Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 100, pp. 1–19. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, DOI: 10.58997/ejde.2025.100 STRUCTURE AND STABILITY OF GLOBAL ATTRACTORS FOR A CAHN-HILLIARD TUMOR GROWTH MODEL WITH CHEMOTAXIS SEMA YAYLA Abstract. In this article, we analyze the long-time dynamics of a Cahn-Hilliard tumor growth model, focusing on the geometric structure and stability of its global attractors. Using a Lojasiewicz-Simon type inequality, we first prove that every full trajectory in the global attrac- tor converges to a single stationary point as t → ∞ and to another stationary point as t → −∞. As a result, we show that the global attractor is the union of the unstable manifolds emanating from the stationary points. We also examine the rate of convergence to these stationary points and provide specific polynomial and exponential rates under certain conditions. Additionally, we demonstrate that the global attractors of the corresponding tumor growth model exhibit upper-semicontinuity with respect to small perturbations of the chemotaxis parameter. Finally, by restricting chemotaxis within a certain interval, we establish the lower-semicontinuity of the global attractors for this model. 1. Introduction Tumor growth and its associated dynamics have long been a subject of significant interest in the fields of biology, medicine and mathematical modeling. One class of mathematical models that has proven particularly useful in this context is the diffuse interface models. The key mathematical equation used in diffuse interface models for tumor growth is the Cahn-Hilliard equation, ϕt − ∆(−∆ϕ + f(ϕ)) = 0. (1.1) Cahn-Hilliard equations were originally developed to describe the phase separation processes in binary materials (see [2]), but they have found applications in wide range of biological systems, including tumor growth. In this context, the following Cahn-Hilliard system was introduced in [18] to describe tumor growth as a continuum-mixture process. ϕt = div(m(ϕ)∇µ) + p(ϕ)(χσσ + χϕ(1 − ϕ) − µ) in (0, T ) × Ω, (1.2) µ = −∆ϕ + Ψ′(ϕ) − χϕσ in (0, T ) × Ω, (1.3) σt = div(n(ϕ)(χσ∇σ − χϕ∇ϕ)) − p(ϕ)(χσσ + χϕ(1 − ϕ) − µ) in (0, T ) × Ω, (1.4) ∂νµ = ∂νϕ = ∂νσ = 0 on (0, T ) × Γ. (1.5) Here, Ω ⊂ R3 is a bounded domain with smooth boundary Γ and ∂ν in (1.5) stands for the normal derivative where ν is the outer unit normal to Γ. Equation (1.2) is a Cahn-Hilliard equation where the order parameter ϕ ranges between −1 and 1, representing the tumorous and healthy phases respectively, with µ denoting the chemical potential for ϕ. Additionally, equation (1.4) is a reaction-diffusion equation where σ represents the chemical concentration acting as a nutrient for the tumor. The terms m(ϕ) and n(ϕ) represent positive mobilities indicating the diffusivity of the binary mixture and the chemicals. Moreover, Ψ is a potential function characterized by two minima at ±1. Lastly, χϕ ≥ 0 stands for the constant representing chemotaxis and active transport, while χσ ≥ 0 denotes the chemical mobility. 2020 Mathematics Subject Classification. 35B40, 35D30, 35K57, 35Q92, 37L30, 92C17. Key words and phrases. Tumor growth model; Cahn-Hilliard equation; long-time dynamics; global attractor. ©2025. This work is licensed under a CC BY 4.0 license. Submitted May 7, 2025. Published October 21, 2025. 1 2 S. YAYLA EJDE-2025/100 To the best of our knowledge, system (1.2)–(1.5) was first analyzed in the mathematical sense in [10] in the case χϕ = 0 and m(ϕ) = n(ϕ) = χσ = 1, where the authors proved the well-posedness of the problem and the existence of a global attractor. Subsequently, problem (1.2)–(1.5) with an additional viscosity regularization term was considered in [8, 9]. In [8], the well-posedness of strong solutions was established, and the long-time behavior of the corresponding dynamical system was studied using the concept of the ω-limit set. In [9], some results from [8] were extended to scenarios where the viscosity parameters are independent of each other. Additionally, we refer to [19], where the formal matched asymptotic limit of a quasi-static variant of (1.2)–(1.5) was explored. Among other valuable contributions in the literature related to similar tumor growth models, we can cite [11, 12, 13, 14]. In [14], a Cahn-Hilliard–Darcy model was introduced for tumor growth with chemotaxis and active transport. In [11] also a Cahn-Hilliard–Darcy system was investigated, and the existence of global weak solutions was established in both two- and three-dimensional cases. In [13], the following tumor growth model was analyzed, ϕt = div(m(ϕ)∇µ) + (λpσ − λa)h(ϕ) in Ω × (0, T ), µ = AΨ′(ϕ) −B∆ϕ− χϕσ in Ω × (0, T ), κσt = div(n(ϕ)(χσ∇σ − χϕ∇ϕ)) − λcσh(ϕ) in Ω × (0, T ), ∇µ · ν = ∇ϕ · ν = 0, n(ϕ)χσ∇σ · ν = K(σ∞ − σ) on Γ × (0, T ). (1.6) Here, κ = 1, A, B and K are positive constants. The parameters λp, λa and λc are nonnegative constants representing, respectively, the proliferation rate, the apoptosis rate of the tumor cells, and the nutrient consumption rate. The function h(ϕ) is an interpolation function, and σ∞ denotes the nutrient amount on the boundary. Additionally, m(ϕ), n(ϕ), χϕ and χσ represent the same quantities as in system (1.2)–(1.5). In that paper, well-posedness of the model (1.6) and its quasi- static version (κ = 0) was established for regular potentials with quadratic growth. On the other hand, in [12], the well-posedness of problem (1.6) with Dirichlet boundary conditions was obtained for regular potentials with higher polynomial growth and even for the singular potentials. Global attractors are essential tools for analyzing the long-time behavior of infinite-dimensional dynamical systems. Since they are compact, invariant sets that attract all bounded trajectories, understanding their structure and stability is key to gaining deeper insight into the overall dy- namics of the system. Some studies have advanced our understanding of global attractors in systems with hyperbolic stationary points, where the number of stationary points is finite [3, 4]. These systems can be described using gradient-like semigroups, where all trajectories move be- tween stationary points, and there are no homoclinic orbits. However, the set of stationary points is generally infinite and those results do not directly apply. To overcome this limitation, [1] shows a novel approach based on the Lojasiewicz-Simon in- equality. In systems where the set of stationary points is infinite, their technique allows one to show that all full trajectories in the global attractor originate from a stationary point as t → −∞ and converge to another as t → ∞. As a