Existence and dimensions of global attractors for a delayed reaction-diffusion equation on an unbounded domain

Nonlinearity Nonlinearity 38 (2025) 075008 (28pp) https://doi.org/10.1088/1361-6544/adde0b Existence and dimensions of global attractors for a delayed reaction-diffusion equation on an unbounded domain Wenjie Hu1,2,∗, Tomás Caraballo3,4and Alain Miranville5,6 1The MOE-LCSM, School of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, People’s Republic of China 2Journal House, Hunan Normal University, Changsha, Hunan 410081, People’s Republic of China 3Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, c/ Tarfia s/n, 41012 Sevilla, Spain 4Department of Mathematics, Wenzhou University, Wenzhou, Zhejiang Province 325035, People’s Republic of China 5School of Mathematics and Statistics, Henan Normal University, Xinxiang, Henan Province 453007, People’s Republic of China 6Laboratoire de Mathématiques Appliquées du Havre (LMAH), Université Le Havre Normandie, 25, rue Philippe Lebon BP 1123, 76063 Le Havre cedex, France E-mail: [email protected],[email protected] and [email protected] Received 3 August 2024; revised 31 March 2025 Accepted for publication 28 May 2025 Published 5 June 2025 Recommended by Dr Andrej Zlato Abstract The purpose of this paper is to investigate the existence and Hausdorff dimension as well as fractal dimension of global attractors for a delayed reactiondiffusion equation on an unbounded domain. The noncompactness of the domain causes the Laplace operator to have a continuous spectrum, the semigroup generated by the linear part and the Sobolev embeddings are no longer compact, making the problem more difficult compared with the bounded domain case. We first obtain the existence of an absorbing set for the infinite dimensional dynamical system generated by the equation thanks to a priori estimates of the solution. Then, we show the asymptotic compactness of the solution semiflow by uniform a priori estimates for far-field values of solutions together with the Arzel` a–Ascoli theorem, which facilitates us to show the existence of global attractors. By decomposing the solution into three parts and establishing squeezing properties of each part, we obtain the explicit ∗Author to whom any correspondence should be addressed. © 2025 IOP Publishing Ltd & London Mathematical Society. All rights, including for text and data mining, AI training, and similar technologies, are reserved. 1 Nonlinearity 38 (2025) 075008 W Hu et al upper bounds of both Hausdorff dimension and fractal dimension of the global attractors, which only depend on the inner characteristics of the equation, while not related to the entropy number compared with the existing literature. Keywords: Hausdorff dimension, fractal dimension, unbounded domain, global attractors, delay, reaction-diffusion equation Mathematics Subject Classification numbers: 35R10, 35B41, 47H20 1. Introduction Consider the following delayed reaction-diffusion equation on RN (∂u ∂t(x,t)=∆u(x,t)−µu(x,t) + σu(x,t−τ) + f(u(x,t−τ)) + g(x),t>0,x∈RN, u0(x,s) = ϕ(x,s),−τ⩽s⩽0,x∈RN,(1.1) where Nis a positive integer, u(·,t)∈X≜L2(RN), the square Lesbegue integrable function on RNwith usual inner product (·,·)and norm ∥·∥ for any t⩾0, ∆is the Laplacian operator on RN,µ,σ and τare positive constants, g∈X. Furthermore, ut∈C is defined by ut(x,θ) = u(x,t+θ),−τ⩽θ⩽0, where Cis the space of continuous functions from [−τ,0] to Xequipped with the supremum norm ∥φ∥C=supθ∈[−τ,0]∥φ(θ)∥for any φ∈C. The initial datum φ∈C and f:R→Ris a nonlinear function. Equation (1.1) arises largely from the real world physical, chemical and biological processes with time delays. For instance, in the case σ=0 and g≡0, (1.1) can be used to describe the evolution of the mature population of species, such as birds, whose immature individuals do not move around but the mature ones do and hence have drawn much attention from the mathematical biology community in the past decades. Most existing works are only concerned with the existence and qualitative properties of traveling wave front solutions, which may explain the invasion of species. See, for instance, [22,24,33,34] and the references therein. However, the asymptotic behaviour of (1.1) has not received enough attention due to the non-compactness of the spatial domain, which causes many existing method cannot be applied directly. In order to overcome this problem, Yi et al [41] proposed a new dynamical system approach to study the existence and global attractivity of a positive steady state by delicately constructing a priori estimates for nontrivial solutions under the compact open topology. The methods have been