Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 50, pp. 1–13. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: https://doi.org/10.58997/ejde.2023.50 EVOLUTION EQUATIONS ON TIME-DEPENDENT LEBESGUE SPACES WITH VARIABLE EXPONENTS JACSON SIMSEN Communicated by Claudianor O. Alves Abstract. We extend the results in Kloeden-Simsen [CPAA 2014] to p(x, t)- Laplacian problems on time-dependent Lebesgue spaces with variable expo- nents. We study the equation ∂uλ ∂t (t)− div ( Dλ(t, x)|∇uλ(t)|p(x,t)−2∇uλ(t) ) + |uλ(t)|p(x,t)−2uλ(t) = B(t, uλ(t)) on a bounded smooth domain Ω in Rn, n ≥ 1, with a homogeneous Neumann boundary condition, where the exponent p(·) ∈ C(Ω̄ × [τ, T ],R+) satisfies min p(x, t) > 2, and λ ∈ [0,∞) is a parameter. We establish the existence and upper semicontinuity of pullback attractors for this equation under the assumption, amongst others, that B is globally Lipschitz in its second variable and Dλ ∈ L∞([τ, T ]×Ω,R+) is bounded from above and below, monotonically nonincreasing in time and continuous in the parameter λ. 1. Introduction We consider the problem ∂uλ ∂t (t)− div ( Dλ(t, x)|∇uλ(t)|p(x,t)−2∇uλ(t) ) + |uλ(t)|p(x,t)−2uλ(t) = B(t, uλ(t)) uλ(τ) = uτλ, (1.1) on a bounded smooth domain Ω in Rn for n ≥ 1 with a homogeneous Neumann boundary condition. The exponent p(·) ∈ C(Ω̄× [τ, T ],R) satisfies p+ := max (x,t)∈Ω̄×[τ,T ] p(x, t) ≥ p− := min (x,t)∈Ω̄×[τ,T ] p(x, t) > 2 and the initial condition uλ(τ) ∈ H := L2(Ω). For the mapping B : [τ, T ]×H → H we use the following assumptions: 2020 Mathematics Subject Classification. 35K55, 35K92, 35A16, 35B40, 35B41, 37B55. Key words and phrases. Non-autonomous parabolic problems; variable exponents; p-Laplacian; pullback attractors; upper semicontinuity. ©2023. This work is licensed under a CC BY 4.0 license. Submitted February 18, 2023. Published July 24, 2023. 1 2 J. SIMSEN EJDE-2023/50 (A1) there exists L ≥ 0 such that ‖B(t, x1)−B(t, x2)‖H ≤ L‖x1 − x2‖H for all t ∈ [τ, T ] and x1, x2 ∈ H; (A2) for all x ∈ H the mapping t 7→ B(t, x) belongs to L2(τ, T ;H); (A3) the function t 7→ ‖B(t, 0)‖H is nondecreasing, absolutely continuous and bounded on compact subsets of R. For each λ ∈ [0,∞), the mapping Dλ : [τ, T ] × Ω → R belongs to L∞([τ, T ] × Ω) and we use the following assumptions: (A4) there are positive constants, β and M independent of λ such that 0 < β ≤ Dλ(t, x) ≤M for almost all (t, x) ∈ [τ, T ]× Ω; (A5) Dλ → Dλ1 in L∞([τ, T ]× Ω) as λ→ λ1; (A6) Dλ(t, x) ≥ Dλ(s, x) for each x ∈ Ω and t ≤ s in [0, T ]. PDEs with variable exponents have application in electrorheological fluids (see [9, 17, 18]), image process (see [8, 12]), flow in porous media [2, 3], magnetostatics [5], and capillarity phenomena [4]. In this article, from the point of view of D-pullback attractor theory [13, 15, 16], we study the asymptotic behavior and analyze the sensitivity of this nonlinear non-autonomous problem for large time varying the diffusion coefficients Dλ. The study of existence of attractors for parabolic problems with spatially variable exponents is a very recent research issue. To the best of our knowledge, the first results for autonomous problems were published in [19] and the first result on existence of a pullback attractor for a non-autonomous problem were published in [14]. The main motivation of this work is that the exponents also depends on time, differently from the previous works [14, 19] which considered only spatially variable exponents. This article is organized as follows. In Section 2 we present the operator which depends on time and show some of its properties. In Section 3 we prove existence and uniqueness of the solution of (1.1). Section 4 is used to prove the existence of the D-pullback attractors. Section 5 is devoted to prove the upper semicontinuity of pullback attractors. 