advection term in the left-hand side vanishes as in Remark 2.2. Hence, d dt|un|2+ 2kunk2= 2(f(un), un)+2hh(t), uni,a.e. t∈(τ, tn). Using (2), we deduce that d dt|un|2+ 2kunk2+ 2β2kunkq Lq(Ω) ≤2κ|Ω|+ 2hh(t), uni,a.e. t∈(τ, tn). 4 Hence, from the Young inequality, we have d dt|un|2+kunk2+ 2β2kunkq Lq(Ω) ≤2κ|Ω|+kh(t)k2 ∗,a.e. t∈(τ, tn). Integrating the last estimate from τto twith t∈(τ, tn), we obtain |un(t)|2+ˆt τ kun(s)k2ds + 2β2ˆt τ kun(s)kq Lq(Ω)ds ≤ |uτ|2+ 2κ|Ω|(T−τ) + ˆT τ kh(s)k2 ∗ds. (8) From the Gronwall lemma, this estimate implies that solutions to (7) are defined in the whole interval [τ, T]. Hence, by (8), we deduce that {un}is bounded in L∞(τ, T ;L2(Ω)) ∩L2(τ, T ;H1 0(Ω)) ∩Lq(τ, T;Lq(Ω)). Using (3) and the fact that {un}is bounded in Lq(τ, T ;Lq(Ω)) we conclude that {f(un)}is bounded in Lq0(τ, T;Lq0(Ω)). In addition, since 2p≤q, by (5) it holds that {~α·∇up n}is also bounded in L2(τ, t;H−1(Ω)). Step 3. A second estimate. Now we provide a uniform estimate for the sequence {u0 n}in a suitable space. To achieve this, consider the projection operators Pn:L2(Ω) →Vn:= span{ω1, . . . , ωn}and e P:H−1(Ω) → H−1(Ω) by Pnu= n X j=1 (u, ωj)ωj,∀u∈L2(Ω), and H−1(Ω) 3v7→ e Pn(v)∈H−1(Ω) where he Pn(v), φi:= hv, Pnφi,∀φ∈H1 0(Ω). Let A=−∆be the Dirichlet-Laplacian operator, which is an isomorphism from H1 0(Ω) into H−1(Ω). Since Ais a positive self-adjoint operator in L2(Ω), one can define its fractional powers as follows (cf. [30]) D(Aδ/2) =   u∈L2(Ω) : ∞ X j=1 λδ j(u, ωj)2<+∞  , where δ≥1is such that Hδ 0(Ω) ,→Lq(Ω). Thus we can define the projection b P:Lq0(Ω) →D(A−δ/2)by Lq0(Ω) 3v7→ b Pn(v)∈D(A−δ/2)where hb Pn(v), φiD(A−δ/2),D(Aδ/2):= hv, Pnφi,∀φ∈D(Aδ/2). We note that e Pnand b Pnare the continuous extensions of Pnin H−1(Ω) and Lq0(Ω), respectively. In the following, we denote both projections by Pn. Since {f(un)}is bounded in Lq0(τ, T;Lq0(Ω)), we have that {Pn(f(un))}is bounded in Lq0(τ, T ;H−δ(Ω)). In addition, {Pnh}is bounded in L2(τ, T;H−1(Ω)). Therefore, we conclude the sequence {u0 n}is bounded in Lq0(τ, T ;H−δ(Ω)). Step 4. Passage to the limit. Using the above estimates and the Aubin-Lions compactness theorem (e.g., cf. [29, Corollary 4]), we can conclude that (up to a subsequence, relabelled the same) 5 un ∗ * u in L∞(τ, T ;L2(Ω)), un* u in L2(τ, T ;H1 0(Ω)), un* u in Lq(τ, T ;Lq(Ω)), u0 n* u0in L2(τ, T;H−δ(Ω)), un→uin L2(τ, T;L2(Ω)) and a.e. in (τ, T)×Ω, ~α · ∇up n* ζ in L2(τ, T;H−1(Ω)), f(un)* ξ in Lq0(τ, T ;Lq0(Ω)). By using [21, Lemme 1.3], we deduce that ξ=f(u). Moreover, as up n→upa.e. on (τ, T)×Ωas well, from the uniform estimate of {un}in Lq(τ, T;Lq(Ω)) and since Lq(τ, T;Lq(Ω)) ,→L2p(τ, T;L2p(Ω)), we obtain by the same argument that up n* upin L2(τ, T;L2(Ω)). Therefore, ˆT τ h~α · ∇up n(s), v(s)ids =−ˆT τ (up n(s), ~α · ∇v(s))ds → − ˆT τ (up(s), ~α · ∇v(s))ds =ˆT τ h~α · ∇up(s), v(s)ids, (9) for all v∈L2(τ, T ;H1 0(Ω)). Hence, we conclude that ζ=~α · ∇up. Therefore