Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 96, pp. 1–35. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2025.96 CONTROLLABILITY UNDER POSITIVITY CONSTRAINTS FOR NON-LINEAR AND NON-LOCAL PARABOLIC PDES MIGUEL R. NUÑEZ-CHÁVEZ Abstract. This article studies the control of a non-local and non-linear parabolic PDE. The tools to develop the control study are: regularity in Hölder spaces, parabolic regularity, Carleman and Observability inequalities, compactness, stability and the Kakutani Fixed Point method. We obtain three results about controllability. first, local results in Hölder spaces; second, restriction on the control signal with target trajectories; and third, positivity of the minimal controllability time. 1. Introduction Control Theory is a branch of Differential Equations with foundational results dating back approximately 60 years. Early studies in this field focused on optimizing (controlling) resources (such as time, space, finances, personnel, and material quality) through the framework of the Calculus of Variations. The development of the HUM method and observability inequality were pivotal advancements, enabling the study of controllability in elliptic, parabolic, and hyperbolic equations. Prominent contributions to Control Theory, both in theoretical and numerical aspects, were made by researchers such as Lions, Alekseev, Fursikov, Imanuvilov, Yamamoto, Coron, Puel, Guerrero, Zuazua, Fernández-Cara and others. Studies on controllability involving nonlocal terms were carried out by Fernández-Cara, Ĺımaco and Menezes [4] in 2012; Fernández-Cara, Clark, Ĺımaco and Medeiros [3] in 2013; Fernández- Cara, Liu and Zuazua [6] in 2015; Fernández-Cara, Ĺımaco, Nina-Huamán and Nuñez-Chávez [5] in 2019; Prouvée and Ĺımaco [16] in 2019; Lopes and Ĺımaco [12] in 2022; and Costa, Ĺımaco, Lopes and Provée [2] in 2023. Furthermore, studies addressing signal-constrained controllability of the solution or control were conducted by Lohéac, Trelát and Zuazua [11] in 2017; Pighin and Zuazua [14] in 2018 and [15] in 2019; and Nuñez-Chávez [13] in 2021. The importance of nonlocal terms in differential equations is highlighted in models such as: • Population dynamics with a ( ∫ Ω y(x, t) dx, ∫ Ω ∇y(x, t) dx ) , • Reaction-diffusion systems with a ( ∫ Ω l(x)y(x, t) dx ) , • Wave theory with ∫ Ω K(x, t)y(x, t) dx. The present work on controllability integrates the concepts of nonlocal terms and sign preservation of the control. Let Ω ⊂ RN , with N ≥ 1 an integer, be a non-empty, open, bounded, connected set with a sufficiently regular boundary ∂Ω. For each T > 0, we denote the sets QT := Ω× (0, T ) and ΣT := ∂Ω× (0, T ). 2020 Mathematics Subject Classification. 93C20, 35K55, 93C10, 93B05, 93B07. Key words and phrases. Nonlocal parabolic equation; nonlinear parabolic equation; exact controllability to trajectories; control with restriction; positivity time of the control. ©2025. This work is licensed under a CC BY 4.0 license. Submitted May 29, 2025. Published October 14, 2025. 1 2 M. R. NUÑEZ-CHÁVEZ EJDE-2025/96 Let ω, ω1 ⊂ Ω be two non-empty open sets such that ω1 ⊂ ω. We will analyze the control behavior for the nonlinear, nonlocal parabolic system yt − a (∫ Ω y(x′, t) dx′ ) ∆y = vϱω in QT , y(x, t) = 0 on ΣT , y(x, 0) = y0(x) in Ω, (1.1) where v(x, t) is the control, y(x, t) is the solution associated with the control v, and ϱω ∈ C∞ 0 (Ω) is such that ϱω ≥ 0 in Ω, ϱω = 0 in Ω\ω, and ϱω = 1 in ω1. Let a = a(·) ∈ C2(R) be a real-valued function, and suppose there exist positive constants a0, a1, and M such that 0 < a0 ≤ a(r) ≤ a1 and |a′(r)|+ |a′′(r)| ≤M, ∀r ∈ R. (1.2) Remark 1.1. Concerning the function a(·), by the Main Value Theorem, for any r1, r2 ∈ R, we have |a(r1)− a(r2)| ≤M |r1 − r2|, |a′(r1)− a′(r2)| ≤M |r1 − r2|. For each k, l ∈ N0 := N ∪ {0} and θ ∈ (0, 1), we define: Ck,l(QT ) := { z ∈ C(QT ) : ∂ σ x z(·, t) ∈ C(Ω), ∀|σ| ≤ k,∀t ∈ [0, T ], ∂jt z(x, ·) ∈ C([0, T ]), ∀j ≤ l, ∀x ∈ Ω } . Ck+θ,l+ θ 2 (QT ) := { z ∈ Ck,l(QT ) : sup |σ|=k sup (x1,t1 )̸=(x2,t2) |∂σx∂ltz(x1, t1)− ∂σx∂ l tz(x2, t2)| (|x1 − x2|+ |t1 − t2|1/2)θ < +∞ } Ck+θ(Ω) := { z ∈ Ck(Ω) : sup |σ|=k sup x1 ̸=x2 |∂σx z(x1)− ∂σx z(x2)| |x1 − x2|θ < +∞ } , all three are Banach spaces with their canonical norms. To simplify notation, we denote: • ∥ · ∥p : norm of the Banach space Lp(Ω), 1 ≤ p ≤ ∞, • (·, ·) : inner product in the Hilbert space L2(Ω), • ∥ · ∥k,l : norm of continuous space Ck,l(QT ), • ∥ · ∥p,q,s1,s2 : norm of the Bochner space Lp((s1, s2);L q(Ω)), 1 ≤ p, q ≤ ∞, p ̸= q • ∥ · ∥p,s1,s2 : norm of the Bochner space Lp((s1, s2);L p(Ω)), 1 ≤ p ≤ ∞, • ∥ · ∥k+θ : norm of Hölder space Ck+θ(Ω), • ∥ · ∥k+θ,l+ θ 2 : norm of Hölder space Ck+θ,l+ θ 2 (QT ). Remark 1.2. Note that, if y0 ∈ C2+ 1 2 (Ω) satisfies the first-order compatibility condition and v ∈ C1/2,1/4(QT ), with ∥y0∥2+ 1 2 and ∥v∥1/2,1/4 sufficiently small, then system (1.1) has exactly one solution y ∈ C2+ 1 2 ,1+ 1 4 (QT ) satisfying ∥y∥2+ 1 2 ,1+ 1 4 ≤ C(Ω,M, a0, a1) ( ∥v∥ 1 2 , 1 4 + ∥y0∥2+ 1 2 ) . The proof will be provided in Appendix 5.1. 1.1. Important results. Although we have local existence and uniqueness results, the present paper will demonstrate global results regarding controllability along trajectories. Definition 1.3. Let y0 = y0(x) and v = v(x, t) are sufficiently regular. The function y = y(x, t) is said to be a target trajectory for the system (1.1) if it is a solution of the system yt − a (∫ Ω y(x′, t) dx′ ) ∆y = vϱω in QT , y(x, t) = 0 on ΣT , y(x, 0) = y0(x) in Ω. (1.3) The function v is called a target control. EJDE-2025/96 CONTROLLABILITY FOR NON-LINEAR AND NON-LOCAL PARABOLIC PDES 3 Remark 1.4. The good definition of a target trajectory is based on the existence and uniqueness of the solution to the system (1.3), for a particular case, we can observe the one provided by Remark 1.2 when the initial data y0 ∈ C2+ 1 2 (Ω) and the control v ∈ C1/2,1/4(QT ) are sufficiently small. In fact, we have the following result: ∥y∥2+ 1 2 ,1+ 1 4 ≤ C(Ω,M, a0, a1) ( ∥v∥1/2,1/4 + ∥y0∥2+ 1 2 ) . We will present the first main result, which is actually the key result of the work, local control- lability along target trajectories. For each δ > 0, let us denote Dδ := { y0 ∈ C2+ 1 2 (Ω) : ∥y0∥2+ 1 2 ≤ δ } . Theorem 1.5. Let y0 ∈ C2+ 1 2 (Ω) and v ∈ C1/2,1/4(QT ). We denote by y(x, t) the target trajectory for the system (1.1). Assuming condition (1.2) on the real function a(·). For each T > 0 and each initial data y0 ∈ C2+ 1 2 (Ω) with y0 − y0 ∈ Dδ where δ > 0 is sufficiently small, we can find a control v ∈ C1/2,1/4(QT ) such that ∥v − v∥1/2,1/4 ≤ C(Ω,M)∥y0 − y0∥2+ 1 2 , where the associated state y(x, t), the