result, the attractor can be described as the union of the unstable manifolds of all stationary points. This article introduces new results concerning the structure and the stability of the global at- tractors for problem (1.2)–(1.5), in the case χϕ > 0, differing from the approach in [10] and related works. Some foundational results concerning the existence of a global attractor and its charac- terization as an unstable manifold emanating from the set of stationary points were previously established in [15], in the case χϕ > 0. In this paper, we aim to present new findings regarding the geometric structure and stability of the global attractors for problem (1.2)–(1.5). The main contributions of this paper can be summarized as follows. • Inspired by [1], and using a Lojasiewicz-Simon type inequality, we prove that every full trajectory within the global attractor converges to a single stationary point as t → ∞ and to another stationary point as t → −∞. We then show that the global attractor is equal to a union of the unstable manifolds emanating from the stationary points, even when the set of stationary points is infinite (see Remark 3.3). EJDE-2025/100 STRUCTURE AND STABILITY OF GLOBAL ATTRACTORS FOR A TUMOR MODEL 3 • We obtain the rate of convergence to equilibrium for the full trajectories in the global attractor, in the space (H1(Ω))∗ × (H1(Ω))∗ (see Proposition 4.1). Then, by imposing suitable restrictions on the chemotaxis, we obtain that the every full trajectory in the global attractor converges exponentially to equilibrium (Corollary 4.2). • In the case where the Lojasiwicz-Simon exponent 0 < θ < 1 2 , we prove that the full trajectories in the global attractor maintain the same convergence rate, established in (H1(Ω))∗ × (H1(Ω))∗ also in the space ZM . This result is not straightforward and requires careful analysis and detailed estimates beyond standard arguments. • By considering the chemotaxis and active transport parameter χϕ as a perturbation parame- ter, we obtain a family of global attractors. We first prove that this family of global attractors is uppersemicontinuous as χϕ → 0. Moreover, restricting the chemotaxis on a suitable interval, we obtain the lower semicontinuity of the global attractors as χϕ → 0. This article is organized as follows. In Section 2, we introduce the problem and present several foundational results from [15]. In Section 3, we conduct a detailed analysis of the geometric properties of the global attractor, along with convergence of trajectories to equilibrium. Section 4 is dedicated to examining the rate of convergence to equilibrium. Finally, in Section 5, we investigate the upper and lower-semicontinuity properties of the global attractors. 2. Setting of the problem and the previous results In the rest of this paper, to simplify the notation, we will denote chemotaxis and active trans- port by χ instead of χϕ. Furthermore, since the choice of χσ does not affect the subsequent mathematical analysis, we will assume χσ = 1. Additionally, we consider the mobilities to be constant, i.e. m(ϕ) = n(ϕ) = 1. Hence, we reformulate the problem (1.2)–(1.5) as follows. ϕχ t = ∆µχ + p(ϕχ)(σχ + χ(1 − ϕχ) − µχ), in (0, T ) × Ω, (2.1) µχ = −∆ϕχ + Ψ′(ϕχ) − χσχ, in (0, T ) × Ω, (2.2) σχ t = ∆σχ − χ∆ϕχ − p(ϕχ)(σχ + χ(1 − ϕχ) − µχ), in (0, T ) × Ω, (2.3) ∂νµ χ = ∂νϕ χ = ∂νσ χ = 0, on (0, T ) × Γ, (2.4) where χ ≥ 0. We consider this problem with the following assumptions, which are identical to those in [15]. (A1) The potential Ψ ∈ C2(R) can be written as Ψ(s) = Ψ0(s) + λ(s) (2.5) where Ψ0 ∈ C2(R) and λ ∈ C2(R) satisfies |λ′′(s)| ≤ α, for all s ∈ R, and for some constant α ≥ 0. Moreover, we assume that c1(1 + |s|ρ−2) ≤ Ψ′′ 0(s) ≤ c2(1 + |s|ρ−2), (2.6) Ψ(s) ≥ R1|s|2 −R2 (2.7) for all s ∈ R, with c1, c2, R1 > 2χ2, R2 ∈ R and with ρ ∈ [2, 6). (A2) The interpolation function p ∈ C0,1 loc (R) satisfies p > 0 and |p′(s)| ≤ c4(1 + |s|q−1) (2.8) for all s ∈ R, with c4 > 0 and with q ∈ [1, 4]. We also assume the following in the proof of the Lojasiewicz-Simon type inequality. (A3) The potential Ψ ∈ C∞(R) is an analytic function. Remark 2.1. Assumption (2.7) is only needed if ρ = 2. Notation. We now introduce some notation that will be used throughout the paper. For a Banach space X, we denote its norm by ∥ · ∥X , its dual space by X∗, and the duality pairing 4 S. YAYLA EJDE-2025/100 between X∗ and X by ⟨⟨·, ·⟩⟩. Additionally, we will use ⟨·, ·⟩X for inner product in X. The inner product in L2(Ω) will be written as ⟨·, ·⟩. For every u ∈ (H1(Ω))∗, we define the mean value as: ⟨u⟩ := 1 |Ω| ⟨⟨u, 1⟩⟩ for every u ∈ (H1(Ω))∗. Moreover, we introduce the operator A : H1(Ω) → (H1(Ω))∗ such that Au = −∆u + u and D(A) = {ϕ ∈ H2(Ω) : ∂νϕ = 0 on Γ}. It is worth noting that the restriction of A on D(A) is an isomorphism from D(A) onto L2(Ω), i.e. D(A) = A−1(L2(Ω)). In addition, the following identities hold: ⟨⟨Au,A−1v∗⟩⟩ = ⟨⟨v∗, u⟩⟩ for every u ∈ H1(Ω), v∗ ∈ (H1(Ω))∗, ⟨⟨u∗, A−1v∗⟩⟩ = ⟨u∗, v∗⟩(H1(Ω))∗ for every u∗, v∗ ∈ (H1(Ω))∗, where ⟨·, ·⟩(H1(Ω))∗ is the dual inner product in (H1(Ω))∗ corresponding to the usual inner product in H1(Ω). Domain of the inverse operator A−1 is defined as D(A−1) = (D(A))∗ (see [22]). Also, we have ⟨⟨v∗, u⟩⟩ = ∫ Ω v∗u if v∗ ∈ L2(Ω), and d dt ∥v∗∥2(H1(Ω))∗ = 2⟨⟨∂tv∗, A−1v∗⟩⟩ for every v∗ ∈ H1(0, T ; (H1(Ω))∗). We now define the weak and strong solutions to the problem (2.1)–(2.4). Definition 2.2 ([15]). Let (ϕ0, σ0) ∈ H1(Ω) × L2(Ω) and T ∈ (0,∞) be given. A pair (ϕχ, σχ), satisfying the properties (ϕχ, σχ) ∈ L∞(0, T ;H1(Ω) × L2(Ω)) ∩ L2(0, T ;H2(Ω) ×H1(Ω)), (ϕχ t , σ χ t ) ∈ Lr(0, T ;D(A−1) ×D(A−1)), µχ := −∆ϕχ + Ψ′(ϕχ) − χσχ ∈ L2(0, T ;H1(Ω)), (ϕχ(0), σχ(0)) = (ϕ0, σ0), for some r > 1, is called a weak solution of problem (2.1)–(2.4) on [0, T ] × Ω, if ⟨⟨ϕχ t , η⟩⟩ + ⟨∇µχ,∇η⟩ = ⟨p(ϕχ)(σχ + χ(1 − ϕχ) − µχ), η⟩ ⟨⟨σχ t , ξ⟩⟩ + ⟨(∇σχ − χ∇ϕχ),∇ξ⟩ = −⟨p(ϕχ)(σχ + χ(1 − ϕχ) − µχ), ξ⟩ (2.9) holds on (0, T ) × Ω, for every η, ξ ∈ D(A). If the pair (ϕχ, σχ) also satisfies the properties ϕχ ∈ L∞(0, T ;H3(Ω)), ϕχ t ∈ L2(0, T ;H1(Ω)), σχ ∈ L∞(0, T ;H1(Ω)) ∩ L2(0, T ;H2(Ω)), σχ t ∈ L2(0, T ;L2(Ω)), µχ ∈ L∞(0, T ;H1(Ω)) ∩ L2(0, T ;H3(Ω)), then it is called a strong solution of problem (2.1)–(2.4) on [0, T ] × Ω. The following results on the well posedness of solutions was obtained in [15]. Theorem 2.3 ([15]). Let conditions (A1), (A2) hold. Then for every (ϕ0, σ0) ∈ H1(Ω) × L2(Ω) and for every T > 0, problem (2.1)–(2.4) has a weak solution such that ϕχ ∈ L2(0, T ;H3(Ω)), Ψ(ϕχ) ∈ L∞(0, T ;L1(Ω)), ∇Nσχ ∈ L2(0, T ;L2(Ω)), √ p(ϕχ)(Nσχ − µχ) ∈ L2(0, T ;L2(Ω)), ϕχ t ∈ L2(0, T ; (H1(Ω))∗), σχ t ∈ L2(0, T ; (H1(Ω))∗). Moreover, the following