generalized to investigate the existence and structure of attractors for deterministic and stochastic nonlocal delayed reaction-diffusion equations on a semi-infinite domain with a Dirichlet boundary condition at the finite end in [30,42] and have been adopted to tackle similar problems with Neumann boundary condition in [29]. One naturally wonders, what is the dynamics of (1.1) in the natural phase space Cunder the usual supremum norm provided the conditions on fused in [41] is removed? Since (1.1) generates an infinite dimensional dynamical system in C, as a first step, if we can obtain the existence of attractors for (1.1), then its essential dynamics can be reduced to a compact set. Furthermore, if the attractors have finite dimension, then one is likely to apply the analysis and computation tools of finite dimension dynamical systems to study the complex structure of the attractors and hence give a global picture of the dynamics of (1.1). Indeed, the study of existence and estimation of Hausdorff dimension as well as fractal dimension of attractors for both deterministic and random infinite dimensional dynamical systems generated by evolution equations has drawn much attention from both dynamical system and applied physics 2 Nonlinearity 38 (2025) 075008 W Hu et al community in the past decades due to the significant roles they played in the study of the long time behaviour of nonlinear dynamical systems. The theory of existence of attractors for deterministic infinite dimensional dynamical systems has been well established, see the monographs [2,25,37]. Criteria for the finite Hausdorff dimensionality of attractors for deterministic fluid dynamics models were firstly derived by Douady and Oesterle [14], which was later generalized by Constantin, Foias and Temam [10,11] (see also the book of Temam [37]). Since then, the methods have been widely and extensively adopted to investigate dimensions of global attractors for various partial differential equations (PDEs) on different domains. For PDEs on bounded domains, [5] studied the existence and dimension estimation of attractors for autonomous N–S equations, [12] studied the dimension of the attractors in two-dimensional turbulence. Moreover, [8] extended the results to the nonautonomous case. There is a huge literature in this field to be covered here and more works can be found in monographs [2,25,37] and the references therein. On the other hand, when the domain is unbounded, several problems arise. As pointed in [7], the Laplace operator has a continuous spectrum, H1(Rn)is not compactly embedded in L2(Rn)and the solution semiflow does not have compact absorbing sets in the original topology, causing the method of obtaining compactness by a priori estimates and compactness of the Sobolev embeddings for the bounded case ineffective. In order to overcome these difficulties, [1] established a method to study the existence and dimensions estimation of attractors for the autonomous reaction diffusion equations on unbounded domain in a weighted space, which was then adopted by [16] to explore dimensions of reaction diffusion equations, used by [21] to explore global attractors for degenerate parabolic equations and extended to the nonautonomous case in [7]. However, when working in weighted spaces one has to impose an additional condition ensuring that the initial data and forcing term also belong to the corresponding spaces [23]. Thus, [38] established a new method to obtain the results in the usual Hilbert space based on uniform a priori estimates for far-field values of solutions following the ideas in [3,4], where global attractors of generalized semiflows and the Navier–Stokes equations as well as the damped semilinear wave equations were studied. The above mentioned theory and methods are mainly established for PDEs in Hilbert spaces. Nevertheless, the natural phase space Cof (1.1) is a Banach space. The lack of smooth inner product yields that the above theoretical results cannot be directly applied. Thus, for the purpose of obtaining the existence and dimensions estimation of global attractors for delayed PDEs, So and Wu [36] recast general semilinear partial functional differential equations into a product Hilbert space and obtained topological dimensions of attractors by a semigroup approach. Su and Qin [35] also recast the 2D Navier–Stokes–Voight equations with a distributed delay in a product Hilbert space to establish the existence and estimation of Hausdorff dimensions of attractors by virtue of energy estimates as well as compact embedding techniques. Both works consider equations on bounded domains and recast functional differential