solution of the system (1.1) with initial data y0, satisfies y(x, T ) = y(x, T ) in Ω. Remark 1.6. In [5], the authors proved a local controllability result similar to Theorem 1.5 with control on Lp spaces, but we will need much more regularity (Hölder spaces) in the local control v for the following results. Let us additionally suppose that the target trajectory satisfies the following: ∥y∥L∞(0,+∞;H5(Ω)) ≤ a0 2M |Ω|1/2CΩ < +∞, (1.4) where |Ω| is the measure (volume) of Ω, and CΩ = max { C(Ω), [C(Ω)]2, [C(Ω)]3, [C(Ω)]4, [C(Ω)]5 } = max{C(Ω), [C(Ω)]5}, with C(Ω) the constant from the Poincaré Inequality ∥ · ∥2 ≤ C(Ω)∥∇ · ∥2. We have the second main result, global controllability for large-time target trajectories while preserving the control signal. Theorem 1.7. Consider the dimension N ≤ 3, let y0 = y0(x) and v = v(x, t) be sufficiently regular and sufficiently small. We denote by y(x, t) the target trajectory for the system (1.1). Assuming the conditions in (1.2) for the real function a(·), the condition (1.4) on the target trajectory y, and that there exists a constant η > 0 such that v(x, t) ≥ η, ∀(x, t) ∈ Ω× [0,+∞). (1.5) Then, for each initial data y0 ∈ C2+ 1 2 (Ω) ∩ (H4(Ω) ∩H1 0 (Ω)), there exists a real number T0 > 0 such that for every T ≥ T0, we can find a control v ∈ L∞(QT ) with v(x, t) ≥ 0, ∀(x, t) ∈ QT , such that the associated state y(x, t), the solution of the system (1.1), satisfies y(x, T ) = y(x, T ) in Ω. The method for obtaining controllability in the previous subsection is not the only one. So the natural question arises: Can we obtain the result stated in Theorem 1.7 for a small time (that is, for any time)? We have an answer to this question, regardless of the method that is applied, one thing is certain, we will need to wait a not so small time for this to happen. Let us consider the state- control (y, v) solution to (1.1) with initial datum y0 ∈ C2+ 1 2 (Ω); and let us consider the target trajectory y solution to (1.3) with control v ∈ C1/2,1/4(QT ) such that v ≥ η > 0 in QT and initial datum y0 ∈ C2+ 1 2 (Ω). Let us denote the set A = Ay0,y0,y,v := { T > 0 : ∃v ∈ L∞(QT ), v ≥ 0 such that y(T ) = y(T ) in Ω } . 4 M. R. NUÑEZ-CHÁVEZ EJDE-2025/96 By Theorem 1.7, we have A ≠ ∅, and it is clear that T = 0 is a lower bound for set A. So, we can define the minimal controllability time as Tmin = inf A. We have the third main result: positivity of Tmin. Theorem 1.8. Consider the dimension N ≤ 3. If y0 ̸= y0, then Tmin > 0. In fact, we have Tmin ≥ T0 > 0 for some T0 > 0. 1.2. Outline. The rest of this paper is organized as follows. In Section 2, will be proved the local exact controllability for the system (1.1) with control in a Hölder space. Here, the compactness technique of [6] is used to show the Carleman and Observability inequalities. Then, the control regularization method of [10] is used to obtain the control in Holder space. Finally, Kakutani’s Fixed Point Theorem is applied to obtain the desired result. In Section 3, will be proved the global exact controllability with suitable trajectories preserving the control signal for a large time. Here, a stabilization result is proved for the system in H3-norm. The construction of the non- negative control is using the local controllability result with Hölder space control. In Section 4, will be proved the positivity of the minimal controllability time. The proof is done in two cases: in the first case the regularity of parabolic systems and the principle of comparison are used, in the second case the method called ”proof by contradiction” is applied, for this an initial data is constructed in the associated adjoint system arriving at a contradiction. In Section 5 (Appendix), the regularity results (existence and uniqueness) for system (1.1) will be proved. Furthermore, an important and fundamental lemma about spectral theory will be proved. 2. Proof of Theorem 1.5 The proof follows from the well-know Kakutani Fixed Point Theorem. The approach to solving this type of problem is to simplify the expression; in other words, we will perform a change of variables to analyze the null controllability of a system equivalent to system (1.1). To do this, let us denote y(x, t) = z(x, t) + y(x, t), v(x, t) = u(x, t) + v(x, t) and y0(x) = z0(x) + y0(x). Then, from systems (1.1) and (1.3), we obtain the system zt − a (∫ Ω (z + y) dx′ ) ∆z − [ a (∫ Ω (z + y) dx′ ) − a (∫ Ω y dx′ )] ∆y = uϱω in QT , z(x, t) = 0 on ΣT , z(x, 0) = z0(x) in Ω. (2.1) It can be verified that the null controllability of system (2.1) is equivalent to the exact local trajectory controllability of system (1.1). To study this type of nonlinear problem, we must work with the linearized version of system (2.1), as suggested by the ideas in[6]. As a first step, let us fix w ∈ Z := C1,1(QT ) and consider the linearized system zt − α(t, w)∆z + β(t, w) (∫ Ω z(x′, t) dx′ ) ∆y = uϱω in QT , z(x, t) = 0 on ΣT , z(x, 0) = z0(x) in Ω, (2.2) where α(t, w) := a (∫ Ω ( w(x′, t) + y(x′, t) ) dx′ ) , β(t, w) := ∫ 1 0 a′ (∫ Ω ( λw(x′, t) + y(x′, t) ) dx′ ) dλ. Proofs of the next 2 propositions will be provided in the appendix. Proposition 2.1. For each fixed w ∈ Z, the linear system (2.2) has a unique solution in Sobolev spaces. That is, given z0 ∈ L2(Ω) and u ∈ L2(QT ), there exists a unique solution z(x, t) to the system (2.2) with the regularity z ∈ L2(0, T ;H1 0 (Ω)) ∩ L∞(QT ) and zt ∈ L2(0, T ;H−1(Ω)). EJDE-2025/96 CONTROLLABILITY FOR NON-LINEAR AND NON-LOCAL PARABOLIC PDES 5 Proposition 2.2. For each fixed w ∈ Z, the linear system (2.2) has a unique solution in Hölder spaces. That is, given z0 ∈ C2+ 1 2 (Ω) and u ∈ C1/2,1/4(QT ), there exists a unique solution z(x, t) to the system (2.2) with the regularity z ∈ C2+ 1 2 ,1+ 1 4 (QT ) satisfying ∥z∥2+ 1 2 ,1+ 1 4 ≤ C(Ω,M, ∥w∥Z , ∥y∥2+ 1 2 ,1+ 1 4 ) ( ∥z0∥2+ 1 2 + ∥u∥1/2,1/4 ) . The adjoint system of (2.2) is defined as −φt − α(t, w)∆φ+ β(t, w) ∫ Ω ∆y(x′, t)φ(x′, t) dx′ = 0 in QT , φ(x, t) = 0 on ΣT , φ(x, T ) = φT (x), in Ω, (2.3) where φT ∈ L2(Ω). Proposition 2.3. For each fixed w ∈ Z, the adjoint system (2.3) has a unique solution in Sobolev spaces. That is, given φT ∈ L2(Ω), there exists a unique solution φ(x, t) to the system (2.3) with the regularity φ ∈ L2(0, T ;H1 0 (Ω) ∩H2(Ω)) and φt ∈ L2(0, T ;H−1(Ω)). The proof of the above proposition is analogous to proof of Proposition 2.1. Remark 2.4. We can obtain more regularity for the system (2.3). Thus, if φT ∈ H2(Ω)∩H1 0 (Ω), then φ ∈ L2(0, T ;H3(Ω) ∩H1 0 (Ω)) and φt ∈ L2(0, T ;H1 0 (Ω)). 