using the above convergences, we can pass to the limit in (7). Then, using the fact that {ωj}∞ j=1 is dense in H1 0(Ω) ∩Lq(Ω), we conclude that (4) holds for all v∈H1 0(Ω) ∩Lq(Ω). Step 5. Initial data. Now, we aim to prove that u(τ) = uτ. To do this, we will use a standard argument. Consider ϕ∈H1(τ, T)with ϕ(T)=0and ϕ(τ)6= 0, and ω∈Vn. Multiplying (7) by ϕand integrating from τto T, we obtain −(uτ, ω)ϕ(τ)−ˆT τ (un(t), ω)ϕ0(t)dt −ˆT τ (up n(t), ~α · ∇ω)dt +ˆT τ ((un(t), ω))ϕ(t)dt =ˆT τ (f(un(t)), ω)ϕ(t)dt +ˆT τ hh(t), ωiϕ(t)dt. Hence, taking the limit as n→ ∞ and using the above convergences, we deduce that −(uτ, ω)ϕ(τ)−ˆT τ (u(t), ω)ϕ0(t)dt −ˆT τ (up(t), ~α · ∇ω)dt +ˆT τ ((u(t), ω))ϕ(t)dt =ˆT τ (f(u(t)), ω)ϕ(t)dt +ˆT τ hh(t), ωiϕ(t)dt. (10) 6 On the other hand, multiplying (4) by ϕand integrating from τto T, we obtain −(u(τ), ω)ϕ(τ)−ˆT τ (u(t), ω)ϕ0(t)dt −ˆT τ (up(t), ~α · ∇ω)dt +ˆT τ ((u(t), ω))ϕ(t)dt =ˆT τ (f(u(t)), ω)ϕ(t)dt +ˆT τ hh(t), ωiϕ(t)dt. (11) Combining (10) and (11) and using the fact that ϕ(τ)6= 0, we conclude that u(τ) = uτ. The proof is complete.  3. Abstract theory of multi-valued process and pullback attractors For the sake of completeness, we recall some basic facts on multi-valued nonautonomous dynamical systems. Along this section, (X, dX)will be a metric space, R2 d={(t, s)∈R2:t≥s}and P(X)the class of all nonempty subsets of X. We also define the Hausdorff semidistance of two nonempty subsets B1, B2⊂X as distX(B1, B2) := sup y1∈B1 inf y2∈B2 ky1−y2kX. Amulti-valued process (also called multi-valued nonautonomous dynamical system) Uon Xis a family of mappings U(t, τ) : X→ P(X)for any pair (t, τ)∈R2 dsuch that (i) U(τ, τ)x={x},∀τ∈R, x ∈X. (ii) U(t, τ)⊂U(t, s)(U(s, τ)x),∀τ≤s≤t, x ∈X, where U(t, τ)W:= Sy∈WU(t, τ)∀W⊂X. If the relation in (ii) is an equality, we say that the multi-valued process is strict. A multi-valued process Uon Xis upper semicontinuous if the mapping U(t, τ)is upper semicontinuous from Xinto P(X)for all (t, τ)∈R2 d, i.e., for any x∈Xand for every neighbourhood Nin Xof the set U(t, τ)x, there exists ε > 0such that U(t, τ)y⊂ N provided that dX(x, y)< ε. Now, we consider a universe D, that is a nonempty class of families parameterized in time b D={D(t) : t∈R}⊂P(X)and a family of nonempty sets b D0={D0(t) : t∈R}⊂P(X). A universe Dis inclusion closed if given b D∈ D and b C={C(t) : t∈R}⊂P(X)with C(t)⊂D(t)for all t∈R, then b C∈ D. A family b D0={D0(t) : t∈R}is said to be pullback D-absorbing for a multi-valued process Uif for every t∈Rand b D∈ D, there exists τ(b D, t)≤tsuch that U(t, τ)D(τ)⊂D0(t),∀τ≤τ(b D, t). Given a family of nonempty sets b D, a multi-valued process Uon Xis pullback b D-asymptotically compact if for any t∈Rand any sequences τn⊂(−∞, t]with τn→ −∞ and {xn} ⊂ Xwith xn∈D(τn), any sequence {yn}is relatively compact in X, where yn∈U(t, τn)xnfor all n. If a multi-valued process is pullback b D-asymptotically compact for