2. Operator Aλ and its properties Let Ω ⊂ Rn, n ≥ 1, be a bounded smooth domain. In this work we will use the following notation Lp(·,t)(Ω) := { u : Ω→ R : u is measurable, ∫ Ω |u(x)|p(x,t) dx <∞ } . We define ρt(u) := ∫ Ω |u(x)|p(x,t) dx and ‖u‖p(·,t) := inf { λ > 0 : ρt (u λ ) ≤ 1 } for u ∈ Lp(·,t)(Ω). The generalized Sobolev space, defined as W 1,p(·,t)(Ω) = { u ∈ Lp(·,t)(Ω) : |∇u| ∈ Lp(·,t)(Ω) } , is a Banach Space with the norm ‖u‖W 1,p(·,t)(Ω) := ‖u‖p(·,t) + ‖∇u‖p(·,t). EJDE-2023/50 EVOLUTION EQUATIONS ON TIME-DEPENDENT LEBESGUE SPACES 3 Let us consider H := L2(Ω), Yt := Lp(·,t)(Ω) and Xt := W 1,p(·,t)(Ω) with p(x, t) ∈ [p−, p+] ⊂ (2,∞) for all (x, t) ∈ Ω × R. Then Xt ⊂ H ⊂ X∗t with continuous and dense embeddings. Consider the operator Aλ(t)u : Xt → R such that to each u ∈ Xt associate the following element of X∗t , Aλ(t)u(v) := ∫ Ω Dλ(t, x)|∇u(x)|p(x,t)−2∇u(x)∇v(x) dx+ ∫ Ω |u(x)|p(x,t)−2u(x)v(x) dx. Lemma 2.1 ([11]). For u ∈ Lp(·,t)(Ω) we have: (i) ‖u‖p(·,t) < 1 (= 1;> 1) if and only if ρt(u) < 1 (= 1;> 1); (ii) If ‖u‖p(·,t) > 1, then ‖u‖p − p(·,t) ≤ ρ t(u) ≤ ‖u‖p + p(·,t); (iii) If ‖u‖p(·,t) < 1, then ‖u‖p + p(·,t) ≤ ρ t(u) ≤ ‖u‖p − p(·,t). Lemma 2.2 ([1]). Let λ, µ be arbitrary nonnegative numbers. For every positive α, β, α ≥ β, it holds λα + µβ ≥ 1 2α { (λ+ µ)α, if λ+ µ < 1, (λ+ µ) β , if λ+ µ ≥ 1. (2.1) Using the previous lemmas we can obtain the following estimates. Lemma 2.3. Let u ∈ Xt. For each t ≥ 0 we have 〈Aλ(t)u, u〉X∗ t ,Xt ≥ min{1, β} 2p+ ‖u‖Xp+t , if ‖u‖Xt < 1 ‖u‖ Xp − t , if ‖u‖Xt ≥ 1. . (2.2) Proof. For an arbitrary u ∈ Xt, we denote λt = ‖∇u‖p(·,t) and µt = ‖u‖p(·,t). From Lemmas 2.1 and 2.2 we have 〈Aλ(t)u, u〉X∗ t ,Xt = ∫ Ω Dλ|∇u|p(x,t) dx+ ∫ Ω |u|p(x,t) dx ≥ βρt(∇u) + ρt(u) ≥ {‖∇u‖p+p(·,t), ‖∇u‖ p− p(·,t)}+ min{‖u‖p+p(·,t), ‖u‖ p− p(·,t)} ≥ min{1, β} ( min{λp+t , λp−t }+ min{µp+t , µp−t } ) ≥ min{1, β} 2p+ { (λt + µt) p+ , if λt + µt < 1 (λt + µt) p− , if λt + µt ≥ 1 = min{1, β} 2p+ { ‖u‖p + Xt , if ‖u‖Xt < 1 ‖u‖p − Xt , if ‖u‖Xt ≥ 1. � Lemma 2.4. The operator Aλ(t) : Xt → X∗t is monotone for each t ∈ [τ, T ]. Proof. By [10], (|ξ|p−2ξ − |η|p−2η)(ξ − η) ≥ (1 2 )p|ξ − η|p, p ≥ 2, ξ, η ∈ RN . (2.3) Fix t ∈ [τ, T ] and let u, v ∈ Xt. Using (2.3) for each fixed x ∈ Ω, we obtain 〈Aλ(t)u−Aλ(t)v, u− v〉X∗ t ,Xt 4 J. SIMSEN EJDE-2023/50 = ∫ Ω Dλ(t, x) ( |∇u|p(x,t)−2∇u− |∇v|p(x,t)−2∇v ) · (∇u−∇v) dx + ∫ Ω ( |u|p(x,t)−2u− |v|p(x,t)−2v ) (u− v) dx ≥ β ∫ Ω (1 2 )p(x,t)|∇u(x)−∇v(x)|p(x,t) dx + ∫ Ω (1 2 )p(x,t)|u(x)− v(x)|p(x,t) dx ≥ 0. The proof is complete. � Remark 2.5. The operator Aλ(t) : Xt → X∗t is coercive and hemicontinuous for each t ∈ [τ, T ]. We will show that AλH(t) is the subdifferential of the convex, proper, and lower semicontinuous map ϕ Dλ(t,·) p(·,t) : L2(Ω)→ R ∪ {+∞}, given by ϕ Dλ(t,·) p(·,t) (u) := {[ ∫ Ω Dλ(t,x) p(x,t) |∇u| p(x,t) dx+ ∫ Ω 1 p(x,t) |u| p(x,t) dx ] if u ∈W 1,p(·,t)(Ω) +∞, otherwise. Lemma 2.6. The map ϕ Dλ(t,·) p(·,t) is convex and