2.1. Null controllability of the linearized system (2.2). It is well known from the results of Fursikov and Imanuvilov [7] (see also [8]) that to demonstrate the null controllability of a linear system, it suffices to establish the observability inequality. This is precisely the reason the adjoint system to the linear system was presented in this work. The following lemma guarantees the existence of the Fursikov function, which plays an important role in developing the observability inequality. Its proof can be found in the book [7]. Lemma 2.5 (Fursikov function). There exists a function σ0 ∈ C2(Ω) that satisfies σ0(x) > 0, ∀x ∈ Ω, σ0(x) = 0, ∀x ∈ ∂Ω, |∇σ0(x)| > 0, ∀x ∈ Ω\ω0. Continuing with the development of control in the linear case, we will introduce the following functions (weight functions): σ(x, t) = e4λ∥σ0∥∞ − eλ(2∥σ0∥∞+σ0(x)) l(t) , ξ(x, t) = eλ(2∥σ0∥∞+σ0(x)) l(t) , with l(t) = { T 2 4 , 0 ≤ t ≤ T 2 t(T − t), T 2 ≤ t ≤ T. Let us denote by λ1, λ2, . . . , (respectively ϕ1, ϕ2, . . . ) the eigenvalues (respectively the unit eigenfunctions in L2) of the Dirichlet Laplacian operator in Ω. We recall some important proper- ties: 0 < λ1 < λ2 ≤ λ3 ≤ · · · ≤ λm ≈ m2/N as m→ +∞, and ϕ1(x) > 0 in Ω. We also have the following result from spectral theory, whose proof can be found in Appendix 5.4. 6 M. R. NUÑEZ-CHÁVEZ EJDE-2025/96 Lemma 2.6. For any w ∈ Z, if the function a(·) satisfies the conditions (1.2), then there exist two positive constants R0 := R0(Ω, ω, T ) and C0 = C0(Ω, ω, T, a0, a1, ∥w∥Z) such that for every f ∈ L2(Ω), we have +∞∑ j=1 e−2R0 √ λj |(f, ϕj)|2 ≤ C0 ∫∫ ω1×(0,T ) e−2sσξ3 ∣∣∣+∞∑ j=1 e−λj ∫ T t α(s,w) ds(f, ϕj)ϕj(x) ∣∣∣2 dx dt. With the previous results, we can demonstrate the controllability result for the linear system. Proposition 2.7. Assuming that a(·) satisfies the conditions (1.2), for any z0 ∈ L2(Ω) that verifies the first-order compatibility condition, there exists a control u(x, t) ∈ C1/2,1/4(QT ) such that the solution z(x, t) associated with the linearized system (2.2) satisfies z(x, T ) = 0 in Ω. Moreover, we have the following estimate for the control u(x, t), ∥u∥1/2,1/4 ≤ C1∥z0∥2, where C1 := C1(Ω, ω, T, a0, a1, ∥w∥Z) > 0. Proof. It is well known in the context of linear systems that the observability inequality for the adjoint system resolves the control problem; therefore, it suffices to prove the following estimates: ∥φ(0)∥22 ≤ C ∫∫ ω1×(0,T ) e−2sσξ3|φ|2 dx dt, ∀φT ∈ L2(Ω) (2.4) and ∫∫ QT e−2sσξ3|φ|2 dx dt ≤ C ∫∫ ω1×(0,T ) e−2sσξ3|φ|2 dx dt, ∀φT ∈ L2(Ω) (2.5) for every φ(x, t) solution of the adjoint system (2.3). Let us prove each of the inequalities presented above separately using similar arguments as in [5, 6]. Proof of (2.4). For φT (x) ∈ L2(Ω), and we denote by φ(x, t) the solution to the adjoint system (2.3). We rewrite the function φ as φ = p+ ζ, where p(x, t) is the unique solution to the system −pt − α(t, w)∆p = 0 in QT p(x, t) = 0 on ΣT p(x, T ) = φT (x), in Ω. (2.6) Thus, the function ζ(x, t) is a solution to the system −ζt − α(t, w)∆ζ + β(t, w) ∫ Ω ∆y(x′, t)ζ(x′, t) dx′ = −β(t, w) ∫ Ω ∆y(x′, t)p(x′, t) dx′ in QT ζ(x, t) = 0 on ΣT ζ(x, T ) = 0, in Ω. (2.7) From the Carleman inequalities for the system (2.6), that is,∫∫ QT e−2sσ(sξ)−1(|∆p|2 + |pt|2) dx dt+ ∫∫ QT e−2sσξ3|p|2 dx dt ≤ C∥φT ∥2h, (2.8) where C := C(Ω, ω, T, a0, a1, ∥w∥Z) > 0, we have that the mapping φT 7→ ∥φT ∥h := (∫∫ ω1×(0,T ) e−2sσξ3|p|2 dx dt )1/2 , is a norm in L2(Ω). EJDE-2025/96 CONTROLLABILITY FOR NON-LINEAR AND NON-LOCAL PARABOLIC PDES 7 By the spectral decomposition of the function p(x, t), p(x, t) = +∞∑ j=1 e−λj ∫ T t α(s,w) ds(φT , ϕj)ϕj(x), from Lemma 2.6, we have +∞∑ j=1 e−2R0 √ λj |(φT , ϕj)|2 ≤ C∥φT ∥2h. (2.9) The observability inequality (2.4) will be a consequence of the two estimates ∥φT ∥2h ≤ C ∫∫ ω1×(0,T ) e−2sσξ3|φ|2 dx dt, ∀φT ∈ L2(Ω) (2.10) and ∥φ(0)∥22 ≤ C∥φT ∥2h, ∀φT ∈ L2(Ω). (2.11) We show estimate (2.10) by contradiction, using the uniqueness and compactness method from [6]. Assuming that the estimate (2.10) does not hold, we have that for every n ∈ N, we can find functions φT n (x) ∈ L2(Ω) such that 1 = ∥φT n∥2h > n ∫∫ ω1×(0,T ) e−2sσξ3|φn|2 dx dt, (2.12) where φn(x, t) is the solution to the adjoint system (2.3) associated with φT n (x). This means that∫∫ ω1×(0,T ) e−2sσξ3|φn|2 dx dt→ 0, as n→ ∞. (2.13) Denoting by pn(x, t) (respectively ζn(x, t)) the solution to the system (2.6) corresponding to φT (x) = φT n (x) (respectively the solution to the system (2.7) with p(x, t) = pn(x, t)), we have 1 = ∥φT n∥2h = ∫∫ ω1×(0,T ) e−2sσξ3|pn|2 dx dt ≤ 2 ∫∫ ω1×(0,T ) e−2sσξ3|φn|2 dx dt+ 2 ∫∫ ω1×(0,T ) e−2sσξ3|ζn|2 dx dt. In the last estimate, we can observe that if∫∫ ω1×(0,T ) e−2sσξ3|ζn|2 dx dt→ 0, as n→ ∞, then we would have a contradiction, since the right-hand side would converge to zero while the left-hand side is equal to 1. For the reason mentioned above, we will demonstrate the following result: ζn → 0 strongly in L2(QT ), (2.14) To do this, we need to estimate the right-hand side of the equation (2.7)1 and use Remark 2.1 for the function ζn. We then state that∥∥β(·, w)∫ Ω ∆y(x′, ·)pn(x′, ·) dx′ ∥∥2 2,0,T ≤ C, ∀n ∈ N. (2.15) Indeed, the spectral decompositions of the functions y, ∆y and pn are: y(x′, t) = +∞∑ j=1 e−λj ∫ t 0 α(s,0) ds(y0, ϕj)ϕj(x ′), ∆y(x′, t) = − +∞∑ j=1 λje −λj ∫ t 0 α(s,0) ds(y0, ϕj)ϕj(x ′), pn(x ′, t) = +∞∑ j=1 e−λj ∫ T t α(s,w) ds(φT n , ϕj)ϕj(x ′). 8 M. R. NUÑEZ-CHÁVEZ EJDE-2025/96 Computing the integral Ky,n(t) := ∫ Ω ∆y(x′, t)pn(x ′, t) dx′ we have Ky,n(t) = − ∫ Ω (+∞∑ j=1 λje −λj ∫ t 0 α(s,0) ds(y0, ϕj)ϕj(x ′) )(+∞∑ l=1 e−λl ∫ T t α(s,w) ds(φT n , ϕl)ϕl(x ′) ) dx′ = − +∞∑ k=1 λke −λk ∫ t 0 α(s,0) dse−λk ∫ T t α(s,w) ds(y0, ϕk)(φ T n , ϕk). It follows that ∥∥β(·, w)∫ Ω ∆y(x′, ·)pn(x′, ·)], dx′ ∥∥2 2,0,T = ∫ T 0 ∥β(t, w)Ky,n(t)∥22dt ≤M2|Ω| ∫ T 0 |Ky,n(t)|2dt. Therefore, using estimate (2.9), we can conclude that∫ T 0 |Ky,n(t)|2dt = ∫ T 0 ∣∣∣ +∞∑ k=1 λke −λk ∫ t 0 α(s,0) dse−λk ∫ T t α(s,w) ds(y0, ϕk)(φ T n , ϕk) ∣∣∣2dt ≤ ∫ T 0 ( +∞∑ k=1 λ2ke −2λk ∫ t 0 a0dte−2λk ∫ T t a0dte2R0 √ λk |(y0, ϕk)|2 )( +∞∑ k=1 e−2R0 √ λk |(φT n , ϕk)|2 ) dt ≤ ∫ T 0 ( +∞∑ k=1 λ2ke −2λka0te−2λka0(T−t)e2R0 √ λk∥y0∥22∥ϕk∥22 )( C∥φT n∥2h ) dt ≤ CT ∥y0∥22 ∥φT n∥2h ( +∞∑ k=1 λ2ke −2λka0T e2R0 √ λk ) . Let us examine the convergence of the series that appears on the right side. To do this, we recall the behavior of the eigenvalues λk, specifically, λm ≈ m2/N as m → +∞. Therefore, for a sufficiently large m0, there exist two positive constants C1 and C2 such that C1m 2/N ≤ λm ≤ C2m 2/N , ∀m ≥ m0. Then +∞∑ k=1 λ2ke −2λka0T e2R0 √ λk = m0−1∑ k=1 λ2ke −2λka0T e2R0 √ λk + +∞∑ k=m0 λ2ke −2λka0T e2R0 √ λk ≤ m0−1∑ k=1 λ2ke −2λka0T e2R0 √ λk + C2 2 +∞∑ k=m0 k 4 N e−2C1a0Tk2/N e2 √ C2R0k 1 N . Denoting m̃0 := max { m0, (2√C2R0 C1a0T )N/5} , it is easy to verify that ∞∑ k=m0 k 4 N e−2C1a0Tk2/N e2 √ C2R0k 1 N ≤ m̃0∑ k=m0 k 4 N e−2C1a0Tk2/N e2 √ C2R0k 1 N + C1,2 +∞∑ k=m̃0+1 e−C1a0Tk2/N . Since e−r ≤ N ! rN for all N ∈ N, we have that +∞∑ k=m̃0+1 e−C1a0Tk2/N ≤ +∞∑ k=m̃0+1 N ! (C1a0Tk2/N )N EJDE-2025/96 CONTROLLABILITY FOR NON-LINEAR AND NON-LOCAL PARABOLIC PDES 9 = N ! (C1a0T )N +∞∑ k=m̃0+1 1 k2 < N ! (C1a0T )N (π2 6 ) . We conclude that∫ T 0 |Ky,n(t)|2dt ≤ C(N,T, a0, m̃0)∥y0∥22 ∥φT n∥2h = C(N,T, a0, m̃0)∥y0∥22. (2.16) Therefore, from (2.16) we obtain (2.15), specifically,∥∥β(·, w)∫ Ω ∆y(x′, ·)pn(x′, ·) dx′ ∥∥2 2,0,T ≤ C(N,T, a0, m̃0, |Ω|)∥y0∥22. (2.17) By Remark 2.1 applied to the system (2.7) with ζ = ζn and p = pn, there exists a unique solution ζn ∈ W̃ where W̃ := { z ∈ L2(0, T ;H1 0 (Ω)) ∩ L∞(0, T ;L2(Ω)) : zt ∈ L2(0, T ;H1 0 (Ω)) } satisfying ∥ζn∥W̃ ≤ C(M,w, y) ∥∥β(·, w)∫ Ω ∆y(x′, ·)pn(x′, ·) dx′ ∥∥ 2,0,T ≤ C(M,w, y) C(N,T, a0, m̃0, |Ω|)∥y0∥2. Thus, we obtain that the sequence of functions ζn is bounded in L2(0, T ;H1 0 (Ω)) and the sequence of functions ζt,n is bounded in L2(0, T ;L2(Ω)). We can then assume that ζn(x, t) converges strongly in L2(QT ) to a function ζ(x, t). To conclude the proof, from (2.14) it would suffice to show that the function ζ is equal to zero. More specifically, using the equation (2.7)1, we need to demonstrate that β(t, w) ∫ Ω ∆y(x′, t)φ(x′, t)dx′ = 0. (2.18) If condition (2.18) holds, we would have a homogeneous linear system for the function ζ with initial data and boundary conditions equal to zero, which would imply that ζ is equal to zero by the uniqueness of the solution. Indeed, from the estimate (2.8), we have∫∫ QT−δ |φn|2 dx dt ≤ 2 ∫∫ QT−δ |pn|2 dx dt+ 2 ∫∫ QT−δ |ζn|2 dx dt ≤ Cδ ∫∫ QT e−2sσξ3|pn|2 dx dt+ C∥φT n∥2h ≤ CδC1∥φT n∥2h + C∥φT n∥2h := Cδ,1. Repeating the previous procedure, we obtain∫∫ QT−δ |∆φn|2 dx dt ≤ Cδ,2, ∫∫ QT−δ |φt,n|2 dx dt ≤ Cδ,3. Taking the limit in the linear system (2.3) in QT−δ, we obtain φn ⇀ φ weakly in L2(QT−δ), ∀δ > 0. (2.19) From (2.13), let us consider the subset ω1 × (0, T − δ), thus ∥φn∥2L2(ω1×(0,T−δ)) := ∫∫ ω1×(0,T ) |φn|2 dx dt→ 0, as n→ ∞. (2.20) Thus, from (2.19) and (2.20) we have φ(x, t) = 0 in ω1 × (0, T − δ). Since φ satisfies the equation −φt − α(t, w)∆φ = −β(t, w) ∫ Ω ∆y(x′, t)φ(x′, t) dx′ in QT , 10 M. R. NUÑEZ-CHÁVEZ EJDE-2025/96 in particular, it holds in ω1 × (0, T − δ). Since β(t, w) ∫ Ω ∆y(x′, t)φ(x′, t) dx′ is independent of the spatial variable, it means that β(t, w) ∫ Ω ∆y(x′, t)φ(x′, t) dx′ = 0 in t ∈ (0, T − δ). (2.21) Thus, φ satisfies the equation −φt − α(t, w)∆φ = 0, which implies that φ = 0 in QT−δ. Since the function ζn converges in L2(QT ), it is therefore bounded in L2(QT ). Thus,∥∥β(·, w)∫ Ω ∆y(x′, ·)ζn(x′, ·) dx′ ∥∥2 2,0,T ≤ ∫ T 0 |β(t, w)|2 (∫ Ω |∆y(x′, t)||ζn(x′, t)| dx′ )2 ∥1∥22dt ≤M2|Ω| ∥y∥2L∞(0,T ;H2(Ω))∥ζn∥ 2 2,0,T ≤ C(M, |Ω|)∥y∥2L∞(0,T ;H2(Ω)). From (2.15) and the previous result, we obtain∥∥β(·, w)∫ Ω ∆y(x′, ·)φn(x ′, ·) dx′ ∥∥ 2,0,T ≤ ∥∥β(·, w)∫ Ω ∆y(x′, ·)pn(x′, ·) dx′ ∥∥ 2,0,T + ∥∥β(·, w)∫ Ω ∆y(x′, ·)ζn(x′, ·) dx′ ∥∥ 2,0,T ≤ C(N,T, a0, m̃0, |Ω|)∥y0∥2 + C(M, |Ω|)∥y∥L∞(0,T ;H2(Ω)). It follows that β(·, w) ∫ Ω ∆y(x′, ·)φn(x ′, ·) dx′ ⇀ ψ(·) weakly in L2(QT ). From (2.21), we can deduce that β(·, w) ∫ Ω ∆y(x′, ·)φn(x ′, ·) dx′ ⇀ 0 weakly in L2(QT ), and consequently, the estimate (2.18) holds. □ Estimate (2.11). First, by applying the energy estimates to the system (2.6), we have that ∥p(0)∥22 ≤ ∥p(t)∥22, ∀t ∈ (0, T ). Using the Carleman inequality (2.8), we obtain ∥p(0)∥22 ≤ 2 T ∫ 3T 4 T 4 ∥p(t)∥22dt ≤ C ∫∫ Ω×(T 4 , 3T4 ) e−2sσξ3|p|2 dx dt ≤ C∥φT ∥2h. (2.22) Secondly, applying the energy estimates to the system (2.7), we have ∥ζ(0)∥22 + ∥ζ∥22,0,T ≤ C ∥∥β(t, w)∫ Ω ∆y(x′, t)p(x′, t) dx′ ∥∥2 2,0,T . (2.23) By performing a procedure analogous to the proof of the estimate (2.15) on the function ζ(x, t), we conclude that ∥ζ(0)∥22 + ∥ζ∥2L2(QT ) ≤ C∥φT ∥2h. (2.24) Finally, since we have the relation φ = p+ ζ it suffices to combine the estimates (2.22) and (2.24) to obtain the estimate (2.11). This concludes the proof of the observability inequality (2.4). □ Proof of (2.5). From (2.8), (2.23) and (2.24), we have∫∫ QT e−2sσξ3|φ|2 dx dt ≤ ∫∫ QT e−2sσξ3|p|2 dx dt+ ∫∫ QT e−2sσξ3|ζ|2 dx dt ≤ C∥φT ∥2h + C∥ζ∥22,0,T ≤ C∥φT ∥2h + C ∥∥β(·, w)∫ Ω ∆y(x′, ·)p(x′, ·). dx′ ∥∥2 2,0,T ≤ C∥φT ∥2h. EJDE-2025/96 CONTROLLABILITY FOR NON-LINEAR AND NON-LOCAL PARABOLIC PDES 11 This concludes the proof of the Carleman inequality (2.5). □ Let us prove the regularity of the control v(x, t) using similar arguments as in [10]. For each ϵ > 0, we consider the functional Fϵ : L 2(Ω) → R defined as Fϵ(φ T ) := 1 2 ∫∫ QT e−2sσξ3|φ|2ϱω dx dt+ ϵ∥φT ∥2 + ∫ Ω z0(x)φ(x, 0) dx, where φT ∈ L2(Ω) and z, φ are the solutions of systems (2.2) and (2.3). It is clear that Fϵ is strictly convex. Let us show that Fϵ is lower semi-continuous. Indeed, let (φT n )n∈N be a sequence in L2(Ω) such that φT n ⇀ φT with φT ∈ L2(Ω). Denote φn as the solution of the system (2.3) with initial data φT n ∈ L2(Ω). By Proposition 2.1, we have φn ∈ W̃ with ∥φn∥W̃ ≤ C∥φT n∥2. Thus φn ⇀ φ in W̃ ∩ C([0, T ];L2(Ω)). Then φn(0)⇀ φ(0) in L2(Ω) ⇐⇒ ∫ Ω φn(x, 0)ϕ(x) dx→ ∫ Ω φ(x, 0)ϕ(x) dx, ∀ϕ ∈ L2(Ω). Therefore Jϵ(φ T ) ≤ lim inf Jϵ(φ T n ). Let us show that Jϵ is coercive. Indeed, we have∣∣ ∫ Ω z0(x)φ(x, 0)dx ∣∣ ≤ ∥z0∥2∥φ(0)∥2 ≤ ∥z0∥2 ( C ∫∫ ω×(0,T ) e−2sσξ3|φ|2 dx dt )1/2 ≤ 1 4 ∫∫ QT e−2sσξ3|φ|2ϱω dx dt+ C∥z0∥22. Then Fϵ(φ T ) ≥ 1 2 ∫∫ QT e−2sσξ3|φ|2ϱω dx dt+ ϵ∥φT ∥2 − ∣∣ ∫ Ω z0(x)φ(x, 0) dx ∣∣ ≥ 1 4 ∫∫ QT e−2sσξ3|φ|2ϱω dx dt+ ϵ∥φT ∥2 − C∥z0∥22 ≥ ϵ∥φT ∥2 − C∥z0∥22. Therefore lim ∥φT ∥2→+∞ Fϵ(φ T ) = +∞. Thus, since Fϵ is strictly convex, lower semicontinuous and coercive, there exists a unique φT ϵ ∈ L2(Ω) such that Fϵ(φ T ϵ ) = min φT∈L2(Ω) Fϵ(φ T ). Now, let us show that Fϵ is differentiable. Indeed, given φT , ψT ∈ L2(Ω), we have ⟨F ′ ϵ(ψ T ), φT ⟩ = ( e−sσξ 3 2ψ, e−sσξ 3 2φ ) + ϵ ( ψT ∥ψT ∥2 , φT ) + (z0, φ(0)). Then, since φT ϵ is the minimum of Fϵ, we have that either φT ϵ = 0 or ⟨F ′ ϵ(φ T ϵ ), φ T ⟩ = 0, ∀φT ∈ L2(Ω). Assuming that φT ϵ ̸= 0, we obtain∫∫ QT e−2sσξ3φϵφϱω dx dt+ ϵ ( φT ϵ ∥φT ϵ ∥2 , φT ) + (z0, φ(0)) = 0, ∀φT ∈ L2(Ω), (2.25) where φϵ(x, t) is the solution of the adjoint system (2.3) corresponding to φT (x) = φT ϵ (x). 