any b D∈ D, then we say it is pullback D-asymptotically compact. A family AD={AD(t) : t∈R} ⊂ P(X)is a pullback D-attractor for a multi-valued process Uon Xif: 7 (1) AD(t)is a compact set of Xfor every t∈R; (2) ADis negatively invariant, i.e., AD(t)⊂U(t, τ)AD(τ),∀(t, τ)∈R2 d; (3) ADis pullback D-attracting, i.e., for any t∈Rand for any b D∈ D, lim τ→−∞ distX(U(t, τ)D(τ),AD(t)) = 0. A pullback D-attractor is said to be minimal if it satisfies the condition that if there exists another family b Cof closed sets such that it is pullback D-attracting, then AD(t)⊂C(t)for all t∈R. Remark 3.1. It is worth emphasizing that pullback D-attractors without minimal property are not unique due to the fact that the attractor does not belong to Din general (cf. [24]). The result of existence of pullback D-attractors is stated as follows (see [7,24,25]). Theorem 3.2. Suppose that Uis an upper semicontinuous multi-valued process on Xwith closed values, b D0⊂ P(X)is a pullback D-absorbing family and also suppose that Uis b D0-pullback-asymptotically compact. Then, the family AD={AD(t) : t∈R}defined by AD(t) = [ b D∈D Λ( b D, t) X ,∀t∈R, where Λdenotes the pullback omega-limit Λ( b D, t) = \ s≤t[ τ≤s U(t, τ)D(τ) X , is the minimal pullback D-attractor and AD(t)⊂D0(t)Xfor all t∈R. If AD∈ D, then it is the unique family of closed subsets in Dthat satisfies the properties (2)-(3) given above. In addition, if the multi-valued process Uis strict, then ADis strictly invariant under the process U, i.e., U(t, τ)AD(τ) = AD(t),∀(t, τ)∈R2 d. We denote by DX Fthe universe of fixed nonempty bounded subsets of X, i.e., the class of all families b D of the form b D={D(t) = B:t∈R}, where Bis a fixed nonempty bounded subset of X. A direct consequence is the existence of a pullback attractor (see [25]). Corollary 3.3. Under the assumption of Theorem 3.2, if DX F⊂ D, then ADX F={ADX F(t) : t∈R}, where ADX F(t) = [ B∈DX F Λ(B, t) X ,∀t∈R, is the minimal pullback DX F-attractor for the multivalued process Uand the following relation holds: ADX F(t)⊂ AD(t),∀t∈R. 8 Corollary 3.4. Under the assumption of Corollary 3.3, if there exists T∈Rsuch that the set St≤TD0(t) is bounded in X, then ADX F(t) = AD(t),∀t≤T. 4. Existence of minimal pullback attractors In this section, the existence of a minimal pullback attractor in L2(Ω) is investigated. In what follows, we use Φ~α(τ, uτ)to represent the set of weak solutions to (1) with advective vector ~α over the interval [τ, ∞)with initial datum uτ∈L2(Ω). In view of Theorem 2.3, we can define a multi-valued map U~α :Rd×L2(Ω) → P(L2(Ω)) as U~α(t, τ)uτ={u(t) : u∈Φ~α(τ, uτ)}, τ ≤t, uτ∈L2(Ω).(12) Remark 4.1. Observe that when ~α =~ 0,U~ 0(t, τ)is the multi-valued process associated with the weak solutions of the classical reaction-diffusion equation with time-dependent force. Lemma 4.2. Suppose that (2)holds. Then, the multivalued map U~α given by (12)is a strict multi-valued process on L2(Ω). Proof. It is achieved similarly to [16, Lemma 6.3]. For brevity, we omit the proof.  