proper. Proof. Let u ∈ Xt. Then, u ∈ Yt = Lp(·,t)(Ω) and ∇u ∈ Yt. So,∫ Ω Dλ(t, x) p(x, t) |∇u|p(x,t) dx+ ∫ Ω 1 p(x, t) |u|p(x,t) dx ≤ 1 2 [ M ∫ Ω |∇u|p(x,t) dx+ ∫ Ω |u|p(x,t) dx ] <∞. Therefore ϕ Dλ(t,·) p(·,t) is proper. Since the application γp is convex for a γ > 0 given, u, v ∈ Xt, and 0 ≤ λ ≤ 1 we have ϕ Dλ(t,·) p(·,t) (λu+ (1− λ)v) = ∫ Ω Dλ(t, x) p(x, t) |∇(λu+ (1− λ)v)|p(x,t) dx+ ∫ Ω 1 p(x, t) |λu+ (1− λ)v|p(x,t) dx ≤ ∫ Ω Dλ(t, x) p(x, t) ( λ|∇u|p(x,t) + (1− λ)|∇v|p(x,t) ) dx + ∫ Ω 1 p(x, t) ( λ|u|p(x,t) + (1− λ)|v|p(x,t) ) dx = λ ∫ Ω Dλ(t, x) p(x, t) |∇u|p(x,t) dx+ λ ∫ Ω 1 p(x, t) |u|p(x,t) dx + (1− λ) ∫ Ω Dλ(t, x) p(x, t) |∇v|p(x,t) dx+ (1− λ) ∫ Ω 1 p(x, t) |v|p(x,t) dx = λϕ Dλ(t,·) p(·,t) (u) + (1− λ)ϕ Dλ(t,·) p(·,t) (v). Therefore ϕ Dλ(t,·) p(·,t) is convex. � Lemma 2.7. The map ϕ Dλ(t,·) p(·,t) is lower semicontinuous. EJDE-2023/50 EVOLUTION EQUATIONS ON TIME-DEPENDENT LEBESGUE SPACES 5 Proof. Let (un) be a sequence such that un → u in H. We have to show that ϕ Dλ(t,·) p(·,t) (u) ≤ lim inf n→∞ ϕ Dλ(t,·) p(·,t) (un) if un → u in H. If lim infn→∞ ϕ Dλ(t,·) p(·,t) (un) = +∞, then ϕ Dλ(t,·) p(·,t) (u) ≤ +∞ = lim inf n→∞ ϕ Dλ(t,·) p(·,t) (un). On the other hand, if lim infn→∞ ϕ Dλ(t,·) p(·,t) (un) = a < +∞ then there is a subse- quence (unj ) ⊂ Xt of (un) such that lim j→∞ ϕ Dλ(t,·) p(·,t) (unj ) = lim j→∞ (∫ Ω Dλ(t, x) p(x, t) |∇unj |p(x,t) dx+ ∫ Ω 1 p(x, t) |unj |p(x,t) dx ) = a. Since ϕ Dλ(t,·) p(·,t) (unj )→ a as j →∞ we have that ϕ Dλ(t,·) p(·,t) (unj ) is bounded, i.e., there exists δ > 0 such that |ϕDλ(t,·) p(·,t) (unj )| ≤ δ, for all j ∈ N. We have ϕ Dλ(t,·) p(·,t) (unj ) ≥ ∫ Ω β p(x, t) |∇unj |p(x,t) dx+ ∫ Ω 1 p(x, t) |unj |p(x,t) dx ≥ ∫ Ω 1 p(x, t) |unj |p(x,t) dx ≥ 1 p+ ∫ Ω |unj | p(x,t) dx ≥ 1 p+ ρt(unj ). Consequently, 1 p+ ρt(unj ) ≤ ϕ Dλ(t,·) p(·,t) (unj ) ≤ δ. Then ρt(unj ) ≤ p+δ. (2.4) Similarly we have that ρt(∇unj ) ≤ p+δ β . (2.5) Using this inequality,(2.4), and Lemma 2.1, we obtain ‖unj‖p(·,t) ≤ { (p+δ) 1/p− , if ‖unj‖p(·,t) ≥ 1, (p+δ) 1/p+ , if ‖unj‖p(·,t) < 1. and ‖∇unj‖p(·,t) ≤  ( p+δ β )1/p− , if ‖∇unj‖p(·,t) ≥ 1,( p+δ β )1/p+ , if ‖∇unj‖p(·,t) < 1. Therefore, ‖unj‖Xt is a bounded sequence in the reflexive Banach space Xt. So, (unj ) has a subsequence (which we still denote by (unj )) such that unj ⇀ v in Xt for some v ∈ Xt. As H∗ ⊂ X∗t we have unj ⇀ v in H and by the uniqueness of the weak limit u = v ∈ Xt. Considering the subdifferential ∂ϕ Dλ(t,·) p(·,t) of ϕ Dλ(t,·) p(·,t) we have 〈∂ϕDλ(t,·) p(·,t) (u), unj − u〉X∗ t ,Xt ≤ ϕDλ(t,·) p(·,t) (unj )− ϕ Dλ(t,·) p(·,t) (u) 6 J. SIMSEN EJDE-2023/50 for all j ∈ N. We observe that ϕ Dλ(t,·) p(·,t) is Gateaux differentiable at u. So, it follows from [6, Example 1, p. 54], that u ∈ D(∂ϕ Dλ(t,·) p(·,t) ) and ∂ϕ Dλ(t,·) p(·,t) (u) consists of a single element, namely the Gateaux differential of ϕ Dλ(t,·) p(·,t) at u, i.e., ∇uϕDλ(t,·) p(·,t) (u) = ∂ϕ Dλ(t,·) p(·,t) (u). As unj ⇀ u in Xt and ∅ 6= ∂ϕ Dλ(t,·) p(·,t) (u) ∈ X∗t we obtain 〈∂ϕDλ(t,·) p(·,t) (u), unj − u〉X∗ t ,Xt → 0 as j →∞. Therefore, ϕ Dλ(t,·) p(·,t) (u) ≤ lim j→∞ ϕ Dλ(t,·) p(·,t) (unj ) = a = lim inf n→∞ ϕ Dλ(t,·) p(·,t) (un). � Theorem 2.8. The realization AλH(t) of Aλ(t) in H is the subdifferential ∂ϕ Dλ(t,·) p(·,t) of ϕ Dλ(t,·) p(·,t) . Proof. The realization AλH(t) of Aλ(t) in H and ∂ϕtp(·) are both maximal monotone operators in H, so it is sufficient to show that for any u ∈ H, AλH(t)u ⊂ ∂ϕDλ(t,·) p(·,t) (u). Let u ∈ D(AλH(t)) := {u ∈ Xt;A λ(t)u ∈ H} and v := AλH(t)u = Aλ(t)u. So, for all ξ ∈ Xt we have 〈v, ξ − u〉X∗ t ,Xt = 〈Aλ(t)u, ξ − u〉X∗ t ,Xt = ∫ Ω Dλ(t, x)|∇u|p(x,t)−2∇u · (∇ξ −∇u) dx+ ∫ Ω |u|p(x,t)−2u(ξ − u) dx = ∫ Ω Dλ(t, x)|∇u|p(x,t)−2∇u · ∇ξ dx− ∫ Ω Dλ(t, x)|∇u|p(x,t) dx + ∫ Ω |u|p(x,t)−2uξ dx− ∫ Ω |u|p(x,t) dx. Considering q(x, t) such that 1 p(x,t) + 1 q(x,t) = 1, we have 〈v, ξ − u〉X∗ t ,Xt + ∫ Ω Dλ(t, x)|∇u|p(x,t) dx+ ∫ Ω |u|p(x,t) dx = ∫ Ω Dλ(t, x)|∇u|p(x,t)−2∇u∇ξ dx+ ∫ Ω |u|p(x,t)−2uξ dx ≤ ∫ Ω Dλ(t, x)|∇u|p(x,t)−1|∇ξ|dx+ ∫ Ω |u|p(x,t)−1|ξ|dx ≤ ∫ Ω Dλ(t, x) q(x, t) |∇u|(p(x,t)−1)q(x,t) + Dλ(t, x) p(x, t) |∇ξ|p(x,t) dx + ∫ Ω 1 q(x, t) |u|(p(x,t)−1)q(x,t) + 1 p(x, t) |ξ|p(x,t) dx = ∫ Ω Dλ(t, x) q(x, t) |∇u|p(x,t) dx+ ∫ Ω Dλ(t, x) p(x, t) |∇ξ|p(x,t) dx + ∫ Ω 1 q(x, t) |u|p(x,t) dx+ ∫ Ω 1 p(x, t) |ξ|p(x,t) dx. EJDE-2023/50 EVOLUTION EQUATIONS ON TIME-DEPENDENT LEBESGUE SPACES 7 Then 〈v, ξ − u〉X∗ t ,Xt + ∫ Ω Dλ(t, x) ( 1− 1 q(x, t) ) |∇u|p(x,t) dx + ∫ Ω ( 1− 1 q(x, t) ) |u|p(x,t) dx ≤ ∫ Ω Dλ(t, x) p(x, t) |∇ξ|p(x,t) dx+ ∫ Ω 1 p(x, t) |ξ|p(x,t) dx. So, we conclude that 〈v, ξ − u〉X∗ t ,Xt ≤ ϕDλ(t,·) p(·,t) (ξ)− ϕDλ(t,·) p(·,t) (u), (2.6) for all ξ ∈ Xt. If ξ ∈ H −Xt, then ϕ Dλ(t,·) p(·,t) (ξ) = ∞ and consequently (2.6) holds. Therefore, AλH(t)(u) = v ∈ ∂ϕDλ(t,·) p(·,t) (u). � 3. Existence of solutions We consider the problem du dt (t) + ∂ϕtu(t) = f(t), t > τ, u(τ) = uτ ∈ H. (3.1) The following definition was introduced by Yotsutani as [21, Definition 2.1]. Definition 3.1. A function u : [τ, T ] → H is called a strong solution of (3.1) on [τ, T ] if the following holds: (i) u is in C([τ, T ];H), (ii) u is strongly absolutely continuous on any compact subset of (τ, T ), (iii) u(t) is in D(ϕt) for a.e. t ∈ [τ, T ] and satisfies (3.1) for a.e. t ∈ [τ, T ]. For T > τ we introduce the following assumptions: (A7) There is a set τ /∈ Z ⊂ [τ, T ] of zero measure such that ϕt is a lower semicontinuous proper convex function from H into (−∞,∞] with a non- empty effective domain for each t ∈ [τ, T ]− Z; (A8) For any positive integer r there exist a constant Kr > 0, an absolutely continuous function gr : [τ, T ] → R with g′r ∈ Lβ(τ, T ) and a function of bounded variation hr : [τ, T ] → R such that if t ∈ [τ, T ] − Z, w ∈ D(ϕt) with |w| ≤ r and s ∈ [t, T ] − Z then there exists an element w̃ ∈ D(ϕs) satisfying |w̃ − w| ≤ |gr(s)− gr(t)|(ϕt(w) +Kr) α, (3.2) ϕs(w̃) ≤ ϕt(w) + |hr(s)− hr(t)|(ϕt(w) +Kr) (3.3) where α is some fixed constant 0 ≤ α ≤ 1 and β := { 2 if 0 ≤ α ≤ 1 2 , 1 1−α if 1 2 ≤ α ≤ 1. Theorem 3.2 ([21]). Suppose that (A7), (A8) are satisfied. Then, for each f ∈ L2(τ, T ;H) and uτ ∈ D(ϕτ ) equation (3.1) has a unique strong solution u on [τ, T ] with u(τ) = uτ . 8 J. SIMSEN EJDE-2023/50 Using the monotonicity of the operator and the Gronwall Lemma we obtain the following Lemma 3.3. If f, g ∈ L2(τ, T ;H) and u, v are the solutions of the equations du dt (t) + ∂ϕtu(t) = f(t), u(τ) = uτ ∈ H and dv dt (t) + ∂ϕtv(t) = g(t), v(τ) = vτ ∈ H, then for τ ≤ s ≤ t ≤ T , we have ‖u(t)− v(t)‖H ≤ ‖u(s)− v(s)‖H + ∫ t s ‖f(r)− g(r)‖Hdr. Using the monotonicity of the operator, Assumptions (A1) and (A2), Theo- rem 3.2, Lemma 3.3, proceeding as in [20] we obtain the following result. Theorem 3.4. If B : [τ, T ] ×H → H satisfies Assumptions (A1), (A2) and uτ ∈ D(ϕτ ), then there exists a unique u ∈ C([τ, T ];H), such