energy identity holds for the weak solutions, Eχ(ϕχ(t), σχ(t)) + ∫ t 0 ∫ Ω (|∇µχ|2 + |∇Nσχ |2 + p(ϕχ)(Nσχ − µχ)2) dx dτ = Eχ(ϕ0, σ0) (2.10) EJDE-2025/100 STRUCTURE AND STABILITY OF GLOBAL ATTRACTORS FOR A TUMOR MODEL 5 where Eχ(ϕ, σ) = 1 2 ∥∇ϕ∥2L2(Ω) + ∫ Ψ(ϕ) + 1 2 ∥σ∥2L2(Ω) + χ ∫ σ(1 − ϕ) and Nχ σ = σχ + χ(1 − ϕχ). Furthermore, for every initial data (ϕ0, σ0) ∈ H3(Ω) × H1(Ω) with ∂νϕ0 = 0 on Γ and for every T > 0, problem (2.1)–(2.4) has a strong solution. Theorem 2.4 ([15]). Let conditions (A1), (A2) hold. Then, for every initial data (ϕ0, σ0) ∈ H1(Ω)×L2(Ω) and for every T > 0, the weak solution of problem (2.1)–(2.4) specified by Theorem 2.3 is unique. Moreover, if (ϕχ i , σ χ i ), i = 1, 2 are weak solutions of problem (2.1)–(2.4) with initial data (ϕ0i, σ0i) ∈ H1(Ω) × L2(Ω), respectively, then ∥ϕχ 2 (t) − ϕχ 1 (t)∥(H1(Ω))∗ + ∥σχ 2 (t) − σχ 1 (t)∥(H1(Ω))∗ + ∥ϕχ 2 (t) − ϕχ 1 (t)∥L2(0,t;H1(Ω)) + ∥σχ 2 (t) − σχ 1 (t)∥L2(0,t;L2(Ω)) ≤ Λ(t) ( ∥ϕχ 02 − ϕχ 01∥(H1(Ω))∗ + ∥σχ 02 − σχ 01∥(H1(Ω))∗ ) where Λ is a continuous positive function which depends on the norms of the initial data and Ψ, p, Ω and T . As a result of Theorems 2.3 and 2.4, problem (2.1)–(2.4) generates a weakly continuous semi- group {Sχ(t)}t≥0 in H1(Ω)×L2(Ω), according to the formula Sχ(t)(ϕ0, σ0) = (ϕχ(t), σχ(t)). Here, (ϕχ(t), σχ(t)) denotes the weak solution determined by Theorem 2.3. Exploiting (2.4), it is easy to see that problem (2.1)–(2.4) satisfies the total mass conservation as ∫ Ω (ϕχ(t) + σχ(t)) dx = ∫ Ω (ϕ0 + σ0) dx ∀t ≥ 0. Hence, we need to introduce the following subspaces: ZM := { (ϕ, σ) ∈ H1(Ω) × L2(Ω) : ∫ Ω (ϕ + σ) dx = |Ω|M } , (2.11) Zr M := { (ϕ, σ) ∈ H2r+1(Ω) ×Hr(Ω) : ∫ Ω (ϕ + σ) dx = |Ω|M } , r > 0, (2.12) which are equipped with the usual norms of H1(Ω) × L2(Ω) and H2r+1(Ω) × Hr(Ω), respec- tively. Thus, restricting the phase space of the problem to ZM , we obtain the dynamical system (ZM , Sχ M (t)), where {Sχ M (t)}t≥0 denotes the restriction of {Sχ(t)}t≥0 on ZM . Moreover, we will denote the weak solution of problem (2.1)–(2.4) in ZM by Sχ M (t)(ϕ0, σ0) = (ϕχ M (t), σχ M (t)). Now, let us define the set of stationary points, Nχ M = {(ϕ, σ) ∈ ZM : Sχ M (t)(ϕ, σ) = (ϕ, σ),∀t ≥ 0}, for problem (2.1)–(2.4) in ZM . The set of stationary points Nχ M , as indicated in [15], is a nonempty, bounded subset of ZM , consisting of solutions to the stationary problem −∆ϕχ + Ψ′(ϕχ) − χσχ = µχ 0 , σχ + χ(1 − ϕχ) = µχ 0 , ∈Ω (ϕχ + σχ) dx = |Ω|M, (2.13) where µχ 0 = 1 |Ω| ∫ Ω Ψ′(ϕχ) dx− χ|Ω| ∫ Ω σχ dx. For convenience of the reader, we state some results from [15], which are used in some steps of this paper. Proposition 2.5 (Asymptotic compactness, [15, Lemma 5.1]). Assume that (A1), (A2) are satis- fied and B is a bounded subset of H1(Ω)×L2(Ω). Then, every sequence of the form {Sχ(tk)(ϕk, σk)}∞k=1, where {(ϕk, σk)}∞k=1 ⊂ B, tk → ∞ has a convergent subsequence in H1(Ω) × L2(Ω). Proposition 2.6 (Gradient Property, [15, Lemma 5.2]). Under conditions (A1), (A2), the dy- namical system (H1(Ω) × L2(Ω), Sχ(t)) is a gradient system, i.e. the energy functional Eχ is a strict Lyapunov function on the whole phase space H1(Ω) × L2(Ω). 6 S. YAYLA EJDE-2025/100 Proposition 2.7 ([15, Lemma 5.3]). Assume that conditions (A1), (A2) are satisfied. Then, energy functional Eχ(ϕ, σ) has at least one minimizer (ϕ∗, σ∗) ∈ ZM such that Eχ(ϕ∗, σ∗) = inf (ϕ,σ)∈ZM Eχ(ϕ, σ). Proposition 2.8 ([15, Lemma 5.4]). Let conditions (A1), (A2) hold and (ϕ∗, σ∗) be a minimizer of Eχ(ϕ, σ) in ZM . Then (ϕ∗, σ∗) ∈ H2(Ω) ×H2(Ω) is the strong solution of the problem (2.13). Proposition 2.9 ([15, Lemma 5.5]). Assume that (A1), (A2) are satisfied. Then the set of stationary points Nχ M is nonempty and bounded in ZM . Before stating the main result of [15], we recall the following definitions. Definition 2.10. Let {S(t)}t≥0 be a semigroup on a metric space (X, d). A set A ⊂ X is called a global attractor for the semigroup {S(t)}t≥0, if • A is a compact set. • A is invariant, i.e. S(t)A = A, ∀t ≥ 0. • limt→∞ distX(S(t)B,A) = 0, for each bounded set B ⊂ X, where dist(·, ·) is the Hausdorff semidistance defined as distX(A,B) = sup a∈A inf b∈B d(a, b). Definition 2.11. Let N be the set of stationary points of the dynamical system (X,S(t)). We define the unstable manifold Mu(N ) emanating from the set N as a set of all y ∈ X such that there exists a full trajectory γ = {u(t) : t ∈ R} with the properties u(0) = y and lim t→−∞ distX(u(t),N ) = 0. The following theorem is the main result of [15] regarding the existence and the regularity of the global attractor. Theorem 2.12 ([15, Theorem 5.9, Theorem 5.10]). Assume that (A1), (A2) are satisfied. Then the semigroup {Sχ M (t)}t≥0 generated by the weak solutions of the problem (2.1)–(2.4) possesses a global attractor Aχ M in ZM , and Aχ M = Mu(Nχ M ). Moreover, the global attractor Aχ M is bounded in Z1 M . 3. Further geometric properties of the global attractor Recalling the definition of the unstable manifold and exploiting the result [7, Theorem 7.5.6], we observe that the global attractor Aχ M consists of full trajectories γχ M = {(ϕχ M (t), σχ M (t)) : t ∈ R} such that lim t→∞ distZM ((ϕχ M (t), σχ M (t)),Nχ M ) = 0 and lim t→−∞ distZM ((ϕχ M (t), σχ M (t)),Nχ M ) = 0. (3.1) In this section, we aim to improve (3.1) by proving the following result. Theorem 3.1. Let assumptions (A1)–(A3) hold. Then for any full trajectory γχ M = {(ϕχ M (t), σχ M (t)) : t ∈ R} in the global attractor Aχ M there exist (ϕ∗, σ∗), (ϕ∗∗, σ∗∗) ∈ Nχ M such that lim t→∞ distZr M ((ϕχ M (t), σχ M (t)), (ϕ∗, σ∗)) = 0. (3.2) and lim t→−∞ distZr M ((ϕχ M (t), σχ M (t)), (ϕ∗∗, σ∗∗) = 0, (3.3) where r ∈ [0, 1). As a consequence of Theorem 3.1, we obtain the following geometric property of the global attractor. Corollary 3.2. Under assumptions (A1)–(A3), the global attractor Aχ M equals to the union of the unstable manifolds of its stationary points, i.e., Aχ M = ∪(ϕ∗,σ∗)∈Nχ M Mu((ϕ∗, σ∗)). (3.4) EJDE-2025/100 STRUCTURE AND STABILITY OF GLOBAL ATTRACTORS FOR A TUMOR MODEL 7 Remark 3.3. We already know that when the set of stationary points Nχ M is finite, the equality (3.4) holds (see for example [7, p.361]). In this section, we demonstrate that the global attractor Aχ M of the problem (2.1)–(2.4) retains the property (3.4), even if the set of stationary points Nχ M is infinite. Remark 3.4. In our framework, the dynamical system associated with problem (2.1)–(2.4) is a gradient system, i.e. the energy functional Eχ(ϕχ M , σχ M ) serves as a strict Lyapunov functional (see Proposition 2.6). Hence, the system admits no heteroclinic cycles. The present work goes beyond establishing the existence of a Lyapunov structure; it also aims to characterize the geometric structure of the global attractor. From Theorem 3.1 we obtain that Aχ M consists of full trajectories connecting distinct equilibrium points, which correspond to heteroclinic orbits. It is important to emphasize that this geometric structure of the attractor is not a direct consequence of the gradient property itself. Systems whose attractors possessing the properties (3.2) and (3.3) are often referred to as gradient-like systems. Although many gradient systems become gradient-like under additional analytical tools, such as the existence of Lojasiewicz-Simon- type inequalities, not every gradient system is necessarily gradient-like (see [1]). To prove Theorem 3.1, we first present some properties of the limit sets. Subsequently, we will derive an appropriate uniform Lojasiewicz-Simon type inequality. Finally, by using these establishments, we prove Theorem 3.1. 