equations in an auxiliary product Hilbert space, which is unnatural and may cause redundancy in the dimensions estimation. In our current work, we investigate the existence as well as their Hausdorff dimension and fractal dimension estimation of global attractors for (1.1) directly in the natural phase space, i.e. the Banach space C. In order to show the existence of global attractors, we show the asymptotic compactness of the solutions semiflow by uniform a priori estimates for far-field values of solutions. For the purpose of estimating the dimensions of the obtained attractors, we decompose the solution of (1.1) into a sum of three parts, among which each part fulfills a squeezing property. The idea of decomposing the solutions originates from [43], where fractal dimension of attractors for stochastic non-autonomous reaction-diffusion equation in R3was studied. Unlike [43], where orthogonal projectors with finite rank in Hilbert spaces together 3 Nonlinearity 38 (2025) 075008 W Hu et al with variation techniques are adopted to analyze the finite dimension of the first part and the decay of the other two parts, we adopt the phase space decomposition based on the exponential dichotomy to overcome the barrier caused by the lack of smooth inner product in a Banach space. Here, our approach gives explicit bounds that only depend on the inner characteristics of equation (1.1), while not related to the entropy number as [6,13,18–20,31] did. The remaining of this paper is organized as follows. Section 2is devoted to preliminaries, including notation, definitions and some lemmas. In section 3, we firstly give some a priori estimates of the solution on an unbounded domain and a priori estimates for far-field values of solutions, which shows the existence of absorbing sets. Then, we prove asymptotic compactness of the solution semiflow in section 4, which combined with the existence of absorbing sets implies the existence of global attractors. In section 5, we construct squeezing properties by decomposing the solution into three parts, two on a bounded domain and the third on an infinite domain. Subsequently, we give explicit upper bounds of the Hausdorff and fractal dimensions of the obtained attractors in section 6by the squeezing properties established in section 5. Finally, we summarize the paper by making a detailed comparison of the present work with the methods established by Chepyzhov, Efendiev, Miranville and Zelik in their ground breaking pioneer works [7,16–18] and point out some potential directions for future study. 2. Preliminaries We first introduce more notation used throughout the remaining part of this paper. Define the Hilbert space X1as X1≜{ϕ∈X|∂φ(x) ∂xi∈X,i=1,2,···,N,x∈RN}. For convenience, we take the seminorm ∥ϕ∥X1= [´RN|∇ϕ(x)|2dx]1 2,ϕ ∈X1as the norm of X1. Denote by C1= C([−τ,0],X1)the set of all continuous functions from [−τ,0]to X1equipped with the usual supremum norm ∥φ∥C1=sup{∥φ(ξ)∥X1:ξ∈[−τ,0]}for all φ∈C1. For any given 0 <K<+∞, denote the ball centered at 0 with radius Kby ΩK= {x∈RN:|x|<K}, its boundary ∂ΩKby ∂ΩK=x∈RN:|x|=Kand its complement by ΩC K=x∈RN:|x|⩾Krespectively. Let XΩKbe the Hilbert space L2(ΩK), which is the square Lesbegue integral functions on ΩK. Define X1 ΩKas X1 ΩK≜{ϕ∈ XΩK|∂φ(x) ∂xi∈L2(ΩK),i=1,2,···,N,x∈ΩK}. For convenience, we take the seminorm ∥ϕ∥X1 ΩK = [´ΩK|∇ϕ(x)|2dx]1 2,ϕ ∈X1 ΩKas the norm of X1 ΩKsince it is in fact a norm equivalent to the usual norm of X1 ΩK. Denote by CΩK=C([−τ,0],XΩK)and C1 ΩK=C([−τ,0],X1 ΩK)the set of all continuous functions from [−τ,0]to XΩKand X1 ΩKequipped with the usual supremum norm ∥ψ∥CΩK=sup{∥ψ(ξ)∥XΩK:ξ∈[−τ,0]}and ∥ϕ∥C1 ΩK =sup{∥ϕ(ξ)∥X1 ΩK :ξ∈[−τ,0]} for all ψ∈CΩKand ϕ∈C1 ΩKrespectively. By the Fourier transformation [26,41], we can see that the semigroup S(t) : X→Xgenerated by ∆−µIis defined as    S(0)[φ](x) = φ(x), S(t)[φ](x) = e−µt (4πt)N/2ˆRN φ(y)e−|x−y|2 4tdy,t∈(0,∞),(2.1) for (x,φ)∈RN×X, which is analytic and strongly continuous on X. For later use, we introduce the following results concerning the properties of semigroup {S(t)}t⩾0. The details of the proof can be found in [41, lemmas 2.1, 2.4]. 