Next, we prove a continuity result of weak solutions to the problem (1). The argument follows the idea of [14]. Lemma 4.3. Suppose that (2)holds. If {un τ}is a sequence such that un τ→uτin L2(Ω), then for any sequence {un}with un∈Φ~α(τ, un τ)for each n≥1, there exist a subsequence (relabelled the same) {un}and u∈Φ~α(τ, uτ)such that un(s)→u(s)in L2(Ω),∀s≥τ. (13) Proof. Fix T > τ. Analogously to the proof of Step 2 of Theorem 2.3, we deduce that {un}is bounded in L∞(τ, T ;L2(Ω)) ∩L2(τ, T ;H1 0(Ω)) ∩Lq(τ, T;Lq(Ω)). In addition, {f(un)}is bounded in Lq0(τ, T ;Lq0(Ω)),{~α · ∇up n}is bounded in L2(τ, t;H−1(Ω)) and {u0 n}is bounded in L2(τ, T ;H−1(Ω)) + Lq0(τ, T ;Lq0(Ω)). Using the above estimates and the Aubin-Lions compactness theorem (e.g., [29, Corollary 4]), we obtain (up to a subsequence) that un ∗ * u in L∞(τ, T ;L2(Ω)), un* u in L2(τ, T ;H1 0(Ω)), un* u in Lq(τ, T ;Lq(Ω)), 9 since the multi-valued process U~α is upper semicontinuous with closed values by Lemma 4.4, the universes satisfy DL2 F⊂ DL2 µ, there exists a pullback DL2 µ-absorbing family b D0by Corollary 4.7 and the multi-valued process U~α is pullback DL2 µ-asymptotically compact by Lemma 4.9. The relation A~α DL2 µ (t)⊂D0(t)for any t∈R, the fact that A~α DL2 µ ∈ DL2 µand the strict U~α-invariance of A~α DL2 µ are also consequences of Theorem 3.2 by noting that b D0∈ DL2 µ, the set D0(t)is closed for all t∈R, the universe DL2 µis inclusion-closed and the multi-valued process U~α is strict by Lemma 4.2. Finally, if hfulfills the assumption (29) we have that St≤TRL2(t)is bounded for each T∈R, where RL2(t)is given by (21). Therefore, by Corollary 3.4 we have A~α DL2 F (t) = A~α DL2 µ (t)for all t∈R. Remark 4.11. The above results also applies for the problem (1)with ~α =~ 0. Thus, the minimal pullback DL2 µ-attractor A~ 0 DL2 µ ={A~ 0 DL2 µ (t) : t∈R}for the multi-valued process U~ 0:R2 d×L2(Ω) → P(L2(Ω)) exists (see Remark 4.1) and it is the minimal pullback attractor for the classical reaction-diffusion problem. 5. Upper semicontinuity of pullback attractors as ~α →~α0 In this section, we study the upper semicontinuity of the attractors A~α DL2 µ (t)as ~α →~α0for all t∈R. Lemma 5.1. Suppose that fsatisfies (2). Consider τ∈Rand sequences {~αn} ⊂ RNwith ~αn→~α0, {un τ} ⊂ L2(Ω) such that un τ* uτin L2(Ω) and {un}with un∈Φ~αn(τ, un τ). Then, there exist a subsequence (relabelled the same) {un}and u∈Φ~α0(τ, uτ)such that for any T > τ: un ∗ * u in L∞(τ, T;L2(Ω)), un* u in L2(τ, T;H1 0(Ω)), un* u in Lq(τ, T;Lq(Ω)), u0 n* u0in L2(τ, T;H−1(Ω)) + Lq0(τ, T;Lq0(Ω)), un→uin L2(τ, T;L2(Ω)) and a.e. in (τ, T )×Ω, ~αn· ∇up n* ~α0· ∇upin L2(τ, T ;H−1(Ω)), f(un)* f(u)in Lq0(τ, T;Lq0(Ω)), un(t)→u(t)in L2(Ω) ∀t > τ. (30) Proof. We split the proof into three steps. Fix T > τ (later we use a standard diagonal argument). Step 1. All convergences but last. Since unis a weak solution to (1) (with ~α =~αn) in [τ, T ], from the energy equality (6), using assumption (2) and the Young inequality, we deduce that d dt|un|2+kunk2+ 2β2kunkq Lq(Ω) ≤2κ|Ω|+kh(t)k2 ∗,a.e. t∈[τ, T]. Therefore, as deduced previously in different but similar framework, {un}is bounded in L∞(τ, T ;L2(Ω)) ∩L2(τ, T ;H1 0(Ω)) ∩Lq(τ, T;Lq(Ω)). 