3.1. Properties of limit sets. Let us start with definitions of the limit sets. Definition 3.5. For any (ϕ0, σ0) ∈ ZM the ω-limit set of (ϕ0, σ0) is defined by ωχ M ((ϕ0, σ0)) := { (ϕ∗, σ∗) ∈ Aχ M : ∃{(ϕχ M (tk), σχ M (tk))}∞k=1, such that (ϕχ M (0), σχ M (0)) = (ϕ0, σ0), tk ↗ ∞ and (ϕχ M (tk), σχ M (tk)) → (ϕ∗, σ∗) strongly in ZM } . Moreover, for any (ϕ0, σ0) ∈ Aχ M the α-limit set of (ϕ0, σ0) can be defined as follows: αχ M ((ϕ0, σ0)) := { (ϕ∗∗, σ∗∗) ∈ Aχ M : ∃{(ϕχ M (tk), σχ M (tk))}∞k=1, such that (ϕχ M (0), σχ M (0)) = (ϕ0, σ0), tk ↘ −∞ and (ϕχ M (tk), σχ M (tk)) → (ϕ∗∗, σ∗∗) strongly in ZM } . On account of the above definitions, we deduce the following lemma. Lemma 3.6. Let (ϕ0, σ0) ∈ Aχ M and (ϕχ M (t), σχ M (t)) be the full trajectory passing through (ϕ0, σ0). Then, the sets ωχ M ((ϕ0, σ0)) and αχ M ((ϕ0, σ0)) are nonempty, compact, invariant subsets of ZM and there exist constants Eχ,∞+ M and Eχ,∞− M such that Eχ(ϕ∗, σ∗) = lim t→∞ Eχ(ϕχ M (t), σχ M (t)) = Eχ,∞+ M ∀(ϕ∗, σ∗) ∈ ωχ M ((ϕ0, σ0), Eχ(ϕ∗∗, σ∗∗) = lim t→−∞ Eχ(ϕχ M (t), σχ M (t)) = Eχ,∞− M ∀(ϕ∗∗, σ∗∗) ∈ αχ M ((ϕ0, σ0)). (3.5) Moreover, ωχ M ((ϕ0, σ0)) ⊂ Nχ M and αχ M ((ϕ0, σ0)) ⊂ Nχ M . Proof. By using the asymptotic compactness property stated in Proposition 2.5, one can see that for any (ϕ0, σ0) ∈ Aχ M the sets ωχ M ((ϕ0, σ0)) and αχ M ((ϕ0, σ0)) are nonempty, compact and invariant subset of ZM (see for example [7, p. 339]). On the other hand, since Eχ(ϕχ M (t), σχ M (t)) is a non-increasing functional which is bounded from above and below (cf. Proposition 2.6, Proposition 2.7), we infer that there exist constants Eχ,∞+ M and Eχ,∞− M such that lim t→∞ Eχ(ϕχ M (t), σχ M (t)) = Eχ,∞+ M , lim t→−∞ Eχ(ϕχ M (t), σχ M (t)) = Eχ,∞− M . (3.6) Moreover, for any (ϕ∗, σ∗) ∈ ωχ M ((ϕ0, σ0)) there exists tn → ∞ (as n → ∞) such that lim n→∞ (ϕχ M (tn), σχ M (tn)) = (ϕ∗, σ∗). (3.7) 8 S. YAYLA EJDE-2025/100 Similarly, for any (ϕ∗∗, σ∗∗) ∈ αχ M ((ϕ0, σ0)) there exists t̃n → −∞ (as n → ∞) such that lim n→∞ (ϕχ M (t̃n), σχ M (t̃n)) = (ϕ∗∗, σ∗∗). (3.8) From (3.6)-(3.8) we obtain that (3.5) is satisfied. Now, using the invariance property of ωχ M ((ϕ0, σ0)) and αχ M ((ϕ0, σ0)), from (3.6)-(3.8) it can be deduced that Eχ(Sχ M (t)(ϕ∗, σ∗)) = Eχ,∞+ M ∀(ϕ∗, σ∗) ∈ ωχ M ((ϕ0, σ0)), ∀t ≥ 0, Eχ(Sχ M (t)(ϕ∗∗, σ∗∗)) = Eχ,∞− M ∀(ϕ∗∗, σ∗∗) ∈ αχ M ((ϕ0, σ0)), ∀t ≥ 0, (3.9) which yields Eχ(Sχ M (t)(ϕ∗, σ∗)) = Eχ((ϕ∗, σ∗)) ∀(ϕ∗, σ∗) ∈ ωχ M ((ϕ0, σ0)), ∀t ≥ 0, Eχ(Sχ M (t)(ϕ∗∗, σ∗∗)) = Eχ((ϕ∗∗, σ∗∗)) ∀(ϕ∗∗, σ∗∗) ∈ αχ M ((ϕ0, σ0)), ∀t ≥ 0. Hence, using that Eχ(ϕχ M (t), σχ M (t)) is a strict Lyapunov functional (see Proposition 2.6), we have (ϕ∗, σ∗), (ϕ∗∗, σ∗∗) ∈ Nχ M , which completes the proof. □ 3.2. Lojasiewicz-Simon type inequality. In the proof of Theorem 3.1, Lojasiewicz-Simon type inequality assumes a pivotal position. Its derivation primarily stems from the theoretical frame- work developed in Chill [5] and Chill et al. [6]. Let us now proceed with the definition of the pertinent spaces, which are obtained by setting M = 0 in (2.11) and (2.12): Z0 := { (ϕ, σ) ∈ H1(Ω) × L2(Ω) : ∫ Ω (ϕ + σ) dx = 0 } , Zr 0 := { (ϕ, σ) ∈ H2r+1(Ω) ×Hr(Ω) : ∫ Ω (ϕ + σ) dx = 0 } , r > 0. These spaces are Hilbert spaces equipped with usual norms of H1(Ω) × L2(Ω) and H2r+1(Ω) × Hr(Ω), respectively. Furthermore, introducing the Hilbert space H0 := { (ϕ, σ) ∈ L2(Ω) × L2(Ω) : ∫ Ω (ϕ + σ) dx = 0 } , (3.10) we obtain the Hilbert triple Z0 ↪→ H0 = H∗ 0 ↪→ Z∗ 0 . It can be inferred from Theorem 2.12 that the every full trajectory γχ M = {(ϕχ M (t), σχ M (t)) : t ∈ R} in the global attractor Aχ M , and thus the set of stationary points Nχ M , is bounded in Zr M , r ∈ [0, 1]. Now, for any solution pair (ϕχ M , σχ M ) of the problem (2.1)–(2.4) in the phase space ZM , let us define the shifted pair (ϕχ M , σχ M ) := (ϕχ M − c1, σ χ M − c2) where c1 = (1+χ)M 1+2χ and c2 = χM 1+2χ . Since c1 + c2 = M and χc1 − (1 + χ)c2 = 0, (3.11) the shifted pair (ϕχ M , σχ M ) is in Z0 and solves problem (2.1)–(2.4) written for the functions ΨM := Ψ(· + M 1 + χ ) and pM := p(· + M 1 + χ ) instead of Ψ and p. Hence, without lose of generality we can study on the spaces Zr 0 , r ∈ [0, 1]. Notation. From this point onward, for simplicity of notation, we will denote the solution pair in Z0 as (ϕχ, σχ) instead of (ϕχ 0 , σ χ 0 ), and the full trajectory in Z1 0 as γχ = {(ϕχ(t), σχ(t)) : t ∈ R} instead of γχ 0 = {(ϕχ 0 (t), σχ 0 (t)) : t ∈ R}. Additionally, we will refer to the set of stationary points in Z1 0 as Nχ instead of Nχ 0 . We firstly state the following auxiliary lemma for the energy functional Eχ, which can be proved by using similar assertions in [6, Lemma 6.2] and arguing as in the proof of Proposition 2.8. EJDE-2025/100 STRUCTURE AND STABILITY OF GLOBAL ATTRACTORS FOR A TUMOR MODEL 9 Lemma 3.7. The energy functional Eχ is twice continuously Frechet differentiable. For every (ϕ, σ), (η, ξ) ∈ Z0, it holds ⟨⟨E ′ χ(ϕ, σ), (η, ξ)⟩⟩Z0 = ∫ Ω (∇ϕ∇η + (Ψ′(ϕ) − χσ)η + (σ + χ(1 − ϕ))ξ) dx. (3.12) Moreover, for every (ϕ, σ), (η1, ξ1), (η2, ξ2) ∈ Z0, we have ⟨E ′′ χ(ϕ, σ)(η1, ξ1), (η2, ξ2)⟩Z0 = ∫ Ω (∇η1∇η2 + Ψ′′(ϕ)η1η2 − χξ2η1 + ξ1ξ2 − χξ1η2) dx. (3.13) Now, we will prove a suitable version of the Lojasiewicz-Simon type inequality which can be proved by adapting the abstract result [5, Corollary 3.11]. At this point, it worths to mention that every critical point of the energy functional Eχ in Z0 is a stationary point and also bounded in Z1 0 (see Proposition (2.8)). The proof of the inequality will be done by arguing as in [6, Proposition 6.6]. We will state the proof for the convenience of the reader. Lemma 3.8. Suppose that (A3) is satisfied and (ϕ∗, σ∗) ∈ Z1 0 is a critical point of the functional Eχ. Then, there exist constants