4 Nonlinearity 38 (2025) 075008 W Hu et al Lemma 2.1. Let {S(t)}t⩾0be defined by (2.1). For any given constant a, let a be the constant function defined by a(θ,x)≡a for any (θ,x)∈[−τ,0]×(0,∞), then we have the following results. (i) ∥S(t)φ∥⩽e−µt∥φ∥for all φ∈X, t ∈R+. (ii) {S(t)}t⩾0is an analytic and strongly continuous semigroup on X. (iii) For all t ∈(0,∞)and (x,a)∈(0,∞)×X, there holds S(t)[a](x) = ae−µt. In the remaining part, we define a nonlinear operator f:C →Xby f(φ)(·) = f(φ(−τ,·)) and always assume that the nonlinear term fsatisfies f(0) = 0 and the following Lipschitz conditions. Hypothesis A1 ∥f(φ1)−f(φ2)∥⩽Lf∥φ1−φ2∥Cfor any φ1,φ2∈C. Here, for notation simplicity, we use the same letter fto represent the operator. Following similar techniques as [39, theorem 8], we have the following results on the existence of solutions. Lemma 2.2. Assume that Hypothesis A1 holds. Then, (i) for any φ∈C and T >0, there exists a solution u(·)to problem (1.1) with u ∈ L2(−τ,T;X)∩L2(0,T;X1)∩L∞(0,T;X)∩C([−τ,T];X), (ii) for any φ∈C1and T >0, problem (1.1) admits a strong solution u ∈L2(0,T;X)∩ C([−τ,T];X1). By the variation of constants method, (1.1) is equivalent to the following integral equation with the given initial function    u(t) = S(t)ϕ(0) + σˆt 0 S(t−s)u(s−τ)ds+ˆt 0 S(t−s)[f(u(s−τ)) + g]ds,t>0, u0=ϕ∈C. (2.2) Let χΩKand χΩC Kbe the characteristic functions on ΩKand ΩC Krespectively, that is χΩK(x) = 0,x∈ΩC K, 1,x∈ΩK,(2.3) and χΩC K(x) = 0,x∈ΩK, 1,x∈ΩC K.(2.4) Let v(t,x)and w(t,x)be the solutions of the following two problems ∂v(x,t) ∂t= ∆v(x,t)−µv(x,t) + σv(x,t−τ) + f(u(x,t−τ))χΩK(x) + g(x)χΩK(x), v(x,s) = ϕ(x,s)χΩK(x)≜φ,s∈[−τ,0],x∈RN(2.5) and (∂w(x,t) ∂t= ∆w(x,t)−µw(x,t) + σw(x,t−τ) + f(u(x,t−τ))χΩC K(x) + g(x)χΩC K(x), w(x,s) = ϕ(x,s)χΩC K(x)≜ψ,s∈[−τ,0],x∈RN,(2.6) 5 Nonlinearity 38 (2025) 075008 W Hu et al respectively. Then, we can see u(x,t) = v(x,t) + w(x,t), for any t⩾−τand x∈RN, where u is the solution to (1.1). For later use, we introduce the following lemmas from [25] concerning the existence of global attractors for continuous semigroups and finite cover of finite dimensional subspace F of a Banach space X. Lemma 2.3. Let X be a Polish space and S(t) : X→X be a continuous semigroup for all t ⩾0 which is asymptotically compact. If S(t) admits a bounded absorbing set B⊆X, then it has a global attractor A⊆B, which is given by A=\ s⩾0[ t⩾s S(t)B. Let Fbe a finite dimensional subspace of a Banach space Xand BF r(x)be a ball with center xand radius rin F, i.e. BF r(x) = {y∈F|∥y−x∥⩽r}. By [32, lemma 2.1], we have that the following covering lemma of balls in finite dimensional Banach spaces. Lemma 2.4. Let r1>r2>0be radii of two balls, F be a given finite dimensional subspace of a Banach space X with dimension m and Nr1,BF r2be minimum number of balls needed to cover the ball of radius r1by balls of BF r2calculated in the Banach space X. Then, Nr1,BF r2 is bounded by Nr1,BF r2⩽m2m1+r1 r2m .(2.7) 3. Uniform estimates of solutions Let uϕ∈C([−τ,∞),X)be the global solution obtained in lemma 2.3. Define the solution semigroup Φ(t) : C →C,t∈R+,of (1.1) as Φ(t)φ=uϕ t. The following lemma, which was proved in [28], shows that the dynamical system Φpossesses an absorbing set. For completeness, we also provide the proof here. Lemma 3.1. Assume that f is bounded, that is, there exists a N >0 such that for any φ∈C, we have ∥f(φ)∥⩽N. Assume further that σeµτ −µ < 0, then the dynamical system Φadmits an absorbing set Bdefined by B=φ∈C :∥φ∥C⩽2M µ+Mσeµτ µ(µ−σeµτ ),(3.1) where M =N+∥g∥. Proof. It follows from (2.4), lemma 2.1 and boundedness of fthat ∥u(t)∥⩽∥S(t)ϕ(0)∥+    σˆt 0 S(t−s)u(s−τ)ds    +   ˆt 0 S(t−s)[f(u(s−τ,·)) + g]ds    ⩽e−µt∥ϕ(0)∥+σˆt 0 e−µ(t−s)∥u(s−τ)∥ds+ˆt 0 e−µ(t−s)[∥f(u(s−τ,·))∥+∥g∥]ds ⩽e−µt∥ϕ(0)∥+σˆt 0 e−µ(t−s)∥u(s−τ)∥ds+M1−e−µt µ. (3.2) 6 Nonlinearity 38 (2025) 075008 W Hu et al Thus, for any ξ∈[−τ,0], we have ∥u(t+ξ)∥⩽e−µ(t+ξ)∥ϕ(0)∥+σˆt+ξ 0 e−µ(t+ξ−s)∥u(s−τ)∥ds+ M1−e−µ(t+ξ) µ ⩽eµτ e−µt∥ϕ(0)∥+σeµτ ˆt 0 e−µ(t−s)∥u(s−τ)∥ds+M µ. (3.3) Keeping in mind that ∥ut∥C=sup{∥u(t+ξ)∥:ξ∈[−τ,0]},we have ∥ut∥C⩽eµτ e−µt∥φ∥C+σeµτ ˆt 0 e−µ(t−s)∥us∥Cds+M µ.(3.4) Multiplying both sides of (3.4) by eµtimplies eµt∥ut∥⩽eµτ ∥φ∥C+σeµτ ˆt 0 eµs∥us∥Cds+Meµt µ.(3.5) Applying Gronwall’s inequality yields eµt∥ut∥C⩽M µeµt+eµτ ∥φ∥C+Mβeβt µ(µ−β)he(µ−β)t−1i+eβt−1eµτ ∥φ∥C,(3.6) where β=σeµτ . Multiplying both sides of (3.4) by e−µtgives ∥ut∥C⩽M µ+eµ(τ−t)∥φ∥C+Mβ µ(µ−β)h1−e−(µ−β)ti+eµτ ∥φ∥Ce(β−µ)t ⩽M µ+Mβ µ(µ−β)+eµ(τ−t)∥φ∥C+eµτ ∥φ∥Ce(β−µ)t. (3.7) Since we have assumed that σeµτ −µ < 0, one can see, for any bounded set D⊂Cand φ∈D, there exists a TD>0 such that for all t⩾TD eµ(τ−t)∥φ∥C+eµτ ∥φ∥Ce(β−µ)t⩽M µ+Mβ µ(µ−β),(3.8) indicating that ∥ut∥C⩽2M µ+Mβ µ(µ−β).(3.9) That is, Bis an absorbing set for Φ. This completes the proof. Next, we establish some estimations of the integral of the solution uϕ t(·)to equation (1.1) in C1, which is important to obtain estimations of uϕ t(·)in C1. Lemma 3.2. Assume that Hypothesis A1 and assumptions of lemma 3.1 hold and let TDbe defined in lemma 3.1. Then, the solution uϕ t(·)to equation (1.1) satisfies ˆt+1 t uϕ s(·)  2 C1ds⩽c4,(3.10) for all t >TD+τ, where c4=1 2c2∥φ(0)∥2+(σ+L2 f)c2 3 µ−σ−1+c1 2, c1=1 µ−σ−1(1 µ∥g∥2+2∥f(0)∥2), c2=e(µ−σ−1)τand c3=2(M µ+Mβ µ(µ−β)). 