16 Analogously to (5), we have k~αn· ∇up nkL2(τ,T ;H−1(Ω)) ≤C|~αn|kunkLq(τ,T ;Lq(Ω)). Hence, since {un}is bounded in Lq(τ, T;Lq(Ω)) and ~αn→~α0, we deduce similarly to (5) that ~αn·∇up n* χ in L2(τ, T;H−1(Ω)). Using (2) and again the fact that {un}is bounded in Lq(τ, T;Lq(Ω)), it arises that {f(un)} is bounded in Lq0(t−2, t;Lq0(Ω)). Therefore, {u0 n}is bounded in L2(τ, T ;H−1(Ω))+Lq0(τ, T;Lq0(Ω)). Using standard functional analysis results and the Aubin-Lions compactness theorem, all convergences but last are obtained (we have used [21, Lemme 1.3] to identify χ=~α0· ∇upsimilarly to (9)). In addition, by the dominated convergence theorem, un(t)→u(t)in L2(Ω) a.e. t∈(τ, T).(31) Step 2: u(t)∈U~α0(t, τ)uτwith t∈[τ, T].Since unis a weak solution to (1) with ~α =~αn, we have −ˆT τ (un(t), v)ϕ0(t)dt +ˆT τ ((un(t), v))ϕ(t)dt −ˆT τ (up n, ~αn· ∇v)ϕ(t)dt =ˆT τ (f(un(t)), v)ϕ(t)dt +ˆT τ hh(t), viϕ(t)dt, for all v∈H1 0(Ω) ∩Lq(Ω) and ϕ∈ D(τ, T ). Using the convergences proved above, we can pass to the limit to obtain −ˆT τ (u(t), v)ϕ0(t)dt +ˆT τ ((u(t), v))ϕ(t)dt −ˆT τ (up, ~α0· ∇v)ϕ(t)dt =ˆT τ (f(u(t)), v)ϕ(t)dt +ˆT τ hh(t), viϕ(t)dt. Therefore, uis a weak solution of (1) with ~α =~α0. To conclude that u∈Φ~ 0(τ, uτ), we need to show that u(τ) = uτ. In order to do this, let us consider v∈H1 0(Ω) ∩Lq(Ω) and ϕ∈H1(τ, T)with ϕ(T)=0and ϕ(τ)6= 0. Using these elements as test functions in problem (1) with ~α =~αnand the fact that un τ* uτin L2(Ω) and then taking the limit as n→ ∞, we obtain −(uτ, v)ϕ(τ)−ˆT τ (u(t), v)ϕ0(t)dt +ˆT τ ((u(T), v))ϕ(t)dt −ˆT τ (up, ~α0· ∇v)ϕ(t)dt =ˆT τ (f(u(t)), v)ϕ(t)dt +ˆT τ hh(t), viϕ(t)dt. On the other hand, using the above test functions in problem (1) with ~α =~α0we have −(u(τ), v)ϕ(τ)−ˆT τ (u(t), v)ϕ0(t)dt +ˆT τ ((u(T), v))ϕ(t)dt −ˆT τ (up, ~α0· ∇v)ϕ(t)dt =ˆT τ (f(u(t)), v)ϕ(t)dt +ˆT τ hh(t), viϕ(t)dt. Combining the two last equations and using the fact that ϕ(τ)6= 0, we conclude that u(τ) = uτ. 17 Step 3. Convergence (30)holds. Indeed, using the energy equality (6), we have |un(t)|2≤ |un(r)|2+ 2κ|Ω|(s−r) + 1 2ˆr t kh(σ)k2 ∗dσ, ∀τ≤r≤s≤t. Analogously, |u(t)|2≤ |u(r)|2+ 2κ|Ω|(s−r) + 1 2ˆr t kh(σ)k2 ∗dσ, ∀τ≤r≤s≤t. Thus, defining ˜ Jn(s) = |un(s)|2−2κ|Ω|s−1 2ˆs t−2 kh(σ)k2 ∗dσ, ˜ J(s) = |u(s)|2−2κ|Ω|s−1 2ˆs t−2 kh(σ)k2 ∗dσ, we see that ˜ Jnand ˜ Jare continuous and non-increasing on [τ, T]. Hence, using (31) and similar arguments as in Lemma 4.3, we can deduce that ˜ Jn(s)→˜ J(s),∀s∈(τ, T]. Consequently, lim n→∞ |un(s)|2=|u(s)|2,∀s∈(τ, T].