12 M. R. NUÑEZ-CHÁVEZ EJDE-2025/96 Defining uϵ(x, t) = e−2sσ(x,t)ξ3(x, t)φϵ(x, t)ϱω(x) in the system (2.2), and denoting by zϵ(x, t) the state associated with the control uϵ, from the estimate (2.25) we have∫ Ω ( zϵ(x, T ) + ϵ ∥φT ϵ ∥2 φT ϵ (x) ) φT (x) dx = 0, ∀φT ∈ L2(Ω), which implies that ∥zϵ(·, T )∥2 = ∥∥− ϵ ∥φT ϵ ∥2 φT ϵ ∥∥ 2 = ϵ. (2.26) From estimates (2.25) and (2.4) (observability inequality), we have∫∫ ω×(0,T ) e−2sσξ3|φϵ|2 dx dt+ ϵ∥φT ϵ ∥2 ≤ ∥z0∥2∥φϵ(0)∥2 ≤ ∥z0∥2 ( C ∫∫ ω×(0,T ) e−2sσξ3|φϵ|2 dx dt )1/2 . Then (∫∫ ω×(0,T ) e−2sσξ3|φϵ|2 dx dt )1/2 ≤ C∥z0∥2. Thus ∥esσξ− 3 2uϵ∥2,0,T = (∫∫ QT e2sσξ−3|uϵ|2 dx dt )1/2 = (∫∫ ω×(0,T ) e−2sσξ3|φϵ|2 dx dt )1/2 ≤ C∥z0∥2, (2.27) where C := C(Ω, T, a0, a1, ∥w∥Z) is a positive constant. For each ϵ > 0, the function φϵ is a solution of system (2.3) with initial data φT ϵ ∈ L2(Ω). Therefore, by Proposition 2.1, we have φϵ ∈ W̃ with ∥φϵ∥W̃ ≤ C∥φT ϵ ∥2, and the function zϵ is the solution of system (2.2) with control uϵ ∈ L2(QT ). Thus, by Proposition 2.1, we have zϵ ∈ W̃ with ∥zϵ∥W̃ ≤ C ( ∥uϵ∥2,0,T + ∥z0∥2 ) ≤ C∥z0∥2. Since ϵ→ 0, we conclude that φϵ ⇀ φ in W̃ , zϵ ⇀ z in W̃ . Thus, z is a solution of the linear system (2.2) and φ is a solution of system (2.3) with initial data φ(T ). Therefore, defining u(x, t) := e−2sσ(x,t)ξ3(x, t)φ(x, t)ϱω(x), we obtain esσξ− 3 2uϵ ⇀ esσξ− 3 2u in L2(QT ). Taking the limit as ϵ→ 0 in system (2.2) with zϵ, we obtain zϵ(·, T )⇀ z(·, T ) in L2(Ω), Ĉ∥u∥2,0,T ≤ ∥esσξ− 3 2u∥2,0,T ≤ lim inf ∥esσξ− 3 2uϵ∥2,0,T ≤ C∥z0∥2. (2.28) Thus, from (2.26) and (2.28), we conclude that z(x, T ) = 0 in Ω. The last step will be to prove that u(x, t) ∈ C1/2,1/4(QT ). To this end, we will rely on the theory of second-order linear parabolic PDEs. The following lemma serves as a starting point. Lemma 2.8. For each r > 1, g ∈ Lr(QT ), b ij ∈ C1,1(QT ), c j ∈ C(QT ) and d ∈ C(QT ), where bij = bji (i, j = 1, . . . , N), and for some constant µ > 0: N∑ i,j=1 bi,j(x, t)ϕiϕj ≥ µ|ϕ|2, ∀(x, t, ϕ) = (x, t, ϕ1, . . . , ϕN ) ∈ QT × RN . EJDE-2025/96 CONTROLLABILITY FOR NON-LINEAR AND NON-LOCAL PARABOLIC PDES 13 Then, the linear parabolic PDE ut − N∑ i,j=1 bijuxixj + N∑ j=1 cjuxj + du = g in QT , u(x, t) = 0 on ΣT , u(x, 0) = 0 in Ω, admits a unique strong solution u = u(x, t) with u ∈W 2,1 r (QT ) := { u ∈ Lr(Ω) : Dα xD s tu ∈ Lr(QT ), |α| ≤ 2 and s ≤ 1 } . Moreover, there exists a positive constant C := C(Ω, T, µ, r) such that ∥u∥W 2,1 r (QT ) ≤ C exp [ C(1 + N∑ i,j=1 ∥bi,j∥81,1 + N∑ j=1 ∥cj∥8∞,0,T + ∥d∥4∞,0,T ) ] ∥g∥r,0,T . For a proof of the above lemma see [17, Chapter 9, Section 2, Theorem 9.2.5] Also we have the following result about the immersion of the space W 2,1 r (QT ) with r > 1. Lemma 2.9. The following continuous immersion holds: (1) If N + 2 > 2r, then W 2,1 r (QT ) ↪→ Lr∗(QT ), where r ∗ = (N+2)r N+2−2r . (2) If N + 2 = 2r, then W 2,1 r (QT ) ↪→ Ls(QT ) for any s > 1. (3) If θ = 2− N+2 r is not an integer, then W 2,1 r (QT ) ↪→ Cθ, θ2 (QT ). For a proof of the above lemma, see [9, Chapter II, Lemma 3.3, page 80]. Let δ > 0 and consider (δk)k∈N as an increasing sequence, such that 0 < δk < δ < s 2 , ∀k ∈ N. Let us introduce the notation ξ0(t) = 1 l(t) , σ∗(t) = max x∈Ω σ(x, t) with σ∗(t) < 4 3 σ(x, t) for λ≫ 1, uk(x, t) = e−(s+δk)σ ∗(t)ξ30(t)φ(x, t). Thus, for every k ≥ 1, the function uk(x, t) satisfies the following system −uk,t − α(t, w)∆uk = gk in QT , uk(x, t) = 0 on ΣT , uk(x, T ) = 0 in Ω. (2.29) From equation (2.29)1, let us denote gk = −β(t, w)e−(s+δk)σ ∗ ξ30 (∫ Ω ∆y(x′, t)φ(x′, t) dx′ ) − ( e−(s+δk)σ ∗ ξ30 ) t φ. We can verify that |σ∗ t | ≤ Cξ20 , |ξ0,t| ≤ Cξ20 ,( e−(s+δk)σ ∗ ξ30 ) t = −(s+ δk)σ ∗ t e −(s+δk)σ ∗ ξ30 + 3e−(s+δk)σ ∗ ξ20ξ0,t. (2.30) When k = 1, the function u1(x, t) is the solution to the system −u1,t − α(t, w)∆u1 + β(t, w) ∫ Ω ∆y(x′, t)u1(x ′, t)dx′ = g̃1 in QT , u1(x, t) = 0 on ΣT , u1(x, T ) = 0 in Ω, (2.31) where g̃1 = −(e−(s+δ1)σ ∗ ξ30)tφ. 14 M. R. NUÑEZ-CHÁVEZ EJDE-2025/96 From Carleman’s inequality (2.5) and (2.30), it follows that ∥g̃1∥22,0,T ≤ C ∫∫ ω1×(0,T ) e−2sσξ3|φ|2 dx dt, and, using the estimate (2.27), we have that ∥g̃1∥22,0,T ≤ C∥z0∥22. (2.32) Thus, u1 ∈ L2(0, T ;H1 0 (Ω)), and from estimate (2.32) we can deduce that ∥u1∥L2(0,T ;H1 0 (Ω)) ≤ C∥z0∥2. Now, taking g1 = −β(t, w) ∫ Ω ∆y(x′, t)u1(x ′, t)dx′ + g̃1, we conclude that ∥g1∥2,0,T ≤ C∥z0∥2. Using Lemma 2.8 and estimates (2.27) and (2.32), we can see that the function u1 ∈ W 2,1 2 (QT ). Furthermore, ∥u1∥2W 2,1 2 (QT ) ≤ C∥z0∥22, where C := C(Ω, T, a0, a1, ∥w∥Z) is a positive constant. By Lemma 2.9, for r1 = { 2(N+2) N−2 , N > 2, s > 1, N ≤ 2, we have the following continuous embedding W 2,1 2 (QT ) ↪→ Lr1(QT ) with ∥u1∥r1,0,T ≤ C∥z0∥2. The next step is to find estimates for the functions g2 and u2. Note that g2 = −β(·, w)e−(s+δ2)σ ∗ ξ30 (∫ Ω ∆y(x′, t)u1(x ′, t) dx′ ) − ( e−(s+δ2)σ ∗ ξ30 ) t φ. Similar to the previous step, we obtain tha ∥g2∥r1,0,T ≤ C∥z0∥2. And once again, by Lemma 2.8, we have that the function u2 ∈W 2,1 r1 (QT ) and that ∥u2∥W 2,1 r1 (QT ) ≤ C∥z0∥2. By Lemma 2.9, for r2 = { r1(N+2) N+2−2r1 , N + 2− 2r1 > 0, s > 1, N + 2− 2r1 ≤ 0, we have the following continuous embedding W 2,1 r1 (QT ) ↪→ Lr2(QT ) , from which we can conclude that: ∥u2∥r2,0,T ≤ C∥z0∥2. By repeating the procedure, we can obtain the sequence (rk)k∈N of positive integer numbers such that rk+1 = { rk(N+2) N+2−2rk , N + 2− 2rk > 0, s > 1, N + 2− 2rk ≤ 0. = { 1 rk − 1 rk+1 = 2 N+2 , N + 2− 2rk > 0, s > 1, N + 2− 2rk ≤ 0. By Lemma 2.9, we have the continuous embedding W 2,1 rk (QT ) ↪→ Lrk+1(QT ). EJDE-2025/96 CONTROLLABILITY FOR NON-LINEAR AND NON-LOCAL PARABOLIC PDES 15 Since N is a fixed positive integer, there exists a positive integer k∗ such that N +2−2rk∗ < 0. Thus, rk∗+1 = s for any s > 1. Considering rk∗+1 = 2(N+2) 3 , by Lemma 2.9, we can obtain that W 2,1 rk∗+1 (QT ) ↪→ C1/2,1/4(QT ), with ∥uk∗+1∥1/2,1/4 ≤ C∥z0∥2. (2.33) Finally, u = e−2sσξ3φ = e−2sσξ3 ( e(s+δk∗+1)σ ∗ ξ−3 0 uk∗+1 ) = ( e−2sσξ3e(s+δk∗+1)σ ∗ ξ−3 0 ) uk∗+1. Since δk∗+1 < s 2 , we obtain that e−2sσξ3e(s+δk∗+1)σ ∗ ξ−3 0 < e− 3 2 sσ ∗ e(s+δk∗+1)σ ∗ ( ξ ξ0 )3 ≤ e−( s 2−δk∗+1)σ ∗ ( ξ ξ0 )3 ≤ C. Then ∥u∥1/2,1/4 = ∥e−2sσξ3e(s+δk∗+1)σ ∗ ξ−3 0 uk∗+1∥1/2,1/4 ≤ C∥uk∗+1∥1/2,1/4. (2.34) From estimates (2.33) and (2.34), we have ∥u∥1/2,1/4 ≤ C∥z0∥2, (2.35) where C := C(Ω, T, a0, a1,M, ∥w∥Z) is a positive constant. Therefore, there exists a control u(x, t) ∈ C1/2,1/4(QT ) that satisfies (2.35), such that the associated state z(x, t), which is solution of the system (2.2), satisfies z(x, T ) = 0 in Ω. (2.36) Thus, Proposition 2.7 is proven. □ 2.2. Null local controllability of the nonlinear system (2.1). In this section, we will conclude the proof of Theorem 1.5. Recalling the definition of the space Z = C1,1(QT ), we will define the spaces W := { (u, z) : u ∈ C1/2,1/4(QT ), z ∈ C2+ 1 2 ,1+ 1 4 (QT ) } , K := { ẑ ∈ C2+ 1 2 ,1+ 1 4 (QT ) : ∥ẑ∥2+ 1 2 ,1+ 1 4 ≤ R } ⊂ Z ⊂ L2(QT ). We will introduce the mapping Λ : K → 2K , as Λ(w) := { z = zu,w,z0 ∈ K : (z, u) is the state-control solution of system (2.2) with (u, z) ∈W satisfying estimates (2.35) and (2.36) } . The multi-valued mapping Λ satisfies