7 Nonlinearity 38 (2025) 075008 W Hu et al Proof. Taking inner product of each term of (1.1) with u(t) in X, we obtain 1 2 d∥u(t)∥2 dt=−∥∇u(t)∥2−µ∥u(t)∥2+ˆRN σu(t−τ)u(t)dx +ˆRN f(u(t−τ))u(t)dx+ˆRN gu(t)dx. (3.11) We now estimate each term on the right-hand side of (3.11). By the basic inequality 2ab ⩽ a2+b2and Hypothesis A1, we deduce σˆRN u(t−τ)u(t)dx⩽σ1 2∥ut∥2 C+1 2∥u(t)∥2,(3.12) ˆRN gu(t)dx⩽1 2µ∥g∥2+µ 2∥u(t)∥2(3.13) and ˆRN f(ut)u(t)dx⩽1 2∥f(ut)∥2+1 2∥u(t)∥2 ⩽1 2L2 f∥ut∥2 C+1 2∥u(t)∥2. (3.14) Therefore, by incorporating (3.12)–(3.14) into (3.11), we are led to d∥u(t)∥2 dt⩽−2∥∇u(t)∥2+ (σ−µ+1)∥u(t)∥2+σ+L2 f∥ut∥2 C+1 µ∥g∥2 ⩽(σ−µ+1)∥u(t)∥2+σ+L2 f∥ut∥2 C+1 µ∥g∥2. (3.15) Thanks to the Gronwall inequality, we find, for all t⩾0, ∥u(t)∥2⩽e(σ−µ+1)t∥φ(0)∥2+σ+L2 fˆt 0 e(σ−µ+1)(t−s)∥us∥2 Cds+c1,(3.16) where c1=∥g∥2 µ(µ−σ−1). This implies, for all t∈[0,∞)and ζ∈[−τ,0], ∥u(t+ζ)∥2⩽e(σ−µ+1)(t+ζ)∥ϕ(0)∥2+σ+L2 fˆt+ζ 0 e(σ−µ+1)(t+ζ−s)∥us∥2 Cds+c1 ⩽e(σ−µ+1)(t−τ)∥ϕ(0)∥2+σ+L2 fe(µ−σ−1)τˆt 0 e(σ−µ+1)(t−s)∥us∥2 Cds+c1. (3.17) Noticing that ∥ut∥C=sup{∥u(t+ξ)∥:ξ∈[−τ,0]},we obtain ∥ut∥2 C⩽e(σ−µ+1)(t−τ)∥φ(0)∥2+σ+L2 fe(µ−σ−1)τ ׈t 0 e(σ−µ+1)(t−s)∥us∥2 Cds+c1.(3.18) Let T1⩾TD+τand replace tby T1in (3.18). We then have ∥uT1∥2 C⩽e(σ−µ+1)(T1−τ)∥φ(0)∥2+σ+L2 fe(µ−σ−1)τ ׈T1 0 e(σ−µ+1)(T1−s)∥us∥2 Cds+c1. (3.19) 8 Nonlinearity 38 (2025) 075008 W Hu et al For every t⩾T1, multiplying (3.19) by e(µ−σ−1)(T1−t)gives e(µ−σ−1)(T1−t)∥uT1∥2 C⩽e(σ−µ+1)(t−τ)∥φ(0)∥2+σ+L2 fe(µ−σ−1)τ ׈T1 0 e(σ−µ+1)(t−s)∥us∥2 Cds+c1. (3.20) Applying Gronwall’s inequality to (3.15) on the interval [T1+ζ,t+ζ],ζ ∈[−τ,0], we obtain ∥u(t+ζ)∥2+2ˆt+ζ T1+ζ e(µ−σ−1)(s−t−ζ)∥∇u(s)∥2ds ⩽e(µ−σ−1)(T1−t)∥u(T1+ζ)∥2+σ+L2 fˆt+ζ T1+ζ e(µ−σ−1)(s−t−ζ)∥u(s−τ)∥2ds +c1 µ−σ−1e(µ−σ−1)τ.(3.21) By a change of variable in the second term on the left hand side of (3.21) and keeping in mind that ∥ut∥C=sup{∥u(t+ξ)∥:ξ∈[−τ,0]}, we deduce ∥ut∥2 C+2ˆt T1 e(µ−σ−1)(s−t)∥us∥2 C1ds ⩽1 2e(µ−σ−1)(T1−t)∥uT1∥2 C+1 2σ+L2 fˆt T1−τ e(µ−σ−1)(s−t−τ)∥us∥2 Cds +c1 2(µ−σ−1)e(µ−σ−1)τ. (3.22) It follows from lemma 3.1 that σ+L2 fˆt T1−τ e(µ−σ−1)(s−t−τ)∥us∥2 Cds⩽σ+L2 fc2 3ˆt T1−τ e(µ−σ−1)(s−t−τ)ds ⩽σ+L2 fc2c2 3 µ−σ−1, (3.23) with c3=2(M µ+Mβ µ(µ−β))and from (3.20), (3.22) and (3.23) we derive that ˆt T1 e(µ−σ−1)(s−t)∥us∥2 C1ds⩽1 2e(σ−µ+1)(t−τ)∥φ(0)∥2+σ+L2 fc2c2 3 µ−σ−1+c1 2.(3.24) By replacing tby t+1 and T1by tin (3.24) for s∈[t,t+1]and t⩾TD, we have ˆt+1 t e(µ−σ−1)(s−t)∥us∥2 C1ds⩽1 2e(µ−σ−1)τ∥φ(0)∥2+σ+L2 fc2 3 µ−σ−1+c1 2,(3.25) indicating ˆt+1 t∥us∥2 C1ds⩽1 2e(µ−σ−1)τ∥φ(0)∥2+σ+L2 fc2 3 µ−σ−1+c1 2.(3.26) This completes the proof. The following result provides a uniform estimate for uin C1. 9 Nonlinearity 38 (2025) 075008 W Hu et al which shows that for the given ε > 0, there exists N3=N3(ε)such that, for all n⩾N3, ∥Φ(tn)|ΩK(χΩKφn)−ξ∥2 CΩK ⩽ε. (4.11) Recall that ξ∈X. Therefore, there exists R∗=R∗(ε)such that ˆ|x|⩾R∗|ξ(x)|2dx⩽ε. (4.12) Let ˜ R=max{R(ε),R∗}and N4=max{N1,N2,N3}. By (4.8), (4.11) and (4.12), we find that for all n⩾N4, ∥Φ(tn,φn)−ξ∥2 C⩽ˆ|x|⩽˜ RΦ(tn)|Ω˜ RχΩ˜ Rφn−ξ 2dx+ˆ|x|⩾˜ R|Φ(tn,φn)−ξ|2dx ⩽3ε, (4.13) which shows that Φ(tn,φn)→ξstrongly in C, implying the asymptotic compactness of Φin C. Therefore, it follows from lemmas 2.3 and 3.1 that Φadmits a global attractor. 5. Squeezing property This section is devoted to the squeezing property of the solutions to (3.1) and (3.2). In the remaining part of this paper, we always assume that conditions of theorem 4.1 are satisfied. Consider the following linear part of (3.1) on XΩK,    ∂˜ v(x,t) ∂t= ∆˜ v(x,t)−µ˜ v(x,t) + σ˜ v(x,t−τ), ˜ v(x,s) = φ(x,s)χΩK(x),s∈[−τ,0],x∈ΩK, ˜ v(x,t) = 0,t∈(0,∞),x∈∂ΩK. (5.1) It follows from [40, theorem 2.6, P51] that (5.1) admits a global solution ˜ vϕ(·):[−r,∞]→ XΩK. Define the linear semigroup U(t) : CΩK→CΩKby U(t)φ=˜ vϕ t(·). Let AU:CΩK→CΩKbe the infinitesimal generator of U(t), which is compact. Consider the following eigenvalue problem on ΩK: −∆u(x) = µu(x),u(x)|x∈∂ΩK=0,x∈ΩK,(5.2) which has a family of solutions (eigenfunctions) {em,K}m∈Nwith eigenvalues {µm,K}m∈Nsuch that 0< µ1,K⩽µ2,K⩽···⩽µm,K⩽···, µm,K→+∞as m→+∞. As AUis compact, by from [40, theorem 1.1, P65], we can see the spectrum of AUare point spectra denoted by %1,%2,···. Assume %1,%2,···are of multiplicity n1,n2,···such that ℜ%1> ℜ%2>···. Apparently, the characteristic equation of (5.1) is µ2 m,K−λ+µ−σe−λτ =0,m=1,2,···.(5.3) 16 Nonlinearity 38 (2025) 075008 W Hu et al It follows from [40, Example 1.12, page 72] that the characteristic values %1,%2,··· of the linear operator AUare the roots of (5.3). Clearly, %1is the first eigenvalue of AUdefined as ℜ%1=maxℜλ:µ2 m,K−λ+µ−σe−λτ =0,n=1,2,···.