(32) Similarly to the proof of Lemma 4.9, using the fact that {un}is bounded in C([τ, T]; L2(Ω)) and {u0 n}is bounded in L2(τ, T ;H−1(Ω)) + Lq0(τ, T ;Lq0(Ω)), and the compact embedding L2(Ω) ,→H−1(Ω) + Lq0(Ω), the Ascoli-Arzelà theorem implies that un→uin C([τ, T]; H−1(Ω) + Lq0(Ω)).Hence, since {un}is bounded in C([τ, T ]; L2(Ω)), we deduce that un(s)* u(s)in L2(Ω),∀s∈[τ, T]. Combining the last convergence and (32), we conclude that (30) holds.  Theorem 5.2 (Upper semicontinuity).Suppose that f∈C(R)satisfies (2)and h∈L2 loc(R;H−1(Ω)) fulfills (20). Then, the family of pullback attractors A~α DL2 µ (t)~α∈RNis upper semicontinuous with respect to the advective vector ~α, i.e., if ~α →~α0then lim ~α→~α0 distL2(Ω)(A~α DL2 µ(t),A~α0 DL2 µ (t)) = 0,∀t∈R.(33) Proof. We prove (33) by contradiction. Suppose that there exist  > 0,t∈R, and a sequence ~αn→~α0 such that distL2(Ω)(A~α DL2 µ(t),A~α0 DL2 µ (t)) > , ∀n∈N. By the negative invariance of the pullback attractors, there exists a sequence of solutions {un}of (1) with corresponding advective vectors ~αnsuch that un(t)∈ A~αn DL2 µ (t)and distL2(Ω)(un(t),A~α0 DL2 µ (t)) > , ∀n∈N.(34) 18 By Theorem 4.10, we have A~αn DL2 µ (t)⊂D0(t)for all nand t∈R. Since b D0∈ DL2 µis DL2 µ-absorbing for the multi-valued process U~α0, there exists τ(t, b D0, )< t such that distL2(Ω)(U~α0(t, τ)D0(τ),A~α0 DL2 µ (t)) < 2,∀τ≤τ(t, b D0, ).(35) On the other hand, since the pullback attractors A~αn DL2 µ have bounded sections at any time, in particular at time τ(t, b D0, ), we deduce that {un(τ(t, b D0, ))}is bounded in L2(Ω). Then, there exists a subsequence such that un(τ(t, b D0, )) * uτin L2(Ω). Therefore, by Lemma 5.1, there exists u∈Φ~α0(τ, uτ)such that all convergences there hold in (τ(t, b D0, ), t). In particular, the convergence (30) at time timplies that there exists n0∈Nsuch that |un(t)−u(t)| ≤  2,∀n≥n0. Combining the above with (35) we obtain that distL2(Ω)(un(t),A~α0 DL2 µ (t)) ≤distL2(Ω)(un(t), u(t)) + distL2(Ω)(u(t),A~α0 DL2 µ (t)) ≤ 2+ 2=, ∀n≥n0. This contradicts (34). The proof is complete.  Remark 5.3. (i) In particular, when ~α →~α0=~ 0,the above result shows the upper semicontinuity of the minimal pullback attractors A~α DL2 µ toward the minimal pullback attractor A~ 0 DL2 µ associated to the reactiondiffusion problem (cf. Remark 4.11). (ii) For simplicity in the exposition we have restricted to the case in which the external force h∈ L2 loc(R;H−1(Ω)) fulfills the integrability condition (20). However this can be generalized to h1+h2with h1∈L2 loc(R;H−1(Ω)) and h2∈Lq0 loc(R;Lq0(Ω)),provided that there exists µ∈(0,2λ1)such that ˆ0 −∞ eµs(kh1(s)k2 ∗+kh2(s)kq0 Lq0(Ω))ds < ∞, instead of (20), and all results remain analogous under extra minor changes as in (22). Acknowledgments We would like to thank the anonymous referees for carefully reading our manuscript and for providing constructive comments that improved the quality of this paper. 