θ ∈ (0, 1 2 ] and C, β > 0 depending on (ϕ∗, σ∗) such that |Eχ(ϕ, σ) − Eχ(ϕ∗, σ∗)|1−θ ≤ C∥E ′ χ(ϕ, σ)∥Z∗ 0 (3.14) for all (ϕ, σ) ∈ Z0 such that ∥(ϕ, σ) − (ϕ∗, σ∗)∥Z0 < β. Proof. Firstly, Sobolev embedding theorem yields that Z1 0 ⊂ L∞(Ω) × L2(Ω). Hence, the restric- tion of E ′ χ to Z1 0 is an analytic function with values in H0 (see e.g. [5, Corollary 4.6]). On the other hand, the associated bilinear form of the linearization E ′′ χ(ϕ, σ) is continuous, symmetric, elliptic operator. Hence, applying Lax-Milgram theorem, E ′′ χ(ϕ, σ) has a nonempty resolvent set. Additionally, since the embeddings Z0 ↪→ Z∗ 0 and Z1 0 ↪→ H0 are compact, E ′′ χ(ϕ, σ) and E ′′ χ(ϕ, σ)|Z1 0 have compact resolvents on Z∗ 0 and H0, respectively. Then, from the Fredholm alternative the kernel ker E ′′ χ(ϕ, σ) is finite dimensional and the ranges Rg E ′′ χ(ϕ, σ) and Rg E ′′ χ(ϕ, σ)|Z1 0 are closed in Z∗ 0 and H0. Moreover, Z∗ 0 (resp. H0) is a direct orthogonal sum of ker E ′′ χ(ϕ, σ) and Rg E ′′ χ(ϕ, σ) (resp. RgE ′′ χ(ϕ, σ)|Z1 0 ). Hence we can define a continuous orthogonal projection P : Z0 → Z0 with RgP = ker E ′′ χ(ϕ, σ). Consequently, we can complete the proof by applying the results in [5, Corollary 3.11] with choice of X = Z1 0 , V = Z0, Y = H0, W = Z∗ 0 . □ Now, we will prove the uniform version of the Lojasiewicz-Simon type inequality, which anni- hilates the dependence of the constants on the choice of the critical points, for the limit sets. Lemma 3.9. Assume that (A1)–(A3) are satisfied. (i) For each (ϕ0, σ0) ∈ Z0, there exists an open neighborhood U ⊂ Zr 0 , r < 1 of ωχ 0 ((ϕ0, σ0)) and the constants θ ∈ (0, 1 2 ] and C > 0 such that |Eχ(ϕ, σ) − Eχ,∞+ 0 |1−θ ≤ C∥E ′ χ(ϕ, σ)∥Z∗ 0 (3.15) for all (ϕ, σ) ∈ U , where Eχ,∞+ 0 is already determined in Lemma 3.6. (ii) Moreover, for any (ϕ0, σ0) ∈ Aχ 0 ⊂ Z1 0 , there exists an open neighborhood V ⊂ Zr 0 , r < 1 of αχ 0 ((ϕ0, σ0)) and the constants θ̃ ∈ (0, 1 2 ] and C̃ > 0 such that |Eχ(ϕ, σ) − Eχ,∞− 0 |1−θ̃ ≤ C̃∥E ′ χ(ϕ, σ)∥Z∗ 0 (3.16) for all (ϕ, σ) ∈ V, where Eχ,∞− 0 is already determined in Lemma 3.6. Proof. (i) As a result of Lemma 3.8, for every (ϕ0j , σ0j) ∈ ωχ 0 ((ϕ0, σ0)) there exist constants θj ∈ (0, 1 2 ] and Cj , βj > 0 such that |Eχ(ϕ, σ) − Eχ(ϕ0j , σ0j)|1−θj ≤ Cj∥E ′ χ(ϕ, σ)∥Z∗ 0 for all (ϕ, σ) ∈ Z0 such that ∥(ϕ, σ) − (ϕ0j , σ0j)∥Z0 < βj . Furthermore, since ωχ 0 ((ϕ0, σ0)) is bounded in Z1 0 , it is compact in Zr 0 for all r < 1. Therefore, it can be covered by a finitely many balls in Zr 0 . Therefore, taking U as the union of these balls, 10 S. YAYLA EJDE-2025/100 we achieve the existence of uniform constants θ ∈ (0, 1 2 ] and C > 0 such that (3.15) holds for all (ϕ, σ) ∈ U . (ii) We can establish (3.16), by using the compactness of αχ 0 ((ϕ0, σ0)) in Zr 0 for all r < 1 and arguing as in the proof of i. □ Now, we are in a position to prove Theorem 3.1. Proof of Theorem 3.1. As previously discussed, it is sufficient to consider the case M = 0. Assume that (ϕ0, σ0) ∈ Aχ 0 ⊂ Z1 0 and that {(ϕχ(t), σχ(t)) : t ∈ R} denotes the corresponding full trajectory passing through the point (ϕ0, σ0). To prove the theorem, it is enough to show that ωχ 0 ((ϕ0, σ0)) and αχ 0 ((ϕ0, σ0)) consist of a single point. From Lemma 3.6, it follows that the energy functional Eχ(., .) is constant on ωχ 0 ((ϕ0, σ0)) and αχ 0 ((ϕ0, σ0)). Furthermore, the following holds. Eχ,∞+ 0 := lim t→∞ Eχ(ϕχ(t), σχ(t)) = E(ϕ∗, σ∗), for all (ϕ∗, σ∗) ∈ ωχ 0 ((ϕ0, σ0)), (3.17) Eχ,∞− 0 := lim t→−∞ Eχ(ϕχ(t), σχ(t)) = E(ϕ∗∗, σ∗∗), for all (ϕ∗∗, σ∗∗) ∈ αχ 0 ((ϕ0, σ0)). (3.18) Moreover, we have proved in Lemma 3.9 that there exist open neighborhoods U of ωχ 0 ((ϕ0, σ0)) and V of αχ 0 ((ϕ0, σ0)) such that the inequalities (3.15) and (3.16) are satisfied on U and V, respec- tively. Besides, the definition of the limit sets (Definition 3.5) yields that there exists T0 > 0 such that (ϕχ(t), σχ(t)) ∈ U ⊂ Zr 0 , r < 1, ∀t ≥ T0, (3.19) (ϕχ(−t), σχ(−t)) ∈ V ⊂ Zr 0 , r < 1, ∀t ≥ T0. (3.20) Thus, we obtain from Lemma 3.9 that |Eχ(ϕχ(t), σχ(t)) − Eχ,∞+ 0 |1−θ ≤ C∥E ′ χ(ϕχ(t), σχ(t))∥Z∗ 0 ∀t ≥ T0, (3.21) |Eχ(ϕχ(−t), σχ(−t)) − Eχ,∞− 0 |1−θ̃ ≤ C̃∥E ′ χ(ϕχ(−t), σχ(−t))∥Z∗ 0 ∀t ≥ T0. (3.22) We separate the following part of the proof into two steps. In the first step, we present a detailed proof of forward convergence to equilibrium, namely (3.2) stated in the theorem. In the second step, we will establish the backward convergence (3.3), mainly referring to the method used in the first step. Step 1: Proof of (3.2). First we assume that there exists a time t∗ such that Eχ(ϕχ(t∗), σχ(t∗)) = Eχ,∞+ 0 . Since the energy functional Eχ(ϕχ(t), σχ(t)) is a strict Lyapunov functional, there exists a unique limit (ϕχ ∞, σχ ∞) ∈ Nχ such that (ϕ0, σ0) = (ϕχ(t), σχ(t)) = (ϕχ(t∗), σχ(t∗)) = (ϕχ ∞, σχ ∞). Hence, the proof of (3.2) is complete in this case. Now, we assume that Eχ(ϕχ(t), σχ(t)) > Eχ,∞+ 0 for all t ≥ 0. Integrating by parts on (3.12), we obtain that ⟨E ′ χ(ϕχ, σχ), (η, ξ)⟩Z0 = ∫ Ω µχη dx + ∫ Ω Nχ σ ξ dx (3.23) for every (η, ξ) ∈ Z0. Moreover, since (η, ξ) ∈ Z0, denoting c := ⟨µχ⟩+⟨Nχ σ ⟩ 2 we have∫ Ω µχη dx + ∫ Ω Nχ σ ξ dx = ∫ Ω (µχ − c)η dx + ∫ Ω (Nχ σ − c)ξ dx = 1 2 ∫ Ω (µχ − ⟨µχ⟩ + µχ − ⟨Nχ σ ⟩)η dx + 1 2 ∫ Ω (Nχ σ − ⟨Nχ σ ⟩ + Nχ σ − ⟨µχ⟩)ξ dx ≤ (∥µχ − ⟨µχ⟩∥L2(Ω) + ∥µχ −Nχ σ ∥L2(Ω) + ∥Nχ σ − ⟨Nχ σ ⟩∥L2(Ω))(∥η∥L2(Ω) + ∥ξ∥L2(Ω)). Considering the previous inequality in (3.23), we have ∥E ′ χ(ϕχ, σχ)∥Z∗ 0 ≤ C(∥µχ − ⟨µχ⟩∥L2(Ω) + ∥µχ −Nχ σ ∥L2(Ω) + ∥Nχ σ − ⟨Nχ σ ⟩∥L2(Ω)). (3.24) EJDE-2025/100 STRUCTURE AND STABILITY OF GLOBAL ATTRACTORS FOR A TUMOR MODEL 11 Furthermore, recalling the embedding H2(Ω) ↪→ C(Ω) and (A2), we obtain ∥µχ −Nχ σ ∥L2(Ω) ≤ max x∈Ω ( 1√ p(ϕχ) )∥ √ p(ϕχ)(µχ −Nχ σ )∥L2(Ω) ≤ C∥ √ p(ϕχ)(µχ −Nχ σ )∥L2(Ω). Therefore, applying the Poincare-Wirtinger inequality and considering the last estimate in (3.24), we deduce that ∥E ′ χ(ϕχ(t), σχ(t))∥Z∗ 0 ≤ C ( ∥∇µχ(t)∥L2(Ω) + ∥∇Nχ σ (t)∥L2(Ω) + ∥ √ p(ϕχ)(µχ(t) −Nχ σ (t))∥L2(Ω) ) , (3.25) for all t ∈ R. Considering (3.25) to estimate the right-hand side of (3.21), we have |Eχ(ϕχ(t), σχ(t)) − Eχ,∞+ 0 |1−θ ≤ CΥχ(t) ∀t ≥ T0, (3.26) where Υχ(t) = ( ∥∇µχ(t)∥L2(Ω) + ∥∇Nχ σ (t)∥L2(Ω) + ∥ √ p(ϕχ(t))(µχ(t) −Nχ σ (t))∥L2(Ω) )1/2 . On the other hand, rewriting the energy identity (2.10) in the differential form, we infer that d dt (Eχ(ϕχ(t), σχ(t))) = −(Υχ(t))2 ∀t ∈ R. (3.27) Recalling that Eχ(ϕχ(t), σχ(t)) − Eχ,∞+ 0 > 0 and using (3.27) in (3.26), we obtain d dt ( Eχ(ϕχ(t), σχ(t)) − Eχ,∞+ 0 ) + C ( Eχ(ϕχ(t), σχ(t)) − Eχ,∞+ 0 )2(1−θ) ≤ 0 ∀t ≥ T0. (3.28) Now, let us examine the ordinary differential inequality d dt y(t) + C (y(t))2(1−θ) ≤ 0, y(t) > 0 ∀t ≥ T0. (3.29) In the case θ = 1 2 , d dt y(t) + C (y(t)) ≤ 0, y(t) > 