(5.4) It follows from [40, theorem 1.10, p 71] that, for any given real number γ, then there exists only a finite of number of eigenvalues of AUthat lie on its right, implying that there exists at most a finite number of eigenvalues of AUwhich are positive. Hence, there exists m⩾1 such that ℜ%m<0, and there is a km=n1+n2+···+nm(5.5) dimensional subspace CU ΩKsuch that CΩKis decomposed by AUas CΩK=CU ΩKMCS ΩK.(5.6) Denote by Pkmand Qkmthe projection of CΩKonto CU ΩKand CS ΩKrespectively, i.e. CU ΩK=PkmCΩK(5.7) and CS ΩK= (I−Pkm)CΩK=QkmCΩK.(5.8) By [40, theorem 2.4, p. 78], we can see there exists a positive constant Kmsuch that ∥U(t)Qkmx∥⩽Kmeℜϱmt∥x∥,t⩾0.(5.9) In order to rewrite (2.5) as an integral equation, we need to extend phase space Cto the following space ˆ C ˆ C=φ: [−τ,0]→CΩK;φ|[−τ,0)is continuous and lim θ→0−φ(θ)∈CΩKexists.(5.10) Following the informal variation of constant formula [40, theorem 2.1, p 116], we can see the solution to (2.5) satisfies the following integral equation v(t) = U(t)ϕ+ˆt 0 [U(t−s)X0f(vs)](0)ds,t⩾0,(5.11) with v0=φχΩK≜ϕ, where X0: [−τ,0]→B(CΩK,CΩK)is given by X0(θ) = 0 if −τ⩽θ < 0 and X0(0) = I. Remark. 5.1. Generally, formula (5.11) is undefined as an integral in the phase space Csince the semigroup U(t) is not defined at discontinuous functions. That is why we need to extend Cto ˆ C. Nevertheless, if we follow [9, pp 144–145] to interpret formula (5.11), it does make sense. Let Φ|ΩKbe defined by (4.1) and define Φ|ΩC K:C →C by Φ(t)|ΩC Kϕ=wφ t(5.12) for all ϕ∈C with wφ tbeing solution to (3.2). Therefore, for any ϕ,ψ ∈C, we can decompose ∥Φ(t)ϕ−Φ(t)ψ∥Cinto two parts ∥Φ(t)φ−Φ(t)ψ∥C⩽ Φ(t)|CΩKφ−Φ(t)|CΩKψ CΩK +  Φ(t)|CΩC K φ−Φ(t)|CΩC K ψ  C.(5.13) 17 Nonlinearity 38 (2025) 075008 W Hu et al By theorem 4.1 in our recent work [27], we have the following squeezing property of the first part of (5.13). Lemma 5.1. Let %1and %mbe the first and the mth eigenvalues of AU, Kmbe defined in (5.9). Denote the finite dimension projection Pkmin (5.7) by P. Then we have  PΦ(t)|CΩKϕ−PΦ(t)|CΩKψ CΩK ⩽2e(Lf+ℜϱ1)t∥ϕ−ψ∥C(5.14) and  (I−P)Φ(t)|CΩKϕ−(I−P)Φ(t)|CΩKψ CΩK ⩽Kmeℜϱmt+KmLf ℜ%1+Lf−ℜ%m e(Lf+ℜϱ1)t∥ϕ−ψ∥C(5.15) for any t ⩾0and ϕ,ψ ∈A. In the sequel, we show the squeezing property of the second part of (5.13). Lemma 5.2. Let R(ε)be defined in lemma 3.4 and K is chosen such that K ⩾R(ε). Then, for any t ∈R+, it holds that   Φ(t)|CΩC K φ−Φ(t)|CΩC K ψ  C=∥wϕ t−wψ t∥C⩽√c2e1 2[c2(σ+L2 f)−(µ−σ−1)]t∥ϕ−φ∥C.(5.16) Proof. Take K⩾max{2,R(ε)}and denote by y=wϕ t−wψ t. Then it follows from (3.1) that y is the solution of the following equation    ∂y(x,t) ∂t= ∆y(x,t)−µy(x,t) + σy(x,t−τ) + [f(uϕ(x,t−τ))−f(uψ(x,t−τ))]χΩC K(x), y(x,s)=[ϕ(x,s)−ψ(x,s)]χΩC K(x),s∈[−τ,0],x∈RN. (5.17) Multiplying both sides of (5.17) by χ|x|2 Kyand integrating on ΩC Kimply 1 2 d dtˆΩC K χ|x|2 Ky2(t)dx =ˆΩC K χ|x|2 Ky∆ydx−µˆΩC K χ|x|2 Ky2(t)dx +ˆΩC K χ|x|2 Kσy(t−τ)y(t)dx +ˆΩC K χ|x|2 Kfuϕ(x,t−τ)−fuψ(x,t−τ)χΩC K(x)y(t)dx. (5.18) We estimate each term on the right hand side of (5.18) as follows. First, we have ˆΩC K χ|x|2 Ku(t)∆u(t)dx=−ˆΩC K χ|x|2 K|∇u(t)|2dx.(5.19) 18 Nonlinearity 38 (2025) 075008 W Hu et al Next, by the Young inequality, we deduce σˆΩC K χ|x|2 Ky(t−τ)y(t)dx ⩽σˆΩC K χ|x|2 K1 2|y(t−τ)|2+1 2|y(t)|2dx ⩽σ 2ˆΩC K χ|x|2 K|y(t−τ)|2dx+σ 2ˆΩC K χ|x|2 K|y(t)|2dx, (5.20) and ˆΩC K χ|x|2 Kfuϕ(x,t−τ)−fuψ(x,t−τ)χΩC K(x)y(t)dx ⩽1 2ˆΩC K χ|x|2 K|fuϕ(x,t−τ)−fuψ(x,t−τ)χΩC K(x)|2dx +1 2ˆΩC K χ|x|2 K|y(t)|2dx ⩽1 2L2 fˆΩC K χ|x|2 K|y(t−τ)|2dx+1 2ˆΩC K χ|x|2 K|y(t)|2dx. (5.21) Incorporating (5.20) and (5.21) into (5.18) yields d dtˆΩC K χ|x|2 K|y(t)|2dx⩽(1+σ−µ)ˆΩC K χ|x|2 K|y(t)|2dx +σ+L2 fˆΩC K χ|x|2 K|y(t−τ)|2dx. (5.22) Thanks to Gronwall’s inequality on [0,t], we have ˆΩC K χ|x|2 K|y(t)|2dx⩽e(σ−µ+1)tˆΩC K χ|x|2 K|y(0)|2dx+σ+L2 fˆt 0 e(σ−µ+1)(t−s) ׈ΩC K χ|x|2 K|y(s−τ)|2dxds. (5.23) This implies that, for all t∈[0,∞)and ζ∈[−τ,0], ˆΩC K χ(|x|2 K)|y(t+ζ)|2dx⩽e(σ−µ+1)(t+ζ)ˆΩC K χ(|x|2 K)|y(0)|2dx +(σ+L2 f)ˆt+ζ 0 e(σ−µ+1)(t+ζ−s)ˆΩC K χ(|x|2 K)|y(s−τ)|2dxds ⩽c2e(σ−µ+1)tˆΩC K χ(|x|2 K)|y(0)|2dx +c2(σ+L2 f)ˆt 0 e(σ−µ+1)(t−s)ˆΩC K χ(|x|2 K)|y(s−τ)|2dxds.(5.24) Notice that ∥yt∥C=sup{∥y(t+ζ)∥:ζ∈[−τ,0]},whence ∥yt∥2 C⩽c2e(σ−µ+1)t∥φ−ϕ∥2 C+c2σ+L2 fˆt 0 e(σ−µ+1)(t−s)∥ys∥2 Cds.(5.25) 19 Nonlinearity 38 (2025) 075008 W Hu et al Multiplying both sides of (5.25) by e(µ−σ−1)t, we derive e(µ−σ−1)t∥yt∥2 C⩽c2∥φ−ϕ∥2 C+c2σ+L2 fˆt 0 e−(σ−µ+1)s∥ys∥2 Cds.(5.26) Using Gronwall’s inequality again yields e(µ−σ−1)t∥yt∥2 C⩽c2ec2(σ+L2 f)t∥φ−ϕ∥2 C,(5.27) indicating that ∥yt∥2 C⩽c2e[c2(σ+L2 f)−(µ−σ−1)]t∥φ−ϕ∥2 C⩽c2e[c2(σ+L2 f)−(µ−σ−1)]t∥φ−ϕ∥2 C.(5.28) The proof is complete. 