that du dt (t) + ∂ϕtu(t) = B(t, u(t)) a.e. on [τ, T ] and u(τ) = uτ . Let us consider now the problem with our specific operator du dt (t) + ∂ϕ Dλ(t,·) p(·,t) u(t) = f(t), t > τ, u(τ) = uτ ∈ H = D(ϕ Dλ(τ,·) p(·,τ) ). (3.4) Theorem 3.5. For each f ∈ L2(τ, T ;H) and uτ ∈ H equation (3.4) has a unique strong solution u on [τ, T ] with u(τ) = uτ . Proof. Taking Z as the empty set, ϕ Dλ(t,·) p(·,t) is lower semicontinuous proper convex function for each t ∈ [τ, T ]. Consider r a positive integer, Kr := r and α := 1 2 . We define gr : [τ, T ] → R with gr(t) := t + r, and hr(t) := r. We have that gr is an absolutely continuous function g′r = 1 ∈ L2(τ ;T ) and hr is a bounded variation function. For all t ∈ [τ, T ], w ∈ D(ϕ Dλ(t,·) p(·,t) ) = Xt with ‖w‖ ≤ r and s ∈ [t, T ]. Consider the element w̃ := w ∈ Xs = D(ϕ Dλ(s,·) p(·,s) ). We will check that w̃ satisfies (3.2) and (3.3). Note that∫ Ω Dλ(t, x) p(x, t) |∇w|p(x,t) dx+ ∫ Ω 1 p(x, t) |w|p(x,t) dx ≥ 0 Thus, |w̃ − w| = 0 ≤ |s− t| (∫ Ω Dλ(t, x) p(x, t) |∇w|p(x,t) dx+ ∫ Ω 1 p(x, t) |w|p(x,t) dx+ r )1/2 = |gr(s)− gr(t)|(ϕDλ(t,·) p(·,t) (w) +Kr) α. EJDE-2023/50 EVOLUTION EQUATIONS ON TIME-DEPENDENT LEBESGUE SPACES 9 Now, note that ϕ Dλ(s,·) p(·,s) (w̃) = ∫ Ω Dλ(t, x) p(x, t) |∇w̃|p(x,t) dx+ ∫ Ω 1 p(x, t) |w̃|p(x,t) dx = ∫ Ω Dλ(t, x) p(x, t) |∇w|p(x,t) dx+ ∫ Ω 1 p(x, t) |w|p(x,t) dx ≤ ∫ Ω Dλ(t, x) p(x, t) |∇w|p(x,t) dx+ ∫ Ω 1 p(x, t) |w|p(x,t) dx = ϕ Dλ(t,·) p(·,t) (w). Thus, ϕ Dλ(s,·) p(·,s) (w̃) ≤ ϕDλ(t,·) p(·,t) (w) + |hr(s)− hr(t)|(ϕDλ(t,·) p(·,t) (w) +Kr). Then from Theorem 3.2 we obtain the existence of a global solution to (3.4). � Theorem 3.6. For each λ ∈ [0,∞), problem (1.1) has a unique global strong solution uλ whenever uτλ ∈ H. The above theorem follows from Theorem 3.4 with ∂ϕ Dλ(t,·) p(·,t) . 4. Existence of pullback attractors We begin this section recalling some definitions and results on D-pullback at- tractors theory. Definition 4.1 ([13]). Let {Yt}t∈R be a family of nonempty metric spaces. A family of operators {U(t; τ)}t≥τ , with U(t; τ) : Yτ → Yt, which satisfies (i) U(τ ; τ) = Iτ (Identity operator), for all τ ∈ R; (ii) U(t; τ) = U(t; s)U(s; τ), for all τ ≤ s ≤ t, is called an evolution process in {Yt}t∈R. Furthermore, if for any sequences xτ → x in Yτ and yt = U(t; τ)xτ → y in Yt we have y = U(t; τ)x, then U(t; τ) is called closed. Definition 4.2 ([13]). Let {Yt}t∈R be a family of metric spaces. The universal D with respect to the family of spaces {Yt}t∈R is defined as D = {D = {D(t)}t∈R;D(t) ⊂ Yt is nonempty, ∀t ∈ R}. In particular, the universal D is called inclusion closed whenever for any D ∈ D and C = {C(t)}t∈R such that C(t) ⊂ D(t) for all t ∈ R, then C ∈ D. Definition 4.3 ([13]). Let B = {B(t)}t∈R be a family of time-dependent nonempty sets. If B satisfies: for any t ∈ R and D ∈ D, there exists τ(D; t) ≤ t, such that U(t; τ)D(τ) ⊂ B(t) holds when τ ≤ τ(D; t), then B is called the pullback D- absorbing family of the process U(t; τ) : Yτ → Yt. In addition, B is called the uniformly pullback D-absorbing, if for any D ∈ D, there exists the positive constant e(D) which only depends on D, such that U(t; τ)D(τ) ⊂ B(t) for any τ ≤ t− e(D) and t ∈ R. Definition 4.4 ([13]). For any given time-dependent family D = {D(t)}t∈R ∈ D, the process U(t; τ) is called pullback D-asymptotically compact in {Yt}t∈R, if for any t ∈ R, any sequences {τn} ⊂ (−∞, t] which