0 ∀t ≥ T0, this yields y(t) ≤ Ke−C(t−T0), ∀t > T0, for some constant K depending on T0. In the case 0 < θ < 1 2 , defining v(t) := (y(t))2θ−1, we obtain v′(t) + (2θ − 1)C ≥ 0 ∀t ≥ T0. Integrating from T0 to t, we infer that v(t) ≥ v(T0) + (1 − 2θ)C(t− T0) > (1 − 2θ)C(t− T0), ∀t > T0, since v(T0) > 0. From the last estimate, it follows that y(t) ≤ K̃(t− T0) −1 1−2θ , ∀t > T0, for some constant K̃ depending on the exponent θ and T0. In the following, we will denote by C, K and K̃ the generic constants depending on θ and T0. Since the inequality in (3.28) can be written in the form of (3.29), we deduce that( Eχ(ϕχ(t), σχ(t)) − Eχ,∞+ 0 ) ≤ Ke−C(t−T0) if θ = 1 2 ,( Eχ(ϕχ(t), σχ(t)) − Eχ,∞+ 0 ) ≤ K̃(t− T0) −1 1−2θ if θ ∈ (0, 1 2 ). (3.30) Recalling the weak formulation formula (2.9), we obtain∫ T t ∥∥(ϕχ t (s), σχ t (s)) ∥∥2 (H1(Ω))∗×(H1(Ω))∗ ds ≤ ∫ T t (Υχ(s))2 ds ∀T > t ≥ T0. (3.31) 12 S. YAYLA EJDE-2025/100 Next, integrating (3.27) from t to T , we have∫ T t (Υχ(s))2 ds = Eχ(ϕχ(t), σχ(t)) − Eχ(ϕχ(T ), σχ(T )) ∀T > t ≥ T0. Considering the last equality in (3.31), we infer that∫ T t ∥∥(ϕχ t (s), σχ t (s)) ∥∥2 (H1(Ω))∗×(H1(Ω))∗ ds ≤ Eχ(ϕχ(t), σχ(t)) − Eχ(ϕχ(T ), σχ(T )) ∀T > t ≥ T0. Passing to limit as T → ∞ in the previous estimate, and using (3.30), we deduce that∫ ∞ t ∥∥(ϕχ t (s), σχ t (s)) ∥∥2 (H1(Ω))∗×(H1(Ω))∗ ds ≤ Ke−C(t−T0) ∀t ≥ T0,∫ ∞ t ∥∥(ϕχ t (s), σχ t (s))2(H1(Ω))∗×(H1(Ω))∗ ds ≤ K̃(t− T0) −1 1−2θ ∀t ≥ T0. Then, arguing as in [17, Lemma 3.2, Lemma 3.3], it follows from the last estimate that∫ ∞ t ∥∥(ϕχ t (s), σχ t (s)) ∥∥ (H1(Ω))∗×(H1(Ω))∗ ds ≤ Ke−C(t−T0) if θ = 1 2 ,∫ ∞ t ∥∥(ϕχ t (s), σχ t (s)) ∥∥ (H1(Ω))∗×(H1(Ω))∗ ds ≤ K̃(t− T0) −θ 1−2θ if θ ∈ (0, 1 2 ), (3.32) for all t ≥ T0. Furthermore, we have∥∥(ϕχ(t), σχ(t)) − (ϕχ(T ), σχ(T )) ∥∥ (H1(Ω))∗×(H1(Ω))∗ ≤ ∫ T t ∥∥(ϕχ t (s), σχ t (s)) ∥∥ (H1(Ω))∗×(H1(Ω))∗ ds. Hence, with the help of (3.32) and using Cauchy criterion for the existence of limits, we obtain that there exists (ϕ∗, σ∗) ∈ ωχ 0 ((ϕ0, σ0)) such that lim t→∞ ∥∥(ϕχ(t), σχ(t) − (ϕ∗, σ∗)) ∥∥ (H1(Ω))∗×(H1(Ω))∗ = 0. (3.33) Furthermore, recalling that the global attractor is bounded in Z1 0 and using interpolation between the spaces (H1(Ω))∗ × (H1(Ω))∗ and Z1 0 , we can complete the proof of (3.2). Step 2. By using the analogous observations (3.18), (3.20), (3.22) and following the similar steps demonstrated in Step 1, we can prove the backward convergence to equilibrium (3.3). Conse- quently, proof is complete. □ 4. Rate of convergence to equilibrium From the proof of Theorem 3.1, we can deduce the rate of convergence in the space (H1(Ω))∗× (H1(Ω))∗. Proposition 4.1. Assume that (A1)–(A3) hold, and θ, θ̃ are the Lojasiewicz exponents fixed in Lemma 3.9. Let T0 > 0 be the time fixed in the proof of Theorem 3.1. Then for any full trajectory γχ M = {(ϕχ M (t), σχ M (t)) : t ∈ R} in the global attractor Aχ M , there exist (ϕ∗, σ∗) ∈ Nχ M and positive constants C,K1, K̃1 such that∥∥(ϕχ M (t), σχ M (t) − (ϕ∗, σ∗)) ∥∥ (H1(Ω))∗×(H1(Ω))∗ ≤ K1 e −C(t−T0), if θ = 1 2 ,∥∥(ϕχ M (t), σχ M (t) − (ϕ∗, σ∗)) ∥∥ (H1(Ω))∗×(H1(Ω))∗ ≤ K̃1 (t− T0) −θ 1−2θ if 0 < θ < 1 2 , (4.1) for all t ≥ T0. Moreover, there exist (ϕ∗∗, σ∗∗) ∈ Nχ M and positive constants C̃,K2, K̃2 such that∥∥(ϕχ M (−t), σχ M (−t) − (ϕ∗∗, σ∗∗)) ∥∥ (H1(Ω))∗×(H1(Ω))∗ ≤ K2 e −C̃(t−T0), if θ̃ = 1 2 ,∥∥(ϕχ M (−t), σχ M (−t) − (ϕ∗∗, σ∗∗)) ∥∥ (H1(Ω))∗×(H1(Ω))∗ ≤ K̃2 (t− T0) −θ̃ 1−2θ̃ if 0 < θ̃ < 1 2 , (4.2) for all t ≥ T0. EJDE-2025/100 STRUCTURE AND STABILITY OF GLOBAL ATTRACTORS FOR A TUMOR MODEL 13 Proof. As discussed previously, it is sufficient to prove the case M = 0. We have already obtained in Theorem 3.1 that such equilibrium points exist. We will only prove (4.1), since (4.2) can be established in the same manner. First, recall the inequality∥∥(ϕχ(t), σχ(t)) − (ϕχ(T ), σχ(T )) ∥∥ (H1(Ω))∗×(H1(Ω))∗ ≤ ∫ T t ∥∥(ϕχ t (s), σχ t (s)) ∥∥ (H1(Ω))∗×(H1(Ω))∗ ds, established in the proof of Theorem 3.1. Passing to limit as T → ∞ in this inequality and considering (3.32) proved in Theorem 3.1, we complete the proof of (4.1). □ Now, let λ2 denote the second eigenvalue of Laplace operator with Neumann boundary condi- tion. By imposing an additional assumption on the chemotaxis and active transport term χ, we obtain exponential convergence to equilibrium. Corollary 4.2. Let (A1)–(A3) hold, and let T0 > 0 be the time fixed in the proof of Theorem 3.1. Suppose χ2 < min(λ2 + c1 − α, 1)). Then, for any full trajectory γχ M = {(ϕχ M (t), σχ M (t)) : t ∈ R} in the global attractor Aχ M there exist (ϕ∗, σ∗) ∈ Nχ M and positive constants C,K1 such that∥∥(ϕχ M (t), σχ M (t) − (ϕ∗, σ∗)) ∥∥ (H1(Ω))∗×(H1(Ω))∗ ≤ K1 e −C(t−T0), for all t ≥ T0. Proof. As mentioned previously, we consider the case M = 0. To prove the corollary, it is sufficient to prove that Lojasiewicz-Simon inequality (3.14) is satisfied with the exponent θ = 1 2 . To this end, we will examine the kernel of the linearization E ′′ χ(ϕ, σ). If (η, ξ) ∈ ker E ′′ χ(ϕ, σ), then it solves the problem −∆η + Ψ′′(ϕ) − χξ = 0, (4.3) ξ − χη = 0. (4.4) Testing (4.3) with η and (4.4) with ξ, we obtain ∥∇η∥2L2(Ω) + ∫ Ω Ψ′′(ϕ)|η1|2 dx− 2χ ∫ Ω ηξ dx + ∥ξ∥2L2(Ω) = 0, which yields (λ2 + c1 − α− χ2)∥η∥2 + (1 − χ2)∥ξ∥2 = 0. Recalling the assumptions on the parameter χ, we infer that (η, ξ) = (0, 0), which means that ker E ′′ χ(ϕ, σ) = {(0, 0)}. Hence, using [5, Corollary 3.12], we deduce that the Lojasiewicz-Simon inequality (3.14) is satisfied with the exponent θ = 1 2 . □ Remark 4.3. The stationary point (ϕ∗, σ∗) ∈ Nχ M is called hyperbolic if the linearization E ′′ χ(ϕ∗, σ∗) is invertible, i.e., ker E ′′ χ(ϕ∗, σ∗) = {(0, 0)}. Therefore, under the assumptions of the Corollary 4.2, all stationary points are hyperbolic. 