6. Hausdorff and fractal dimensions This section is devoted to the estimations of Hausdorff and fractal dimensions of the global attractor obtained in theorem 4.1. We first concentrate on the Hausdorff dimension and review the definition of Hausdorff dimension of global attractors for autonomous dynamical systems. Let Abe the global attractor of (1.1) obtained in theorem 4.1. For any ε > 0,d⩾0 and ri⩽ε, define µC(A,d,ε) = inf X i rd i, where the infimum is taken over all coverings of Aby balls of radius ri⩽ε. Then, the ddimensional Hausdorff measure of the set A⊂Cis defined as µC(A,d) = lim ε→0µC(A,d,ε). At last, we give the Hausdorff dimension of the compact set A⊂Cas dC(A) = inf {d:µC(A,d) = 0}. Clearly, there exists a threshold dC(A)∈[0,+∞]such that µC(A,d) = 0 for d>dC(A)and µC(A,d) = ∞for d<dC(A).dC(A)is called the Hausdorff dimension of A. For any φ∈C, define the map Pχ(ΩC K):C →C and Pχ(ΩK):C →CΩKby Pχ(ΩK)φ=χ(ΩK)φ(6.1) and Pχ(ΩC K)φ=χΩC Kφ(6.2) respectively, where χ(ΩK)and χ(ΩC K)are the characteristic functions defined in (2.1) and (2.4). Then, by (5.6), we have C=PkmPχ(ΩK)CMQkmPχ(ΩK)CMPχ(ΩC K)C≜CU ΩKMCS ΩKMCΩC K,(6.3) where Pkmand Qkmare defined by (5.7) and (5.8). Define P=PkmPχ(ΩK),Q=QkmPχ(ΩK),R=Pχ(ΩC K).(6.4) 20 Nonlinearity 38 (2025) 075008 W Hu et al It follows from lemmas 5.1 and 5.2 that ∥P[Φ(t)ϕ−Φ(t)ψ]∥C(ΩK)⩽e(Lf+ℜϱ1)t∥ϕ−ψ∥C,(6.5) ∥Q[Φ(t)ϕ−Φ(t)ψ]∥C(ΩK)⩽Kmeℜϱmt+KmLf ℜ%1+Lf−ℜ%m e(Lf+ℜϱ1)t∥ϕ−ψ∥C,(6.6) and ∥R[Φ(t)ϕ−Φ(t)ψ]∥C(ΩC K)⩽√c2e1 2[c2(σ+L2 f)−(µ−σ−1)]t∥ϕ−ψ∥C.(6.7) Moreover, by (5.5) and (5.6), we can see Phas kmdimension range space, that is PkmCΩKis akmdimension subspace of C. In the following, we prove that the global attractor of (1.1) established in theorem 4.1 has finite Hausdorff dimension and fractal dimension. Theorem 6.1. Let km,%1,%mand Kmbe defined in (5.5) and (5.9) respectively, P,Qand Rbe defined by (6.4). Assume that conditions of lemma 5.1 are satisfied and there exist t0>0and 0< α < 2such that η=2Kmeℜϱmt0+α+2KmLf ℜϱ1+Lf−ℜϱme(Lf+ℜϱ1)t0+2√c2e[c2(σ+L2 f)−(µ−σ−1)]t0 2<1.(6.8) Then, the Hausdorff dimension of global attractor Asatisfies dH(A)<−lnkm−kmln(2+4 α) ln(2Kmeℜϱmt0+(α+2KmLf ℜϱ1+Lf−ℜϱm)e(Lf+ℜϱ1)t0+2√c2e[c2(σ+L2 f)−(µ−σ−1)]t0 2).(6.9) Proof. Let 0 <ε<1 be a given positive real number. As Ais a compact subset of C, there exist r1,...,rNin (0,ε]and ˜ u1,...,˜ uNin Csuch that A⊂ N [ i=1 B(˜ ui,ri),(6.10) where B(˜ ui,ri)is the ball in Cwith center ˜ uiand radius ri. Generally, we can assume that for any i B(˜ ui,ri)∩A=∅.(6.11) If on the contrary, we can delete it from the sequence ˜ u1,...,˜ uN. Hence, we can choose ui,i= 1,2,···,Nsuch that ui∈B(˜ ui,ri)∩A,(6.12) and A⊂ N [ i=1 (B(ui,2ri)∩A).(6.13) By lemmas 5.1 and 5.2, i.e. formulas (6.5) and (6.6), we can see ∥PΦ(t0)u−PΦ(t0)ui∥C⩽2e(Lf+ℜϱ1)t0ri,(6.14) ∥QΦ(t0)u−QΦ(t0)ui∥C⩽2Kmeℜϱmt0+KmLf ℜϱ1+Lf−ℜϱm e(Lf+ℜϱ1)t0ri,(6.15) 21 Nonlinearity 38 (2025) 075008 W Hu et al and ∥RΦ(t0)u−RΦ(t0)ui∥C⩽2√c2e[c2(σ+L2 f)−(µ−σ−1)]t0 2ri,(6.16) for any t0>0 and u∈B(ui,2ri)∩A. It follows from lemma 3.1 that, for any α > 0, we can find y1 i,...,yni isuch that BPC PΦ(t0)ui,2e(Lf+ℜϱ1)t0ri⊂ ni [ j=1 BPC yj i,αe(Lf+ℜϱ1)t0ri(6.17) with ni⩽km2+4 αkm ,(6.18) where kmis the dimension of PC,BPC(y,r)is the ball with radius rand center yin PC. Set uj i=yj i+QΦ(t0)ui+RΦ(t0)ui(6.19) for i=1,...,N,j=1,...,ni. Then, for any u∈B(ui,2ri)∩A, we can find a jsuch that   Φ(t0)u−uj i  C ⩽  PΦ(t0)u−yj i  C+∥QΦ(t0)u−QΦ(t0)ui∥C+∥RΦ(t0)u−RΦ(t0)ui∥C ⩽ 2Kmeℜϱmt0+α+2KmLf ℜϱ1+Lf−ℜϱme(Lf+ℜϱ1)t0+2√c2e[c2(σ+L2 f)−(µ−σ−1)]t0 2!ri. (6.20) Denote by η=2Kmeℜϱmt0+ (α+2KmLf ℜϱ1+Lf−ℜϱm)e(Lf+ℜϱ1)t0+2√c2e[c2(σ+L2 f)−(µ−σ−1)]t0 2. Then, (6.20) implies Φ(1)(B(ui,2ri)∩A)⊂ ni [ j=1 Buj i,ηri.(6.21) Due to the invariance of A, i.e. A= Φ(1)A, we can see A⊂ N [ i=1 ni [ j=1 Buj i,ηri,(6.22) which implies that, for any d⩾0, µH(A,d,ηε)⩽ N X i=1 ni X j=1 ηdrd i⩽km2+4 αkm ηd N X i=1 rd i.(6.23) By taking the infimum over all the coverings of Aby balls of radii less than ε, We deduce µH(A,d,ηε)⩽km2+4 αkm ηdµH(A,d,ε).(6.24) Then, we can obtain µHA,d,(ηε)k⩽"km2+4 αkm ηd#k µH(A,d,ε),(6.25) 22 Nonlinearity 38 (2025) 075008 W Hu et al by applying the formula (6.24) recursively for ktimes. Thus, if d<−lnkm−kmln2+4 α ln2Kmeℜϱmt0+α+2KmLf ℜϱ1+Lf−ℜϱme(Lf+ℜϱ1)t0+2√c2e[c2(σ+L2 f)−(µ−σ−1)]t0 2,(6.26) then km2+4 αkm ηd<1.(6.27) Therefore, by taking k→∞, we have (ηε)k→0, which combined with (6.25) gives rise to µHA,d,(ηε)k→0.