satisfies τn → −∞ as n → ∞ and any sequences yn ∈ D(τn), such that the sequence {U(t; τn)yn} is relatively compact in Yt. 10 J. SIMSEN EJDE-2023/50 Moreover, if for any D ∈ D, the process U(t; τ) is pullback D-asymptotically compact in {Yt}t∈R, then we may call that this process is pullback D-asymptotically compact in {Yt}t∈R. Lemma 4.5 ([13]). Let B = {B(t)}t∈R ∈ D be the pullback D-absorbing family of U(t; τ). The process U(t; τ) is pullback D-asymptotically compact in {Yt}t∈R, if B(t) is compact in Yt for any t ∈ R. Definition 4.6 ([16]). For any time-dependent family D = {D(t)}t∈R ∈ D, the pullback ω-limit set with respect to the process U(t; τ) in the family of spaces {Yt}t∈R is defined as ω(D; t) := ∩s≤t∪τ≤sU(t; τ)D(τ) Yt . Definition 4.7 ([16, 13]). A family of sets A = {A(t)}t∈R is said to be the pullback D-attractor for the process U(t; τ) : Yτ → Yt, if (i) For any t ∈ R, A(t) is a nonempty compact subset of Yt; (ii) A pullback attracts every D ∈ D, lim τ→−∞ distYt(U(t; τ)D(τ), A(t)) = 0, for all t ∈ R; and (iii) A is invariant, i.e., U(t; τ)A(τ) = A(t) for all t ≥ τ . Moreover, if for any pullback D-attracting family C = {C(t)}t∈R composed of nonempty closed set family, there is A(t) ⊂ C(t) for any t ∈ R, then it is said that the pullback D-attractor has the property of minimality. The following theorem will be used to obtain the existence of the pullback D- attractors. Theorem 4.8 ([15]). Let U(t; τ) : Yτ → Yt be the closed process in the fam- ily of time-dependent metric spaces {Yt}t∈R, D is the universal (with respect to {Yt}t∈R). If the process U(t; τ) possesses the pullback D-absorbing families B0 = {B0(t)}t∈R and it is pullback B0-asymptotically compact. Then the family of sets AD = {A(t)}t∈R is the minimal pullback D-attractor of the process U(t; τ), where the sections A(t) are given as follows A(t) = ∪D∈Dω(D, t) Yt , ∀ t ∈ R. If the pullback D-absorbing family B0 is in D, then A(t) = ω(B0; t) ⊂ B0(t) Yt , ∀t ∈ R. Moreover, if B0(t) is a closed subset in Yt for any t ∈ R, and the universal D is inclusion closed, then the pullback D-attractor AD belongs to D. Definition 4.9. A global (or entire) solution of a process U(·, ·) is a function ξ : R → {Yt}t∈R such that for each ` ∈ R ξ(`) ∈ Y` and U(t, s)ξ(s) = ξ(t) for all t ≥ s. Using the invariance of the pullback attractor and proceeding as in [7, Lemma 1.10, page 9] we obtain the following result. Theorem 4.10. The pullback D-attractor in Theorem 4.8 consists of a collection of global solutions. EJDE-2023/50 EVOLUTION EQUATIONS ON TIME-DEPENDENT LEBESGUE SPACES 11 Now consider problem (1.1) but with the initial datum u0λ ∈ Yτ = Lp(·,τ)(Ω) ⊂ H, ∂uλ ∂t (t)− div ( Dλ(t, x)|∇uλ(t)|p(x,t)−2∇uλ(t) ) + |uλ(t)|p(x,t)−2uλ(t) = B(t, uλ(t)) uλ(τ) = uτλ ∈ Yτ . (4.1) Consider Uλ(t; τ) : Yτ → Yt, where Uλ(t; τ) = uλ(t), the global solution of (4.1). Moreover consider the universal D = {D = {D(t)}t∈R;D(t) ⊂ Yt is nonempty, ∀t ∈ R}. Proceeding analogously as in [14] we obtain the following two uniform estimates. Theorem 4.11. Let uλ be a solution of (4.1). Then there exist a constant T1 and a nondecreasing function B1 : R→ R such that ‖uλ(t)‖H ≤ B1(t), ∀ t ≥ T1 + τ and λ ∈ [0,∞). Theorem 4.12. Let uλ(·) ∈ C([τ,∞);H) be the global solution of (4.1). Then there exist a constant T2 > 0 and a nondecreasing function B2 : R→ R such that ‖uλ(t)‖Xt ≤ B2(t), ∀ t ≥ T2 + τ, λ ∈ [0,∞). Theorem 4.13. The evolution process {Uλ(t; τ)}t≥τ associated with problem (4.1) in {Yt}t∈R has a pullback D-attractor {Aλ(t) : t ∈ R} =: Aλ ∈ D. Proof. If xτ → x in Yτ and yt := Uλ(t; τ)xτ → y in Yt we have that xτ → x in H. Using the monotonicity of the main operator and that B is Lipschitz, it is easy to show that yt → Uλ(t; τ)x in H. Since yt → y in Yt implies that yt → y in H, then by the uniqueness of the limit we conclude that y = Uλ(t; τ)x. So, the process is closed. Now consider B0 = {B0(t)}t∈R, B0(t) = BXt(0, B2(t)) Yt . Observe that the sets B0(t) are compact in Yt. Theorem 4.12 shows that B0 is a pullback D-absorbing family of the process Uλ(t; τ) : Yτ → Yt. Then, by Lemma 4.5, Uλ(t; τ) is pullback B0-asymptotically compact in {Yt}t∈R. Thus, the existence of the pullback D- attractor follows from Theorem 4.8. � Remark 4.14. (i) By Theorem 4.10 the pullback D-attractor in the previous the- orem consists of a collection of global solutions and by Theorem 4.8, the sections of the pullback attractor are characterized as Aλ(t) = ωλ(B0; t) ⊂ B0(t) Yt , ∀ t ∈ R. (ii) Using the invariance of the pullback attractors and Theorem 4.12, we have that Aλ(t) ⊂ Xt. As a consequence of Theorem 4.12, we have the following corollary. Corollary 4.15. ∪λ∈[0,∞)Aλ(τ) Yt is a compact subset of Yt and ∪λ∈[0,∞)Aλ(τ) H is a compact subset of H for each τ ∈ R. 12 J. SIMSEN EJDE-2023/50 5. Robustness of the pullback attractors Consider a family of functions Dλ ∈ L∞([τ, T ]×Ω) with 0 < β ≤ Dλ(t, x) ≤M in [τ, T ]× Ω, λ ∈ [0,∞), Dλ → Dλ1 in L∞([τ, T ]× Ω) as λ→ λ1. Our objective in this section is to prove that the family of pullback attractors be- haves as upper semicontinuously with respect to positive finite diffusion parameters. Proceeding as in the proof of [14, Theorem 4.1] we obtain the following. Theorem 5.1. Let {Uλ(t, τ) : t ≥ τ} be the evolution process generated by the problem (4.1). If {uτλ : λ ∈ [0,∞)} is a bounded set in Xτ and uτλ → uτλ1 in H as λ → λ1, then Uλ(t, τ)uτλ → Uλ1 (t, τ)uτλ1 in H as λ → λ1, uniformly for t in compact subsets of R. Theorem 5.2. The family of pullback attractors {Aλ(t) : t ∈ R}, λ ∈ [0,∞) is upper semicontinuous at λ1 in the topology of H. Proof. We will prove that for each t ∈ R, distH ( Aλ(t),Aλ1(t) ) → 0 as λ→ λ1. For t ∈ R and ε > 0, let τ ∈ R be such that distYt ( Uλ1(t, τ)B(τ),Aλ1(t) ) < ε 3 , where ∪λ∈[0,∞)Aλ(τ) ⊂ B(τ) and B(τ) is a nonempty set in Xτ ⊂ Yτ (see Theo- rem 4.12). Once Yt ⊂ H, we have distH ( Uλ1(t, τ)B(τ),Aλ1(t) ) < ε 3 . Using the invariance of the pullback attractors, Theorems 4.12 and 5.1, there exists δ = δ(ε) > 0 such that sup ψλ∈Aλ(τ) ‖Uλ(t, τ)ψλ − Uλ1 (t, τ)ψλ‖H < ε 3 for all |λ− λ1| < δ. Then distH ( Aλ(t),Aλ1(t) ) = distH ( Uλ(t, τ)Aλ(τ),Aλ1(t) ) = sup ψλ∈Aλ(τ) distH ( Uλ(t, τ)ψλ,Aλ1(t) ) ≤ sup ψλ∈Aλ(τ) { distH (Uλ(t, τ)ψλ, Uλ1(t, τ)ψλ) + distH ( Uλ1(t, τ)ψλ,Aλ1(t) )} ≤ ε 3 + ε 3 < ε, for all |λ− λ1| < δ, showing the upper semicontinuity as desired. � Acknowledgements. This work was partially supported by the Brazilian research agency FAPEMIG - Process APQ-01601-21. References [1] C. O. Alves, S. Shmarev, J. Simsen, M. Simsen; The Cauchy problem for a class of parabolic equations in weighted variable Sobolev spaces: existence and asymptotic behavior, J. Math. Anal. Appl., 443 (2016), no. 1, 265–294. EJDE-2023/50 EVOLUTION EQUATIONS ON TIME-DEPENDENT LEBESGUE SPACES 