the hypotheses of Kakutani’s Fixed Point Theorem. Let us examine each of these conditions. Lemma 2.10. The mapping Λ is well-defined, and for every w ∈ K, Λ(w) is non-empty. Proof. For each w ∈ K ⊂ Z, by Proposition 2.2, there exists a solution z(x, t) ∈ C2+ 1 2 ,1+ 1 4 (QT ). According to Proposition 2.7, there exists a control u ∈ C1/2,1/4(QT ) such that z(x, T ) = 0 in Ω. Thus, the pair (u, z) ∈ W is a solution to the system (2.2) satisfying the estimates (2.35) and (2.36). Therefore, z = Λ(w) ∈ C2+ 1 2 ,1+ 1 4 (QT ). If ∥z0∥2+ 1 2 is sufficiently small, by Proposition 2.2, we have that ∥z∥2+ 1 2 ,1+ 1 4 ≤ C(∥w∥Z) ( ∥z0∥2+ 1 2 + ∥u∥1/2,1/4 ) ≤ C(R)∥z0∥2+ 1 2 ≤ R. Thus, z ∈ Λ(w) ⊂ K. Therefore, Λ is well-defined and Λ(w) is non-empty. □ Lemma 2.11. K is convex and compact. Proof. It is clear that in a normed space, closed ball are convex. Therefore, K is convex. Now, we show that K is compact. Let (ẑn)n∈N ⊂ K, meaning ∥ẑn∥2+ 1 2 ,1+ 1 4 ≤ R. Thus, the sequence ẑn is bounded in C2+ 1 2 ,1+ 1 4 (QT ). By the compact embedding of C2+ 1 2 ,1+ 1 4 (QT ) ↪→ L2(QT ), there exists a subsequence ẑn that converges in L2(QT ). Therefore, K is compact in L2(QT ). □ 16 M. R. NUÑEZ-CHÁVEZ EJDE-2025/96 Lemma 2.12. The mapping Λ(w) is convex. Proof. Let z1 = zu1,w,z0 , z2 = zu2,w,z0 ∈ Λ(w), and let λ ∈ [0, 1]. Thus, z1 are z2 solutions of the linear system (2.2) with controls u1 and u2, respectively, satisfying z1(x, T ) = z2(x, T ) = 0 in Ω, ∥z1∥2+ 1 2 ,1+ 1 4 ≤ R, ∥z2∥2+ 1 2 ,1+ 1 4 ≤ R, ∥u1∥1/2,1/4 ≤ C(∥w∥Z)∥z0∥2 ≤ C(R)∥z0∥2, ∥u2∥1/2,1/4 ≤ C(∥w∥Z)∥z0∥2 ≤ C(R)∥z0∥2. Since the system (2.2) is linear, we have that λz1+(1−λ)z2 is a solution of the system (2.2) with control λu1 + (1− λ)u2 ∈ C1/2,1/4(QT ). Thus, we can conclude that (λz1 + (1− λ)z2)(x, T ) = λz1(x, T ) + (1− λ)z2(x, T ) = 0 in Ω, ∥λz1 + (1− λ)z2∥2+ 1 2 ,1+ 1 4 ≤ λ∥z1∥2+ 1 2 ,1+ 1 4 + (1− λ)∥z2∥2+ 1 2 ,1+ 1 4 ≤ λR+ (1− λ)R = R. ∥λu1 + (1− λ)u2∥1/2,1/4 ≤ λ∥u1∥1/2,1/4 + (1− λ)∥u2∥1/2,1/4 ≤ λC(R)∥z0∥2 + (1− λ)C(R)∥z0∥2 = C(R)∥z0∥2. Thus, λz1 + (1− λ)z2 ∈ Λ(w). Therefore, Λ(w) is convex. □ Lemma 2.13. The mapping Λ(w) is compact. Proof. By Lemma 2.10, we have that Λ(w) ⊂ K. By Lemma 2.11, K is compact; it remains to verify that Λ(w) is closed in order to conclude the proof. Let z ∈ Λ(w), which means there exists a sequence (zn)n∈N = (zun,w,y0 )n∈N ⊂ Λ(w) such that zn → z in K. Thus, zn is a solution of the linear solution (2.2) with control un ∈ C1/2,1/4(QT ) such that zn(x, T ) = 0 in Ω and ∥un∥1/2,1/4 ≤ C(∥w∥Z)∥z0∥2 ≤ C(R)∥z0∥2. Thus, we have ∥un∥2,0,T ≤ C(R)∥z0∥2. Without loss of generality, we can consider the sequence un itself such that un ⇀ u in L2(QT ) in the weak sense, with ∥u∥2,0,T ≤ lim inf ∥un∥2,0,T ≤ C(R)∥z0∥2. Taking the limit in the linear system (2.2) with the state-control pair (zn, un), we see that the function z is a solution to the linear system (2.2) with control u ∈ L2(QT ). Now, by Proposition 2.7, we can regularize the control to ensure that u ∈ C1/2,1/4(QT ) such that z(x, T ) = 0 in Ω and ∥u∥1/2,1/4 ≤ C(∥w∥Z)∥z0∥2 ≤ C(R)∥z0∥2. This shows that z = zu,w,z0 ∈ Λ(w), meaning that Λ(w) is closed. Therefore, Λ(w) is compact. □ Lemma 2.14. The mapping Λ has a closed graph in K. Proof. We recall the definition of the graph of a mapping: Graf(Λ) := { (w, z) ∈ K ×K : z ∈ Λ(w) } . Let (w, z) ∈ Graf(Λ), meaning there exists a sequence (wn, zn) ∈ Graf(Λ) such that (wn, zn) → (w, z) in K ×K. Thus, we have that zn = zun,wn,z0 ∈ Λ(wn) satisfies the system zn,t − α(t, wn)∆zn + β(t, wn) (∫ Ω zn(x ′, t) dx′ ) ∆y = unϱω in QT , zn(x, t) = 0 on ΣT , zn(x, 0) = z0(x) in Ω, (2.37) EJDE-2025/96 CONTROLLABILITY FOR NON-LINEAR AND NON-LOCAL PARABOLIC PDES 17 with (un, zn) ∈ W such that ∥un∥1/2,1/4 ≤ C(R)∥z0∥2 and zn(x, T ) = 0 in Ω. We can take the limit in the system (2.37) to obtain the new system zt − α(t, w)∆z + β(t, w) (∫ Ω z(x′, t) dx′ ) ∆y = uϱω in QT , z(x, t) = 0 on ΣT , z(x, 0) = z0(x) in Ω, where the function z satisfies z(x, T ) = 0 in Ω and the control satisfies ∥u∥1/2,1/4 ≤ lim inf ∥un∥1/2,1/4 ≤ C(R)∥z0∥2. Thus, we conclude that z ∈ Λ(w). Therefore, Graf(Λ) is closed in K. □ By Lemmas 2.10-2.14, the hypotheses of Kakutani’s Point Fixed Theorem are satisfied. There- fore, the mapping Λ has at least one fixed point, which we shall denote by z(x, t). Clearly, z(x, t) is the associated state with control u(x, t) such that estimates (2.35) and (2.36) hold. This completes the proof of Theorem 1.5. 3. Proof of Theorem 1.7 Let the target trajectory y(x, t) be the solution to system (1.3) with control v and initial data y0 sufficiently regular and sufficiently small. We will complete the proof in three steps as shown in Figure 1. t y(t) 0 T − τ T y0 y0 (y : v) (y : v) (y : ṽ) y(T − τ) y(T − τ) y(T ) = y(T ) stabilization local control Figure 1. Trajectory of state-control in blue. Target trajectory in red. Step 1: Stabilization of (y − y)(x, t) system. Let τ > 0 be a fixed constant, and consider T > τ sufficiently large. Over the time interval [0, T − τ ], we can control the function y(x, t) using the control v = v. We have the stabilization property in C2+ 1 2 over the time interval [0, T − τ ], that is, ∥y(t)− y(t)∥2+ 1 2 ≤ Ce−λt∥y0 − y0∥H4(Ω), ∀t ∈ [0, T − τ ], (3.1) where the constants C, λ are positive. λ does not depend on T . For N ≤ 3, we have the following continuous embedding Hm(Ω) ↪→ C(m−2)+ 1 2 (Ω), with 2m > N. Therefore, for m = 4 it works, so in fact we will show the following ∥y(t)− y(t)∥H4(Ω) ≤ Ce−λt∥y0 − y0∥H4(Ω), ∀t ∈ [0, T − τ ], (3.2) Subtracting the system (1.3)1 from the system (1.1)1 we have (y − y)t − a (∫ Ω y dx′ ) ∆(y − y)− [ a (∫ Ω y dx′ ) − a (∫ Ω y dx′ )] ∆y = 0 (3.3) 18 M. R. NUÑEZ-CHÁVEZ EJDE-2025/96 Estimate I. Multiplying (3.3) by (y − y) and integrating in Ω, we obtain 1 2 d dt ( ∥y − y∥22 ) + a (∫ Ω y dx′ )∫ Ω |∇(y − y)|2 dx = − [ a (∫ Ω y dx′ ) − a (∫ Ω y dx′ )] ∫ Ω ∇y · ∇(y − y) dx. Using the conditions on the function a(·) in (1.2) and the Poincare inequality ∥ · ∥2 ≤ C(Ω)∥∇ ·∥2, we deduce that ∣∣∣[a(∫ Ω y dx′ ) − a (∫ Ω y dx′ )] ∫ Ω ∇y · ∇(y − y) dx ∣∣∣ ≤M (∫ Ω |y − y| dx′ )(∫ Ω |∇y||∇(y − y)| dx ) ≤M |Ω|1/2∥(y − y)(t)∥2∥∇y(t)∥2∥∇(y − y)(t)∥2 ≤M |Ω|1/2C(Ω)∥∇(y − y)(t)∥2∥∇y(t)∥2∥∇(y − y)(t)∥2 ≤M |Ω|1/2CΩ∥y∥L∞(0,+∞;H1(Ω))∥∇(y − y)(t)∥22, where constants M and CΩ were defined in (1.2) and (1.4) respectively. Combining the previous results and the estimate from (1.4), we obtain 1 2 d dt ( ∥(y − y)(t)∥22 ) + a0∥∇(y − y)(t)∥22 ≤ a0 2 ∥∇(y − y)(t)∥22. Then d dt ( ∥(y − y)(t)∥22 ) + a0∥∇(y − y)(t)∥22 ≤ 0. Thus d dt ( ∥(y − y)(t)∥22 ) + a0 [C(Ω)]2 ∥(y − y)(t)∥22 ≤ 0. Integrating over the time variable from 0 to t, with t ∈ [0, T − τ ], we have shown that ∥y(t)− y(t)∥2 ≤ e−λt∥y0 − y0∥2, ∀t ∈ [0, T − τ ], (3.4) where λ = a0 2[C(Ω)]2 > 0. Estimate II. Multiplying by −∆(y − y) in (3.3) and integrating in Ω, we obtain∫ Ω (y − y)t(−∆(y − y)) dx+ a (∫ Ω y dx′ )∫ Ω ∆(y − y)∆(y − y) dx = − [ a (∫ Ω y dx′ ) − a (∫ Ω y dx′ )] ∫ Ω ∆y ∆(y − y) dx. Using the conditions on the function a(·) in (1.2) and the inequality ∥ · ∥2 ≤ [C(Ω)]2∥∆ · ∥2, we can deduce that ∣∣∣[a(∫ Ω y dx′ ) − a (∫ Ω y dx′ )] ∫ Ω ∆y∆(y − y) dx ∣∣∣ ≤M (∫ Ω |y − y| dx′ )(∫ Ω |∆y||∆(y − y)| dx ) ≤M |Ω|1/2∥(y − y)(t)∥2∥∆y(t)∥2∥∆(y − y)(t)∥2 ≤M |Ω|1/2[C(Ω)]2∥∆(y − y)(t)∥2∥∆y(t)∥2∥∆(y − y)(t)∥2 ≤M |Ω|1/2CΩ∥y∥L∞(0,+∞;H2(Ω))∥∆(y − y)(t)∥22. Combining the previous results and the estimate from (1.4), we obtain: 1 2 d dt ( ∥∇(y − y)∥22 ) + a0∥∆(y − y)∥22 ≤ a0 2 ∥∆(y − y)∥22. Then d dt ( ∥∇(y − y)(t)∥22 ) + a0∥∆(y − y)(t)∥22 ≤ 0. EJDE-2025/96 CONTROLLABILITY FOR NON-LINEAR AND NON-LOCAL PARABOLIC PDES 19 As ∥∇ · ∥2 ≤ C(Ω)∥∆ · ∥2, thus d dt ( ∥∇(y − y)(t)∥22 ) + a0 [C(Ω)]2 ∥∇(y − y)(t)∥22 ≤ 0. Integrating over the time variable from 0 to t, with t ∈ [0, T − τ ], we have shown that ∥∇(y − y)(t)∥2 ≤ e−λt∥∇(y0 − y0)∥2, ∀t ∈ [0, T − τ ]. (3.5) Applying the operator ∆ in (3.3) we obtain ∆(y − y)t − a (∫ Ω y dx′ ) ∆2(y − y)− [ a (∫ Ω y dx′ ) − a (∫ Ω y dx′ )] ∆2y = 0. (3.6) Note that from the equation (1.1)1, for all t ∈ [0, T ] we have a (∫ Ω y(x′, t) dx′ ) ∆y(·, t) = yt(·, t)− v(·, t)ϱω(·) = 0 in ∂Ω. Then ∆y(·, t) = 0 in ∂Ω. Analogously ∆y(·, t) = 0 in ∂Ω. Estimate III. Multiplying (3.6) by ∆(y − y) and integrating in Ω, we obtain∫ Ω ∆(y − y)t∆(y − y) dx+ a (∫ Ω y dx′ )∫ Ω (∇∆(y − y))(∇∆(y − y)) dx = − [ a (∫ Ω y dx′ ) − a (∫ Ω y dx′ )] ∫ Ω (∇∆y)(∇∆(y − y)) dx. Similar to Estimates I and II, using the conditions on the function a(·) in (1.2) and the inequality ∥ · ∥2 ≤ [C(Ω)]3∥∇∆ · ∥2, we deduce that d dt ( ∥∆(y − y)(t)∥22 ) + a0∥∇∆(y − y)(t)∥22 ≤ 0. As ∥∆ · ∥2 ≤ C(Ω)∥∇∆ · ∥2 , thus d dt ( ∥∆(y − y)(t)∥22 ) + a0 [C(Ω)]2 ∥∆(y − y)(t)∥22 ≤ 0. Integrating over the time variable from 0 to t, with t ∈ [0, T − τ ], we have shown that ∥∆(y − y)(t)∥2 ≤ e−λt∥∆(y0 − y0)∥2, ∀t ∈ [0, T − τ ]. (3.7) Estimate IV. Multiplying (3.6) by −∆2(y − y) and integrating in Ω, we obtain∫ Ω (∇∆(y − y)t)(∇∆(y − y)) dx+ a (∫ Ω y dx′ )∫ Ω (∆2(y − y))(∆2(y − y)) dx = − [ a (∫ Ω y dx′ ) − a (∫ Ω y dx′ )] ∫ Ω (∆2y)(∆2(y − y)) dx. Similar to Estimates I and II, using the conditions on the function a(·) in (1.2) and the inequality ∥ · ∥2 ≤ [C(Ω)]4∥∆2 · ∥2, we deduce that d dt ( ∥∇∆(y − y)(t)∥22 ) + a0∥∆2(y − y)(t)∥22 ≤ 0. As ∥∇∆ · ∥2 ≤ C(Ω)∥∆2 · ∥2, thus d dt ( ∥∇∆(y − y)(t)∥22 ) + a0 [C(Ω)]2 ∥∇∆(y − y)(t)∥22 ≤ 0. Integrating over the time variable from 0 to t, with t ∈ [0, T − τ ], we have shown that ∥∇∆(y − y)(t)∥2 ≤ e−λt∥∇∆(y0 − y0)∥2, ∀t ∈ [0, T − τ ]. (3.8) Applying the operator ∆2 in (3.3) we obtain ∆2(y − y)t − a (∫ Ω y dx′ ) ∆3(y − y)− [ a (∫ Ω y dx′ ) − a (∫ Ω y dx′ )] ∆3y = 0. (3.9) 20 M. R. NUÑEZ-CHÁVEZ EJDE-2025/96 Note that applying the operator ∆ to the equation (1.1)1, for all t ∈ [0, T ] we have a (∫ Ω y(x′, t) dx′ ) ∆2y(·, t) = ∆yt(·, t)−∆(v(·, t)ϱω(·)) = 0 in ∂Ω. Then ∆2y(·, t) = 0 in ∂Ω. Analogously ∆2y(·, t) = 0 in ∂Ω. Estimate V. Multiplying (3.9) by ∆2(y − y) and integrating in Ω, we obtain∫ Ω ∆2(y − y)t ∆ 2(y − y) dx+ a (∫ Ω y dx′ )∫ Ω (∇∆2(y − y))(∇∆2(y − y)) dx = − [ a (∫ Ω y dx′ ) − a (∫ Ω y dx′ )] ∫ Ω (∇∆2y) (∇∆2(y − y)) dx. Similar to Estimates I and II, using the conditions on the function a(·) in (1.2) and the inequality ∥ · ∥2 ≤ [C(Ω)]5∥∇∆2 · ∥2, we deduce that d dt ( ∥∆2(y − y)(t)∥22 ) + a0∥∇∆2(y − y)(t)∥22 ≤ 0. As ∥∆2 · ∥2 ≤ C(Ω)∥∇∆2 · ∥2, thus d dt ( ∥∆2(y − y)(t)∥22 ) + a0 [C(Ω)]2 ∥∆2(y − y)(t)∥22 ≤ 0. Integrating over the time variable from 0 to t, with t ∈ [0, T − τ ], we have shown that ∥∆2(y − y)(t)∥2 ≤ e−λt∥∆2(y0 − y0)∥2, ∀t ∈ [0, T − τ ]. (3.10) Now, adding the estimates (3.4), (3.5), (3.7), (3.8) and (3.10), we obtain estimate (3.2). As H4(Ω) ↪→ C2+ 1 2 (Ω), we conclude that ∥y(T − τ)− y(T − τ)∥2+ 1 2 ≤ e−λ(T−τ)∥y0 − y0∥H4(Ω), where λ > 0. (3.11) Step 2: Local control for the y(x, t) system. We will construct the local control at the final time t = T0, where T0 > 0 is sufficiently large. From (3.11), we have that for each ϵ > 0, there exists a T0 := T0(ϵ, τ) > 0 with T0 > ln (∥y0−y0∥H4(Ω) ϵ ) λ + τ, such that ∥y(T0 − τ)− y(T0 − τ)∥2+ 1 2 ≤ e−λ(T0−τ)∥y0 − y0∥H4(Ω) < ϵ. We can consider y(·, T0−τ) as a new initial datum and y(x, t)|Ω×(T0−τ,T0) as a new target trajectory with control v(x, t)|Ω×(T0−τ,T0). By Theorem 1.5, there exists a control ṽ ∈ C1/2,1/4(Ω× [T0 − τ, T0]) with ∥ṽ − v∥ C 1 2 , 1 4 (Ω×[T0−τ,T0]) ≤ C(Ω,M, τ)∥y(T0 − τ)− y(T0 − τ)∥2, ≤ C(Ω,M, τ)∥y(T0 − τ)− y(T0 − τ)∥2+ 1 2 < C(Ω,M, τ)ϵ, where the associated state y(x, t) satisfies y(x, T0) = y(x, T0) in Ω. Taking ϵ = η 2C(Ω,M,τ) > 0, we obtain that T0 := T0(η, τ) and ∥ṽ − v∥C1/2,1/4(Ω×[T0−τ,T0]) ≤ η 2 . By decomposition ṽ = (ṽ − v) + v, we can conclude that for all (x, t) ∈ Ω× (T0 − τ, T0), ṽ(x, t) ≥ −|ṽ(x, t)− v(x, t)|+ v(x, t) ≥ −∥ṽ − v∥C1/2,1/4(Ω×[T0−τ,T0]) + v(x, t) ≥ −η 2 + η = η 2 > 0. EJDE-2025/96 CONTROLLABILITY FOR NON-LINEAR AND NON-LOCAL PARABOLIC PDES 21 Step 3: Global control construction. Finally for T ≥ T0 = T0(η, τ), it is natural to define the desired control over (0, T ) as v(x, t) :=  v(x, t) in Ω× (0, T0 − τ ], ṽ(x, t) in Ω× (T0 − τ, T0], 0 in Ω× (T0, T ) and thus we have completed the proof. 4. Proof of Theorem 1.8 As y0 ̸= y0 in Ω, let us denote Ω0 := {x ∈ Ω : y0(x) = y0(x)}, Ω1 := {x ∈ Ω : y0(x) < y0(x)}, Ω2 := {x ∈ Ω : y0(x) > y0(x)}. So, we have that |Ω0| = 0, and |Ω1| > 0 or |Ω2| > 0. Let us divide the proof in two cases. Case 1: y0 ̸< y0 in Ω. This means that there exists Ω∗ 2 ⊂ Ω2 with |Ω∗ 2| > 0, in other words, y0(x) > y0(x) in Ω∗ 2. Considering the function φ0(x) := 1Ω∗ 2 (x) ∈ L2(Ω), we have (y0 − y0, φ0) = ∫ Ω (y0 − y0)(x)φ0(x) dx = ∫ Ω∗ 2 ( y0(x)− y0(x) ) dx > 0. Let us denote z := z(x, t) as the solution to system zt − a (∫ Ω z(x′, t) dx′ ) ∆z = 0 in QT , z(x, t) = 0 on ΣT , z(x, 0) = y0(x) in Ω. (4.1) As y0 and v are regular enough, by Proposition 2.2 we have that z − y ∈ C2+ 1 2 ,1+ 1 4 (QT ) ⊂ C([0, T ];L2(Ω)) and as (z(0)− y(0), φ0) = (y0 − y0, φ0) > 0, then there exists T1 > 0 such that (z(t)− y(t), φ0) > 0, ∀t ∈ [0, T1). (4.2) Claim 1: Tmin ≥ T1. Suppose by contradiction, if Tmin < T1, then there exists a T∗ ∈ A with T∗ < T1, in other words, we have T∗ ∈ (0, T1) and a control v ∈ L∞(QT∗) with v ≥ 0 em QT∗ such that y = y(x, t) the solution to system (1.1) with control v and initial data y0 satisfying y(·, T∗) = y(·, T∗) in Ω. By the comparison principle, we have y ≥ z in QT∗ . Then, as the function φ0 is nonnegative and from (4.2), we obtain (y(T∗), φ0) ≥ (z(T∗), φ0) > (y(T∗), φ0). Hence y(·, T∗) ̸= y(·, T∗) in Ω. So, T∗ /∈ A and we have a contradiction. Therefore, Tmin ≥ T1 > 0. Case 2: y0 < y0 in Ω. This means that Ω1 = Ω, then |Ω1| > 0, in other words, y0(x) < y0(x) in Ω1 = Ω. Then ∥y0 − y0∥1 = ∫ Ω |y0(x)− y0(x)| dx = ∫ Ω1 (y0(x)− y0(x)) dx > 0 . We consider z = z(x, t) the solution to system (4.1). As y0 ∈ H1 0 (Ω) ∩H2(Ω), by [1, 5], we have that z ∈ C([0, T ], L2(Ω)). 