(6.28) This completes the proof. Remark 6.1. As %1and %mdepend on the delay effect, then the Hausdorff dimension of global attractor Aof equation (5.1) depends on the time delay via the distribution of eigenvalues of the linear part AUof equation (5.1). Moreover, the Hausdorff dimension depends on constants of exponential dichotomy, the Lipschitz constant of the nonlinear term and the spectrum gap of the linear part AU, which indicates that the Hausdorff dimension of global attractor Ais very flexible to be tuned by a variety of parameters. Remark 6.2. At first glance, it seems αis redundant as 2Kmeℜϱmt0+2KmLf ℜϱ1+Lf−ℜϱme(Lf+ℜϱ1)t0+ 2√c2e[c2(σ+L2 f)−(µ−σ−1)]t0 2<1 naturally implies that there exists an α > 0 such that η:= 2Kmeℜϱmt0+ (α+2KmLf ℜϱ1+Lf−ℜϱm)e(Lf+ℜϱ1)t0+2√c2e[c2(σ+L2 f)−(µ−σ−1)]t0 2<1. However, in the proof of theorem 6.1,αis necessary to cover the ball in F. By taking α↑2 and assume 2Kmeℜϱmt0+ (2+2KmLf ℜϱ1+Lf−ℜϱm)e(Lf+ℜϱ1)t0+2√c2e[c2(σ+L2 f)−(µ−σ−1)]t0 2<1, then for all α∈(0,2), we have 2Kmeℜϱmt0+ (α+2KmLf ℜϱ1+Lf−ℜϱm)e(Lf+ℜϱ1)t0+2√c2e[c2(σ+L2 f)−(µ−σ−1)]t0 2<1 and hence we obtain the estimation dH(A)⩽−lnΛ−Λln4 ln 2Kmeℜϱmt0+2+2KmLf ℜϱ1+Lf−ℜϱme(Lf+ℜϱ1)t0+2√c2e[c2(σ+L2 f)−(µ−σ−1)]t0 2!,(6.29) which is independent of α. Nevertheless, this may not be optimal. Next, we study the fractal dimension of attractors for the nonlinear dynamical system Φ(t). Recall that the fractal dimension (or capacity) of Ais defined as dimfA=limsup ε→0 lnNε(A) −lnε,(6.30) where Nε(A)is the minimum number of balls of radius less than εneeded to cover A. 23 Nonlinearity 38 (2025) 075008 W Hu et al Theorem 6.2. Let km,%1,%mand Kmbe defined in (5.4) and (5.9) respectively, P,Qand Rbe defined by (6.4). Assume that conditions of lemma 5.1 are satisfied. Moreover, assume there exists β > 0 such that ζ:= βe(Lf+ℜϱ1)+Kmeℜϱm+KmLf ℜϱ1+Lf−ℜϱme(Lf+ℜϱ1)+ √c2e[c2(σ+L2 f)−(µ−σ−1)] 2<1. Then, the fractal dimension of global attractor Ahas an upper bound dimfA⩽lnkm+kmln2+2 β −lnζ<∞.(6.31) Proof. Let RAbe the diameter of A. As Ais compact, then RAis well defined. Hence, we have A⊆B(u0,RA),(6.32) for any u0∈A, where B(u0,RA)is the ball with center u0and radius RA. By lemmas 5.1 and 5.2, i.e. formulas (6.5) and (6.6), we can see, for any t0>0 and u∈B(ui,2ri)∩A, ∥PΦ(t0)u−PΦ(t0)ui∥C⩽e(Lf+ℜϱ1)t0ri,(6.33) ∥QΦ(t0)u−QΦ(t0)ui∥C⩽Kmeℜϱmt0+KmLf ℜ%1+Lf−ℜ%m e(Lf+ℜϱ1)t0ri,(6.34) and ∥RΦ(t0)u−RΦ(t0)ui∥C⩽√c2e[c2(σ+L2 f)−(µ−σ−1)]t0 2ri.(6.35) By lemma 3.1, for any β > 0, we can find y1 i,...,yni isuch that BPC PΦ(t0)ui,e(Lf+ℜϱ1)ri⊂ ni [ j=1 BPC yj i,βe(Lf+ℜϱ1)ri(6.36) with ni⩽km2km1+1 βkm ,(6.37) where kmis the dimension of PC and we have denoted by BPC(y,r)the ball in PC of radius r and center y. Take t0=1 and set uj 0=yj 0+QΦ(1)u0+RΦ(1)u0(6.38) for j=1,...,n0. Then, for any u∈A∩B(u0,RA), there exists jsuch that   Φ(1)u−uj 0   ⩽  PΦ(1)u−yj 0  +∥QΦ(1)u−QΦ(1)u0∥+∥RΦ(1)u−RΦ(1)u0∥ ⩽ βe(Lf+ℜϱ1)+Kmeℜϱm+KmLf ℜϱ1+Lf−ℜϱm e(Lf+ℜϱ1)+√c2e[c2(σ+L2 f)−(µ−σ−1)] 2!RA. (6.39) 24 Nonlinearity 38 (2025) 075008 W Hu et al Denote by ζ=βe(Lf+ℜϱ1)+Kmeℜϱm+KmLf ℜϱ1+Lf−ℜϱme(Lf+ℜϱ1)+√c2e[c2(σ+L2 f)−(µ−σ−1)] 2. Since Ais invariant, i.e. A=S(1)A, we have A= Φ(1)(A∩B(u0,RA)) ⊆ n0 [ j=1 Buj 0,ζRA.(6.40) Applying the formula recursively for ktimes gives A= Φ(k)(A∩B(u0,RA)) ⊆ n0,n1,···,nk−1 [ j=1 Buj k−1,ζkRA,(6.41) implying that the minimal number Nrk(A)of balls with radius rk=ζkRAcovering Ain C satisfies Nrk(A)⩽n0·...·nk−1≤"km2km1+1 βkm#k .(6.42) Since we have assumed that ζ < 1, then rk→0 as k→∞. Then it follows from (6.30) that dimfA=limsup rk→0 lnNrk(A) −lnrk ⩽limsup k→∞ lnkm2km1+1 βkmk −ln(ζkRA) = lnkm+kmln2+2 β −lnζ<∞. (6.43) 7. Summary We end this paper by making a detailed comparison of the present work with [7,16–18]. In [16], the authors constructed exponential attractors of a reaction-diffusion equation in a weighted Hilbert space without giving explicit fractal dimension of the attractor by the squeeze method originated from [15], which cannot be directly to used to (1.1). In [7,17], the authors adopted the Lyapunov exponent method to obtain sharp estimation in a weighted Hilbert space, although the estimations depend on less parameters and may be optimal than the present work, the methods can be neither be employed to tackle the problem in this paper since Cis not a Hilbert space. In [18] , the authors obtained exponential attractors with explicit fractal dimensions. Specifically, in [18], the obtained fractal dimension depends on a parameter Cin formula (8) in [18] between two Banach spaces Hand H1, which may be different if His differently chosen. As the aforementioned works concerned about the dimension estimation of uniform attractors, one natural generalization of the present work is to study the nonautonomous case, which will be tackled in an upcoming paper. Moreover, as pointed in [10], Lyapunov exponent method will give shaper estimation of dimensions compared with the squeeze method and hence another question is how to establish Lyapunov exponent method to estimate dimension of attractors in Banach spaces, which will be studied in the near future. 25