19 Funding W. Cintra was partially supported by FAPDF grants 00193.00001821/2022-21 and 00193.00001819/2022-5 and FEMAT Proc. 056/2023. M. M. Freitas was supported by CNPq grant 313081/2021-2. T. F. Ma has been supported by CNPq grant 315165/2021-9 and by FAPDF grant 193.00001821/2022-21. P. Marín-Rubio has been supported by MICINN/FEDER-EU project PID2021-122991NB-C21. References [1] Anguiano M, Caraballo T, Real J, Valero J. Pullback attractors for reaction-diffusion equations in some unbounded domain with an H−1-valued non-autonomous forcing term and without uniqueness of solutions. Discrete Contin Dyn Syst Ser B 2010;14(2):307–26. https://doi.org/10.3934/dcdsb.2010.14.307 [2] Anguiano M. Attractors for nonlinear and non-autonomous parabolic PDEs in unbounded domains. Ph.D. Thesis, Universidad de Sevilla; 2011. [3] Cantrell RS, Cosner C. Spatial ecology via reaction-diffusion equations. Wiley Series in Mathematical and Computational Biology. Chichester: John Wiley & Sons, Ltd; 2003. [4] Cantrell RS, Cosner C, Lou Y. Approximating the ideal free distribution via reaction-diffusion-advection equations. J. Differential Equations 2008;245(12):3687–703. https://doi.org/10.1016/j.jde.2008.07.024 [5] Caraballo T, Chueshov I, Marín-Rubio P, Real J. Existence and asymptotic behaviour for stochastic heat equations with multiplicative noise in materials with memory. Discrete Contin Dyn Syst 2007;18(2- 3):253–70. https://doi.org/10.3934/dcds.2007.18.253 [6] Caraballo T, Herrera-Cobos M, Marín-Rubio P. Robustness of nonautonomous attractors for a family of nonlocal reaction-diffusion equations without uniqueness. Nonlinear Dyn 2016;84(1):35–50. https://doi.org/10.1007/s11071-015-2200-4 [7] Caraballo T, Kloeden PE. Non-autonomous attractors for integro-differential evolution equations. Discrete Contin Dyn Syst Ser S 2009;2(1):17–36. https://doi.org/10.3934/dcdss.2009.2.17 [8] Caraballo T, Łukaszewicz G, Real J. Pullback attractors for non-autonomous 2D-Navier- Stokes equations in some unbounded domains. C R Math Acad Sci Paris 2006;342(4):263–8. https://doi.org/10.1016/j.crma.2005.12.015 [9] Carvalho AN, Langa JA, Robinson JC. Attractors for infinite-dimensional non-autonomous dynamical systems. New York: Springer-Verlag; 2013. 20 [10] Chepyzhov VV, Vishik MI. Attractors for equations of mathematical physics. Providence, RI: AMS; 2002. [11] Cintra W, Montenegro M, Suárez A. The logistic equation with nonlinear advection term. Nonlinear Anal Real World Appl 2022;65:103503. https://doi.org/10.1016/j.nonrwa.2021.103503 [12] Cosner C. Reaction-diffusion-advection models for the effects and evolution of dispersal. Discrete Contin Dyn Syst 2014;34(5):1701–45. https://doi.org/10.3934/dcds.2014.34.1701 [13] Crauel H, Debussche A, Flandoli F. Random attractors. J Dynam Differential Equations 1997;9(2):307– 41. https://doi.org/10.1007/BF02219225 [14] García-Luengo J, Marín-Rubio P, Real J. Pullback attractors in Vfor non-autonomous 2D-Navier- Stokes