13 [2] S. Antontsev, S. Shmarev; Evolution PDEs with nonstandard growth conditions. Existence, uniqueness, localization, blow-up, Atlantis Studies in Differential Equations, 4. Atlantis Press, Paris, 2015. [3] S. Antonsev, S. Shmarev; A model porous medium equation with variable exponent of non- linearity: Existence, uniqueness and localization properties of solutions, Nonlinear Anal., 60 (2005), 515–545. [4] M. Avci; Ni-Serrin type equations arising from capillarity phenomena with non-standard growth, Bound. Value Probl., 2013: 55 (2013), 1–13. [5] M. Avci, B. Cekic, A. V. Kalinin, R. A. Mashiyev; Lp(x)(Ω)-estimates of vector fields and some applications to magnetostatics problems, J. Math. Anal. Appl., 389 (2012), No. 2, 838–851. [6] V. Barbu; Nonlinear Semigroups and Differential Equations in Banach Space, Noordhoff International, 1976. [7] A. N. Carvalho, J. A. Langa, J. C. Robinson; Attractors for Infinite-dimensional Non- autonomous Dynamical Systems, Applied Mathematical Sciences 182, Springer-Verlag, 2012. [8] Y. Chen, S. Levine, M. Rao; Variable exponent, linear growth functionals in image restoration, SIAM J. Math., 66 (4) (2006) 1383–1406. [9] L. Diening, P. Harjulehto, P. Hästö, M. Růžička; Lebesgue and Sobolev Spaces with Variable Exponents, Springer-Verlag, Berlin, Heidelberg, 2011. [10] X. L. Fan, Q. H. Zhang; Existence of solutions for p(x)-laplacian Dirichlet problems, Nonlin- ear Anal., 52 (2003) 1843–1852. [11] X. L. Fan, D. Zhao; On the spaces Lp(x)(Ω) and Wm,p(x)(Ω), J. Math. Anal. Appl., 263 (2001), 424–446. [12] Z. Guo, Q. Liu, J. Sun, B. Wu; Reaction-diffusion systems with p(x)−growth for image denoising, Nonlinear Anal. Real World Appl., 12 (2011), 2904–2918. [13] P. E. Kloeden, P. Maŕın-Rubio, J. Real; Pullback attractors for a semilinear heat equation in a non-cylindrical domain, J. Differential Equations, 244 (2008), 2062–2090. [14] P. E. Kloeden, J. Simsen; Pullback attractors for non-autonomous evolution equation with spatially variable exponents, Commun. Pure & Appl. Analysis, 13 , no. 6, (2014), 2543–2557. [15] T. F. Ma, P. Maŕın-Rubio, C. M. S. Chuño; Dynamics of wave equations with moving bound- ary, J. Differential Equations, 262 (2017), 3317–3342. [16] P. Maŕın-Rubio, J. Real; On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems, Nonlinear Analysis, 71 (2009), 3956–3963. [17] K. Rajagopal, M. Růžička; Mathematical modelling of electrorheological fluids, Contin. Mech. Thermodyn., 13 (2001), 59–78. [18] M. Růžička; Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Math., vol. 1748, Springer-Verlag, Berlin, 2000. [19] J. Simsen; A global attractor for a p(x)-Laplacian parabolic problem, Nonlinear Anal., 73 (10) (2010), 3278–3283. [20] J. Simsen, M. J. D. Nascimento, M. S. Simsen; Existence and upper semicontinuity of pullback attractors for non-autonomous p-Laplacian parabolic problems, J. Math. Anal. Appl., 413 (2014), 685–699. [21] S. Yotsutani; Evolution equations associated with the subdifferentials, J. Math. Soc. Japan, 31 (1978), 623–646. Jacson Simsen Instituto de Matemática e Computação, Universidade Federal de Itajubá, 37500-903 Itajubá, Minas Gerais, Brazil Email address: jacson@unifei.edu.br 1. Introduction 2. Operator A and its properties 3. Existence of solutions 4. Existence of pullback attractors 5. Robustness of the pullback attractors Acknowledgements References