4.1. Rate of convergence in ZM for 0 < θ < 1/2. In this section, we will demonstrate that when 0 < θ < 1/2, the full trajectories in the global attractor maintain the same convergence rate as established in (H1(Ω))∗ × (H1(Ω))∗ (see Proposition 4.1) within the space ZM . First, note that (−∆)−1 denotes the inverse of the minus Laplace operator associated with Neumann boundary conditions, acting on functions with zero spatial average. Namely, the norm ∥(−∆)−1/2 · ∥L2(Ω) is a norm on {v ∈ (H1(Ω))∗ : ⟨v⟩ = 0}, which is equivalent to the usual norm of (H1(Ω))∗. Furthermore, ∥(−∆)1/2 · ∥L2(Ω) = ∥∇ · ∥L2(Ω) is equivalent to the usual norm on H1(Ω). In the following theorem, we use the ideas in [21]. Theorem 4.4. Assume (A1)–(A3) hold, and θ, θ̃ ∈ (0, 1 2 ) are the Lojasiewicz exponents fixed in Lemma 3.9. Let T0 > 0 be the time fixed in the proof of Theorem 3.1. Then, for any full trajectory γχ M = {(ϕχ M (t), σχ M (t)) : t ∈ R} in the global attractor Aχ M , there exist (ϕ∗, σ∗) ∈ Nχ M and κ > 0 such that ∥∥(ϕχ M (t), σχ M (t) − (ϕ∗, σ∗)) ∥∥ ZM ≤ κ (t− T0) −θ 1−2θ , (4.5) 14 S. YAYLA EJDE-2025/100 for all t ≥ T0. Moreover, there exist (ϕ∗∗, σ∗∗) ∈ Nχ M and κ̃ > 0 such that∥∥(ϕχ M (−t), σχ M (−t) − (ϕ∗∗, σ∗∗)) ∥∥ ZM ≤ κ̃ (t− T0) −θ̃ 1−2θ̃ , (4.6) for all t ≥ T0. Proof. As previously discussed, it is sufficient to consider the case M = 0. We will only prove (4.5) and do not present the proof of (4.6), as it can be established following the same steps. From Proposition 4.1, we know that for any full trajectory γχ = {(ϕχ(t), σχ(t)) : t ∈ R} in the global attractor Aχ, there exist (ϕ∗, σ∗) ∈ Nχ such that (4.1) is satisfied. Defining ϕ̄χ := ϕχ − ϕ∗, σ̄χ := σχ − σ∗ and µ̄χ := µ− µ0, we obtain from (2.9) that ⟨ϕ̄χ t, η⟩ + ⟨∇µ̄χ,∇η⟩ = ⟨p(ϕχ)(σ̄χ − χϕ̄χ − µ̄χ), η⟩, (4.7) µ̄χ = −∆ϕ̄χ + Ψ′(ϕχ) − Ψ′(ϕ∗) − χσ̄χ, (4.8) ⟨σ̄χ t, ξ⟩ + ⟨∇σ̄χ,∇ξ⟩ = χ⟨∇ϕ̄χ,∇ξ⟩ − ⟨p(ϕχ)(σ̄χ − χϕ̄χ − µ̄χ), η⟩, (4.9) for all η, ξ ∈ H1(Ω). Choosing η = (−∆)−1ϕ̄χ in (4.7) and ξ = (−∆)−1σ̄χ in (4.9), and summing the obtained identities, we infer that 1 2 d dt (∥ϕ̄χ∥2(H1(Ω))∗ + ∥σ̄χ∥2(H1(Ω))∗) + ∥ϕ̄χ∥2H1(Ω) + ∥σ̄χ∥2L2(Ω) + ⟨Ψ′(ϕχ) − Ψ′(ϕ∗), ϕ̄χ⟩ = 2χ ∫ σ̄χϕ̄χ dx + ⟨p(ϕχ)(σ̄χ − χϕ̄χ − µ̄χ), (−∆)−1ϕ̄χ⟩ − ⟨p(ϕχ)(σ̄χ − χϕ̄χ − µ̄χ), (−∆)−1σ̄χ⟩. (4.10) Now, we will evaluate the terms on the right-hand side of (4.10). First, let us observe that for any η ∈ H1(Ω), since ϕχ ∈ H3(Ω) ⊂ L∞, it holds |⟨p(ϕχ)(σ̄χ − χϕ̄χ − µ̄χ), η⟩| ≤ ∥σ̄χ − χϕ̄χ − µ̄χ∥L2(Ω)∥η∥L2(Ω) ≤ C∥η∥H1(Ω). Then |⟨p(ϕχ)(σ̄χ − χϕ̄χ − µ̄χ), (−∆)−1ϕ̄χ⟩| + |⟨p(ϕχ)(σ̄χ − χϕ̄χ − µ̄χ), (−∆)−1σ̄χ⟩| ≤ ∥p(ϕχ)(σ̄χ − χϕ̄χ − µ̄χ)∥(H1(Ω))∗(∥ϕ̄χ∥(H1(Ω))∗ + ∥σ̄χ∥(H1(Ω))∗) ≤ C(∥ϕ̄χ∥(H1(Ω))∗ + ∥σ̄χ∥(H1(Ω))∗). (4.11) On the other hand, for the first term on the right-hand side of (4.10), we have 2χ ∣∣ ∫ Ω σ̄χϕ̄χ dx ∣∣ ≤ 4χ2∥σ̄χ∥2(H1(Ω))∗ + 1 4 ∥ϕ̄χ∥2H1(Ω). (4.12) Moreover, since Ψ′ 0 is monotone and Λ′ is Lipschitz continuous, for the nonlinear term we obtain (Ψ′(ϕχ) − Ψ′(ϕ∗), ϕ̄χ) = (Ψ′ 0(ϕχ) − Ψ′ 0(ϕ∗), ϕ̄χ) + (Λ′(ϕχ) − Λ′(ϕ∗), ϕ̄χ) ≥ −α∥ϕ̄χ∥2L2 ≥ −1 4 ∥ϕ̄χ∥2H1(Ω) − c∥ϕ̄χ∥2(H1(Ω))∗ , (4.13) for some c > 0. In the following, C denotes a generic constant, which may vary from line to line and even within the same line. Now, considering the estimates (4.11), (4.12) and (4.13) in (4.10), we obtain 1 2 d dt ( ∥ϕ̄χ∥2(H1(Ω))∗ + ∥σ̄χ∥2(H1(Ω))∗ ) + ∥ϕ̄χ∥2H1(Ω) + ∥σ̄χ∥2L2(Ω) ≤ C(∥ϕ̄χ∥(H1(Ω))∗ + ∥σ̄χ∥(H1(Ω))∗). (4.14) Next, testing (4.7) with µ̄χ and (4.9) with σ̄χ − χϕ̄χ, we infer that d dt (1 2 ∥ϕ̄χ∥2H1(Ω) + 1 2 ∥σ̄χ∥2L2(Ω) + χ ∫ σ̄χϕ̄χ dx ) + d dt (∫ Ω Ψ(ϕχ) dx− ∫ Ω Ψ(ϕ∗) dx + ∫ Ω Ψ′(ϕ∗)ϕ∗ dx− ∫ Ω Ψ′(ϕ∗)ϕχ dx ) + ∫ Ω ( |∇µ̄χ|2 + |∇σ̄χ − χ∇ϕ̄χ|2 + p(ϕχ)(σ̄χ − χϕ̄χ − µχ)2 ) dx = 0. (4.15) EJDE-2025/100 STRUCTURE AND STABILITY OF GLOBAL ATTRACTORS FOR A TUMOR MODEL 15 Using the Newton-Leibniz formula and recalling that ϕχ, ϕ∗ ∈ H3(Ω) ⊂ L∞(Ω),∣∣∣ ∫ Ω Ψ(ϕχ) dx− ∫ Ω Ψ(ϕ∗) dx + ∫ Ω Ψ′(ϕ∗)ϕ∗ dx− ∫ Ω Ψ′(ϕ∗)ϕχ dx ∣∣∣ ≤ ∣∣∣ ∫ Ω ∫ 1 0 ∫ 1 0 Ψ′′(τsϕ + (1 − τs)ϕ∗)(ϕχ − ϕ∗)2 ds dτ dx ∣∣∣ ≤ C∥ϕ̄χ∥2L2(Ω) ≤ 1 8 ∥ϕ̄χ∥2H1(Ω) + C∥ϕ̄χ∥2(H1(Ω))∗ . (4.16) Furthermore, ∣∣χ∫ σ̄χϕ̄χ dx ∣∣ ≤ C∥ϕ̄χ∥2L2(Ω) ≤ 1 8 ∥ϕ̄χ∥2H1(Ω) + C∥σ̄χ∥2(H1(Ω))∗ . (4.17) Now defining, Φ(t) := 1 2 ∥∇ϕ̄χ∥2L2(Ω) + 1 2 ∥σ̄χ∥2L2(Ω) + χ ∫ σ̄χϕ̄χ dx + ∫ Ω Ψ(ϕχ) dx− ∫ Ω Ψ(ϕ∗) dx + ∫ Ω Ψ′(ϕ∗)ϕ∗ dx − ∫ Ω Ψ′(ϕ∗)ϕχ dx + 1 2 (∥ϕ̄χ∥2(H1(Ω))∗ + ∥σ̄χ∥2(H1(Ω))∗), we deduce from (4.16) and (4.17) that∣∣Φ(t) − C(∥ϕ̄χ∥2H1(Ω) + ∥σ̄χ∥2L2(Ω))| ≤ ∥ϕ̄χ∥2(H1(Ω))∗ + ∥σ̄χ∥2(H1(Ω))∗ . (4.18) Therefore, adding (4.14) and (4.15), we obtain d dt (Φ(t)) + β Φ(t) ≤ C(∥ϕ̄χ∥(H1(Ω))∗ + ∥σ̄χ∥(H1(Ω))∗), (4.19) for some β > 0. Considering (4.1) in (4.19), we have d dt (Φ(t)) + βΦ(t) ≤ C (t− T0) −θ 1−2θ . Using that the exponential function grows faster than any polynomial, it follows from the last inequality that Φ(t) ≤ Φ(T0)e−β(t−T0) + e−βtC ∫ t T0 eβτ (τ − T0) −2θ 1−2θ dτ ≤ Ce−βt + C (t− T0) −θ 1−2θ ≤ κ1 (t− T0) −θ 1−2θ , (4.20) for some κ1 > 0. Then, recalling (4.18), we obtain from (4.1) and (4.20) that ∥ϕ̄χ∥2H1(Ω) + ∥σ̄χ∥2L2(Ω) ≤ Φ(t) + ∥ϕ̄χ∥2(H1(Ω))∗ + ∥σ̄χ∥2(H1(Ω))∗ ≤ κ2 (t− T0) −θ 1−2θ , (4.21) for some κ2 > 0. Hence, we conclude that (4.5) is satisfied in Z0, i.e.,∥∥(ϕχ(t), σχ(t) − (ϕ∗, σ∗)) ∥∥ Z0 ≤ κ (t− T0) −θ 1−2θ , (4.22) for some κ > 0. □ 5. Stability of the Attractors In this section, we consider the chemotaxis and active transport parameter χ as a perturbation parameter. Then, from Theorem 2.12, there exists a family of global attractors {Aχ M}χ≥0 for the family of the semigroups {Sχ M}χ≥0 acting on the phase space ZM . Since M is fixed and has no effect on the following calculations, we omit M from the notation for the sake of simplicity. In the following part of the paper, we will use the notations {Aχ}χ≥0 and (ZM , Sχ) instead of {Aχ M}χ≥0 and (ZM , Sχ M ), respectively. Here, it is worth recalling that A0 and S0 denote the global attractor 16 S. YAYLA EJDE-2025/100 and the semigroup, respectively, in the case where chemotaxis and active transport are neglected (χ = 0). We investigate the stability of the family of the global attractors {Aχ}χ≥0 as χ → 0. Definition 5.1. Let X be a Banach space, and I be a metric space. The family of global attractors {Aα}α∈I is called upper semicontinuous at the point α0 ∈ I if lim α→α0 distX(Aα,Aα0) = 0, and it is called lower semicontinuous at the point α0 ∈ I if lim α→α0 distX(Aα0 ,Aα) = 0, where distX(·, ·) is the Hausdorff semidistance defined in Definition 2.10. We exploit the following abstract result proven in [20] to obtain the upper semicontinuity of the family of global attractors {Aχ}χ≥0 as χ → 0. Theorem 5.2. Suppose that a dynamical system (X,Sα(t)) possesses a global attractor Aα for every α ∈ I, where I is a complete metric space. Assume that the following conditions hold: (1) There exists a compact set K such that ∪ α∈I Aα ⊂ K, (2) If αk → α0, xk ∈ Aαk and xk → x0, then Sαk(t0)xk → Sα0(t0)x0 for some t0 > 0. Then, the family of attractors is upper semicontinuous at the point α0. We begin by proving the following lemma, which states the continuous dependence of the solutions on the parameter χ. Lemma 5.3. Assume that (A1), (A2) are satisfied and {χn}n∈N ∈ [0, χ0], χn → 0 as n → ∞. For any initial datum (ϕ0, σ0) ∈ ZM , we have Sχn(·)(ϕ0, σ0) → S0(·)(ϕ0, σ0) strongly in C([0, T ];L2(Ω) × L2(Ω)). Proof. Let us denote (ϕn, σn) := Sχn(t)(ϕ0, σ0). Applying the same procedure used in the proof of [15, Theorem 3.1] to the problem ⟨⟨ϕχn t , η⟩⟩ + ⟨∇µχn ,∇η⟩ = ⟨p(ϕχn)(σχn + χn(1 − ϕχn) − µχn), η⟩, ⟨µχn , η⟩ = ⟨∇ϕχn ,∇η⟩ + ⟨Ψ′(ϕχn), η⟩ − ⟨χnσ χn , η⟩, ⟨⟨σχn t , ξ⟩⟩ + ⟨(∇σχn − χn∇ϕχn),∇ξ⟩ = −⟨p(ϕχn)(σχn + χn(1 − ϕχn) − µχn), ξ⟩, (5.1) we obtain the uniform bounds ∥ϕn∥L∞(0,T ;H1(Ω))∩L2(0,T ;H3(Ω)) ≤ C, ∥ϕnt∥L2(0,T ;(H1(Ω))∗) ≤ C ∥σn∥L∞(0,T ;L2(Ω))∩L2(0,T ;H1(Ω)) ≤ C, ∥σnt∥L2(0,T ;(H1(Ω))∗) ≤ C, ∥µn∥L2(0,T ;H1(Ω)) ≤ C, where µn := µχn . Therefore, from the Banach-Alaoglu theorem it follows that up to a subsequence, (ϕn, σn) → (ϕ̂, σ̂) weakly-star in L∞(0, T ;H1(Ω) × L2(Ω)), (ϕnt, σnt) → (ϕ̂t, σ̂t) weakly in L2(0, T ; (H1(Ω))∗ × (H1(Ω))∗), (ϕn, σn) → (ϕ̂, σ̂) weakly in L2(0, T ;H3(Ω) ×H1(Ω)), µn → µ̂ weakly in L2(0, T ;H1(Ω)), (5.2) where ϕ̂ ∈ L∞(0, T ;H1(Ω)) ∩ L2(0, T ;H3(Ω)), σ̂ ∈ L∞(0, T ;L2(Ω)) ∩ L2(0, T ;H1(Ω)), µ̂ ∈ L2(0, T ;H1(Ω)), ϕ̂t, σ̂t ∈ L2(0, T ; (H1(Ω))∗). EJDE-2025/100 STRUCTURE AND STABILITY OF GLOBAL ATTRACTORS FOR A TUMOR MODEL 17 From (5.2), using the Aubin-Lions lemma, we infer that (up to a subsequence) ϕn → ϕ̂ strongly in C([0, T ];Lκ(Ω)), for 2 ≤ κ < 6. (5.3) Moreover, from (5.2), using the Lions-Magenes lemma, we infer that (up to a subsequence) σn → σ̂ strongly in C([0, T ];L2(Ω)). (5.4) Then, arguing as in [15, Theorem 3.1], we obtain (up to a subsequences) p(ϕn) → p(ϕ̂), strongly in L2(0, T ;L6/5(Ω)) (5.5) and (σχn + χn(1 − ϕχn) − µχn) → (σ̂ − µ̂), weakly in L2(0, T ;L6(Ω)). (5.6) Hence, we can pass to the limit in the term ⟨p(ϕχn)(σχn +χn(1−ϕχn)−µχn), η⟩ with the help of (5.5) and (5.6). Thanks to all of the convergence results established above, we can pass to the limit in the problem (5.1) and deduce that (ϕ̂, σ̂, µ̂) is a weak solution of the problem (2.1)–(2.4) without chemotaxis and active transport, i.e., with the parameter χ = 0. Then, using the uniqueness of weak solutions to the problem (2.1)–(2.4), we deduce that (ϕ̂, σ̂) = S0(t)(ϕ0, σ0). Consequently, every convergent subsequence of {(ϕn, σn)}n∈N has the same limit, so we conclude that (5.3) and (5.4) are satisfied by the whole sequence. Hence, the proof is completed. □ Now, we prove that the ∪ 0≤χ≤χ0 Aχ is relatively compact. Lemma 5.4. Assume that (A1), (A2) are satisfied and χ ∈ [0, χ0]. Then, the family of the global attractors {Aχ}0≤χ≤χ0 for problem (2.1)–(2.4) is relatively compact in Z1 M . Namely, there exists a compact set K ⊂ ZM such that ∪0≤χ≤χ0 Aχ ⊂ K. Proof. In [15], it was shown that the global attractor Aχ is bounded in Z1 M . Exactly, using the regularization of weak solutions, it can be observed that for all t > 0 (cf. the proof of [15, Theorem 3.3]), ∫ t+1 t ( ∥ϕχ(τ)∥2H3(Ω) + ∥σχ(τ)∥2H1(Ω) ) dτ ≤ C, where C is independent of χ and depends on χ0. Hence, by following the same steps in [15, Theorem 5.10], we readily obtain that the family of the global attractors {Aχ}0≤χ≤χ0 is uniformly bounded in Z1 M . Precisely, there exist B ∈ Z1 M such that ∪0≤χ≤χ0 Aχ ⊂ B. Hence, by choosing K = BH1(Ω)×L2(Ω) , we obtain that K is bounded in H3(Ω) × H1(Ω). Then, by the compact embedding H3(Ω) ×H1(Ω) ↪→ H1(Ω) × L2(Ω), the set K is compact in ZM , and the proof is complete. □ Lemma 5.5. Under the assumptions of Lemma 5.4, for every bounded set B ⊂ ZM , there exists a time T0(B) > 0 such that sup 0≤χ≤χ0 sup (ϕ,σ)∈B ∥Sχ(t)(ϕ, σ)∥H3(Ω)×H1(Ω) ≤ R, ∀t ≥ T0(B), for some constant R > 0. Proof. From Lemma 5.4, we observe that there exists a bounded absorbing set B in H3(Ω)×H1(Ω) such that for all 0 ≤ χ ≤ χ0 and for every bounded set B ⊂ ZM , there exists a time tχ0 (B) > 0 such that Sχ(t)B ⊂ B for all t ≥ tχ0 (B). Since for any t ≥ 0, the state Sχ(t) depends continuously on the parameter χ (Lemma 5.3), we infer that the time tχ0 (B) also depends continuously on χ. Therefore, we deduce that ∪0≤χ≤χ0 Sχ(t)B ⊂ B for all t ≥ T0(B) := max 0≤χ≤χ0 tχ0 (B). We complete the proof of the lemma by choosing R as the diameter of the absorbing set B. □ 18 S. YAYLA EJDE-2025/100 Lemma 5.6. Let the assumptions of Lemma 5.4 hold and let B ∈ ZM be a bounded set. Then, for any (ϕ1, σ1), (ϕ2, σ2) ∈ B, there exists a time T0(B) > 0 such that the semigroup {Sχ(t)}t≥0, generated by the weak solution of the problem (2.1)–(2.4), satisfies ∥Sχ(t)(ϕ1, σ1) − Sχ(t)(ϕ2, σ2)∥H1(Ω)×L2(Ω) ≤ C ( ∥ϕ1 − ϕ2∥1/2H1(Ω) + ∥σ1 − σ2∥1/2L2(Ω) ) ∀t ≥ T0(B), where C is independent of χ. Proof. Using the interpolation inequality, we have ∥Sχ(t)(ϕ1, σ1) − Sχ(t)(ϕ2, σ2)∥2H1(Ω)×L2(Ω) ≤ ∥Sχ(t)(ϕ1, σ1) − Sχ(t)(ϕ2, σ2)∥H3(Ω)×H1(Ω)∥Sχ(t)(ϕ1, σ1) − Sχ(t)(ϕ2, σ2)∥(H1(Ω))∗×(H1(Ω))∗ . From Lemma 5.5, there exists a time T0(B) > 0 such that ∥Sχ(t)(ϕ1, σ1) − Sχ(t)(ϕ2, σ2)∥2H1(Ω)×L2(Ω) ≤ 2R∥Sχ(t)(ϕ1, σ1) − Sχ(t)(ϕ2, σ2)∥(H1(Ω))∗×(H1(Ω))∗ ∀t ≥ T0(B). (5.7) Taking into account Theorem 2.4 in (5.7), we complete the proof. □ Lemma 5.7. Let the assumptions of Lemma 5.4 hold. Then, If χn → 0, (ϕn, σn) ∈ Aχn and (ϕn, σn) → (ϕ0, σ0), then Sχn(t0)(ϕn, σn) → S0(t0)(ϕ0, σ0) for some t0 > 0. Proof. Firstly, from Lemma 5.3, we have Sχn(t)(ϕ0, σ0) → S0(t)(ϕ0, σ0) strongly in L2(Ω) × L2(Ω) ∀t ≥ 0. (5.8) On the other hand, Lemma 5.5 implies that there exists T0 > 0 such that the sequence {Sχn(t)(ϕ0, σ0) : χn ∈ [0, χ0], t ≥ T0} is uniformly bounded in H3(Ω) ×H1(Ω). Therefore, (5.8) yields Sχn(t)(ϕ0, σ0) → S0(t)(ϕ0, σ0) strongly in H1(Ω) × L2(Ω) ∀t ≥ T0. (5.9) Finally, applying the triangle inequality and Lemma 5.6, we have ∥Sχn(t)(ϕn, σn) − S0(t)(ϕ0, σ0)∥H1(Ω)×L2(Ω) ≤ ∥Sχn(t)(ϕn, σn) − Sχn(t)(ϕ0, σ0)∥H1(Ω)×L2(Ω) + ∥Sχn(t)(ϕ0, σ0) − S0(t)(ϕ0, σ0)∥H1(Ω)×L2(Ω) ≤ C ( ∥ϕn − ϕ0∥1/2H1(Ω) + ∥σn − σ0∥1/2L2(Ω) ) + ∥Sχn(t)(ϕ0, σ0) − S0(t)(ϕ0, σ0)∥H1(Ω)×L2(Ω) ∀t ≥ T0. (5.10) Hence, taking into account (5.9) in (5.10) and choosing t0 > T0, we obtain the desired result of the lemma. □ Now, we are in a position to prove the main result of this section. Theorem 5.8. Let the conditions of Lemma 5.4 hold. Then the family of global attractors {Aχ}0≤χ≤χ0 is upper-semicontinuous as χ → 0, i.e., lim χ→0 distZM (Aχ,A0) = 0. Proof. Lemmas 5.4 and 5.7 verify that all assumptions of Theorem 5.2 are satisfied. Therefore, the proof is completed by applying Theorem 5.2 introduced in [20]. □ As mentioned in Remark 4.3, if the chemotaxis parameter satisfies χ2 < min(λ2 + c1 − α, 1)), all stationary points in Nχ are hyperbolic. 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Wu; Convergence to equilibrium for a bulk-surface Allen-Cahn system coupled through a nonlinear robin boundary condition, Discrete Contin. Dyn. Syst., 40 (2020), no. 3, 1847–1878. [22] A. Miranville; The Cahn-Hilliard equation: recent advances and applications, SIAM, 2019. Sema Yayla Department of Mathematics, Faculty of Science, Hacettepe University, Beytepe 06800, Ankara, Turkey Email address: semasimsek@hacettepe.edu.tr 1. Introduction 2. Setting of the problem and the previous results 3. Further geometric properties of the global attractor 3.1. Properties of limit sets 3.2. Łojasiewicz-Simon type inequality 4. Rate of convergence to equilibrium 4.1. Rate of convergence in ZM for 0<<1/2 5. Stability of the Attractors References