equations and their tempered behaviour. J. Differential Equations 2012;252(8):4333–56. https://doi.org/10.1016/j.jde.2012.01.010 [15] He X, Liu L. On the conjecture of the role of advection in a two-species competition-diffusion model. SIAM J Appl Dyn Syst 2022;21(3):1663–85. https://doi.org/10.1137/21M1451713 [16] Herrera-Cobos M. Long-time behaviour of nonlocal partial differential equations. Ph.D. Thesis, Universidad de Sevilla; 2016. [17] Kapustyan AV, Melnik VS, Valero J. Attractors of multivalued dynamical processes generated by phase-field equations. Int J Bifur Chaos Appl Sci Eng 2003;13(7):1969–83. https://doi.org/10.1142/S0218127403007801 [18] Kapustyan AV, Valero J. On the connectedness and asymptotic behaviour of solutions of reactiondiffusion systems. J Math Anal Appl 2006;323(1):614–33. https://doi.org/10.1016/j.jmaa.2005.10.042 [19] Kapustyan AV, Valero J. On the Kneser property for the complex Ginzburg-Landau equation and the Lotka-Volterra system with diffusion. J Math Anal Appl 2009;357(1):254–72. https://doi.org/10.1016/j.jmaa.2009.04.010 [20] Kloeden PE, Rasmussen M. Nonautonomous dynamical systems. Providence, RI: AMS; 2011. [21] Lions JL. Quelques méthodes de résolution des problèmes aux limites non linéaires. Paris: Dunod; 1969. [22] Marín-Rubio P. Attractors for parametric delay differential equations without uniqueness and their upper semicontinuous behaviour. Nonlinear Anal 2008;68(3):3166–74. https://doi.org/10.1016/j.na.2007.03.011 21 [23] Marín-Rubio P, Planas G, Real J. Asymptotic behaviour of a phase-field model with three coupled equations without uniqueness. J. Differential Equations 2009;246(12):4632–52. https://doi.org/10.1016/j.jde.2009.01.021 [24] Marín-Rubio P, Real J. On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems. Nonlinear Anal 2009;71(9):3956–63. https://doi.org/10.1016/j.na.2009.02.065 [25] Marín-Rubio P, Real J. Pullback attractors for 2D-Navier-Stokes equations with delays in continuous and sublinear operators. Discrete Contin Dyn Syst 2010;26(3):989–1006. https://doi.org/10.3934/dcds.2010.26.989 [26] McPeek MA, Holt RD. The evolution of dispersal in spatially and temporally varying environments. American Naturalist 1992;140(6):1010–27. https://www.jstor.org/stable/2462931 [27] Melnik VS, Valero J. On attractors of multi-valued semiflows and differential inclusions. Set-Valued Anal 1998;6(1):83–111. https://doi.org/10.1023/A:1008608431399 [28] Okubo A. Diffusion and ecological problems: mathematical models. An extended version of the Japanese edition, Ecology and diffusion. Translated by Parker GN. Biomathematics, 10. Berlin-New York: Springer-Verlag; 1980. [29] Simon J. Compact sets in the space Lp(0, T ;B). Ann Math Pura Appl (4) 1987;148:65–96. https://doi.org/10.1007/BF01762360 [30] Temam R. Infinite dimensional dynamical systems in mechanics and physics. Second Edition. New York: Springer-Verlag; 1997. 22