22 M. R. NUÑEZ-CHÁVEZ EJDE-2025/96 Denoting ξ = y − z, we have that the function ξ = ξ(x, t) is a solution to the system ξt − α̃(t, y)∆ξ − β̃(t, y, z) (∫ Ω ξ(x′, t) dx′ ) ∆z = vϱω in QT , ξ(x, t) = 0 on ΣT , ξ(x, 0) = 0 in Ω, (4.3) where α̃(t,Ψ) := a (∫ Ω Ψ(x′, t) dx′ ) , β̃(t,Ψ,Φ) := ∫ 1 0 a′ (∫ Ω ( λΨ(x′, t) + (1− λ)Φ(x′, t) ) dx′ ) dλ. As v ∈ C1/2,1/4(QT ) ↪→ H1(0, T ;L2(Ω)), we have that ξ ∈ C([0, T ];L2(Ω)). Denoting ξ = y − z, we have that the function ξ = ξ(x, t) is solution to the system ξt − α̃(t, z)∆ξ − β̃(t, y, z) (∫ Ω ξ(x′, t) dx′ ) ∆y = vϱω in QT , ξ(x, t) = 0 on ΣT , ξ(x, 0) = y0(x)− y0(x) in Ω. (4.4) As v ∈ C1/2,1/4(QT ) and y0 − y0 ∈ C2+ 1 2 (Ω), we have that ξ ∈ C([0, T ];L2(Ω)). Claim 2: There exists T2 > 0, such that, for any T ∈ (0, T2) and for any v ∈ L∞(QT ) with v ≥ 0 in QT , we have ξ(·, T ) ̸= ξ(·, T ) in Ω. Suppose by contradiction, for any T > 0 there exists T∗ ∈ (0, T ) and a control v ∈ L∞(QT∗) with v ≥ 0 in QT∗ , such that ξ(·, T∗) = ξ(·, T∗) in Ω. Let us define the adjoint system to (4.3) as −φt − α̃(t, y)∆φ+ β̃(t, y, z) (∫ Ω φ(x′, t)∆z(x′, t) dx′ ) = 0 in QT∗ , φ(x, t) = 0 on ΣT∗ , φ(x, T∗) = φT∗(x) in Ω. (4.5) By duality results in systems (4.3) and (4.5), we obtain (ξ(T∗), φ T∗) = ∫∫ ω×(0,T∗) vϱωφdx dt. (4.6) Let us conveniently construct an initial data φT∗(·) for the system (4.5). Let ϕ1 be the first eigenfunction of the Dirichlet Laplacian in Ω. We know that ϕ1 > 0 in Ω and ϕ1 ∈ H1 0 (Ω) ∩H2(Ω). For any r ∈ (0,+∞), we define the sets Er := {x ∈ Ω\ω : dist(x, ∂ω) < r}, Ec r := {x ∈ Ω\ω : dist(x, ∂ω) ≥ r}. Then Ω = ω ∪ Er ∪ Ec r for any r > 0. We consider a constant θ > 0 such that∫ Ec d (−ϕ1)(y0 − y0) dx ≤ −θ < 0, where d := dist(∂ω,∂Ω) 2 , and we define the constant Cθ := θ 3∥ϕ1∥∞∥y0−y0∥1 > 0. Let us define as cut-off function ψ ∈ C∞(Ω) as ψ(x) =  −1, x ∈ Ec δ , −1 ≤ ψ(x) ≤ Cθ, x ∈ Eδ, Cθ, x ∈ ω, EJDE-2025/96 CONTROLLABILITY FOR NON-LINEAR AND NON-LOCAL PARABOLIC PDES 23 with δ > 0 small enough. So, we can define φT∗(x) := ψ(x)ϕ1(x) (see Figure 2), where φT∗(x) = Cθϕ1(x) ≥ θ Kω 3∥ϕ1∥∞∥y0 − y0∥1 = θ̃ > 0, ∀x ∈ ω. RN R 0 θ̃ Eδ Eδ ω Ω φT∗ Figure 2. Initial datum φT∗ in Ω. We will arrive at a contradiction with equation (4.6). (a) We will prove that (ξ(T∗), φ T∗) < 0 for T ∗ ∈ (0, T3). Indeed, take T∗ > 0 arbitrary:∫ Ω ξ(x, 0)φT∗(x) dx = ∫ Ω (y0 − y0)φ T∗ dx = ∫ Ec δ (y0 − y0)φ T∗ dx+ ∫ Eδ (y0 − y0)φ T∗ dx+ ∫ ω (y0 − y0)φ T∗ dx. Considering δ > 0 small enough, we obtain the following estimates:∫ Ec δ (y0 − y0)φ T∗ dx = ∫ Ec δ (y0 − y0)(−ϕ1) dx ≤ ∫ Ec d (y0 − y0)(−ϕ1) dx ≤ −θ < 0, ∣∣ ∫ Eδ (y0 − y0)φ T∗ dx ∣∣ ≤ ∫ Eδ |y0 − y0||ψ||ϕ1| dx ≤ C(θ)∥y0 − y0∥1∥ϕ1∥∞|Eδ| ≤ C(θ) θ 3Cθ |Eδ| ≤ θ 3 .∣∣ ∫ ω (y0 − y0)φ T∗ dx ∣∣ = ∣∣ ∫ ω (y0 − y0)Cθϕ1 dx ∣∣ ≤ Cθ∥y0 − y0∥1∥ϕ1∥∞ = θ 3 . Then (ξ(0), φT∗) = ∫ Ω (y0 − y0)φ T∗ dx ≤ −θ 3 < 0. As ξ ∈ C([0, T ];L2(Ω)), for T = T3 > 0 small enough, we can conclude that (ξ(t), φT∗) < 0 for all t ∈ [0, T3). In particular, taking T∗ ∈ (0, T3), (ξ(T∗), φ T∗) < 0. (4.7) (b) We will prove that φ− = 0 in ω × (0, T4) for some T4 > 0. Indeed, by Remark 2.4 as φT ∈ H1 0 (Ω) ∩H2(Ω) we have φ ∈ L2(0, T ;H3(Ω) ∩H1 0 (Ω)), φt ∈ L2(0, T ;H1 0 (Ω)). By continuous embedding, φ ∈ C([0, T ];H2(Ω)). As N ≤ 3, we have φ ∈ C([0, T ];C(Ω)) = C(Ω × [0, T ]) = C(QT ). Then, as φ(·, T ) = φT (·) ≥ θ̃ > 0 in ω, by continuity of the function φ, taking T = T4 > 0 small enough, we have that φ ≥ θ̃ > 0 in ω × (0, T4). 24 M. R. NUÑEZ-CHÁVEZ EJDE-2025/96 Thus, φ− = 0 in ω × (0, T4). In particular, taking T5 = min{T3, T4} > 0, as T∗ ∈ (0, T5): φ− = 0 in ω × (0, T∗) (4.8) Substituting (4.7) and (4.8) in (4.6), we obtain 0 > (ξ(T∗), φ T∗) = ∫∫ ω×(0,T∗) vϱωφdx dt ≥ 0. This is a contradiction. From Claim 2, there exists T2 > 0, such that ξ(·, T ) ̸= ξ(·, T ) in Ω, then y(·, T ) ̸= y(·, T ) in Ω. Therefore Tmin ≥ T2 > 0. Finally, for any case, we obtain Tmin ≥ T0 > 0, where T0 = max{T1, T2}. 5. Appendix 5.1. Proof of Remark 1.2. We will need a classical result from the theory of linear parabolic equations. Lemma 5.1. Assume that α > 0, the functions bi,j, cj, d ∈ Cα,α2 (QT ) and the boundary ∂Ω is sufficiently regular (more precisely, of class C2+α), and for some constant µ > 0, N∑ i,j=1 bi,j(x, t)v̂iv̂j ≥ µ|v̂|2, ∀(x, t, v̂) = (x, t, v̂1, . . . , v̂N ) ∈ QT × RN . Then, for every g ∈ Cα,α2 (QT ), ϕ ∈ C2+α(Ω) and Φ ∈ C2+α,1+α 2 (ΣT ), satisfying the compatibility condition of order ⌈ α 2 + 1 ⌉ , the linear parabolic system ut − N∑ i,j=1 bijuxixj + N∑ j=1 cjuxj + du = g in QT , u(x, t) = Φ(x, t) on ΣT , u(x, 0) = ϕ(x) in Ω, admits a unique solution u ∈ C2+α,1+α 2 (QT ) and we have the estimate ∥u∥2+α,1+α 2 ≤ C(bi,j , cj , d) ( ∥ϕ∥2+α + ∥Φ∥ C2+α,1+α 2 (ΣT ) + ∥g∥α,α2 ) . For a proof of the above lemma, see Ladyzhenskaya’s book [9, Theorems III. 12.2 and IV. 5.2]. We will divide the proof of Remark 1.2 into two parts. Proof of existence. We apply the fixed-point method, introducing the map Λ0 : C1,1(QT ) → C1,1(QT ) such that Λ0(ŷ) = y, where y(x, t) is the solution to the system yt − a (∫ Ω ŷ(x′, t) dx′ ) ∆y = v in QT , y(x, t) = 0 on ΣT , y(x, 0) = y0(x) in Ω, (5.1) with ∥v∥1/2,1/4 ≤ ∥y0∥2+ 1 2 . For every u ∈ C1,1(QT ), the functions a ( ∫ Ω u(x′, t) dx′ ) and a′ ( ∫ Ω u(x′, t) dx′ ) depend only on the temporal variable t, so we can denote them as au(t) and a ′ u(t), respectively. Remark 5.2. Given the functions u1, u2 ∈ C1,1(QT ), let us consider t ∈ [0, T ]. By Remark 1.1, we have |au1 (t)− au2 (t)| = ∣∣∣a(∫ Ω u1(x ′, t) dx′ ) − a (∫ Ω u2(x ′, t) dx′ )∣∣∣ ≤M ∫ Ω |u1(x′, t)− u2(x ′, t)| dx′ ≤M ∫ Ω ∥u1 − u2∥0,0 dx′ EJDE-2025/96 CONTROLLABILITY FOR NON-LINEAR AND NON-LOCAL PARABOLIC PDES 25 ≤M |Ω| ∥u1 − u2∥1,1. By the Mean Value Theorem, we have au1 (t)− au2 (t) = a (∫ Ω u1(x ′, t) dx′ ) − a (∫ Ω u2(x ′, t) dx′ ) = a′ (∫ Ω ( θ(t)u1 + (1− θ(t))u2 ) (x′, t) dx′ )∫ Ω ( u1 − u2 ) (x′, t) dx′ = a′θu1+(1−θ)u2 (t) ∫ Ω ( u1 − u2 ) (x′, t) dx′ := a′θ,u1,u2 (t) ∫ Ω ( u1 − u2 ) (x′, t) dx′, where θ(t) ∈ (0, 1). Lemma 5.3. For every u ∈ C1,1(QT ), we have that au, a ′ u ∈ C1/2,1/4(QT ). Proof. Taking t1, t2 ∈ [0, T ] with t1 ̸= t2. By Remark 1.1, we have |au(t1)− au(t2)| |t1 − t2|1/4 = ∣∣∣a(∫ Ω u(x′, t1) dx ′ ) − a (∫ Ω u(x′, t2) dx ′ )∣∣∣/|t1 − t2|1/4 ≤ M ∫ Ω |u(x′, t1)− u(x′, t2)| dx′ |t1 − t2|1/4 ≤M ∫ Ω ∥u∥1/2,1/4 dx′ ≤M |Ω|∥u∥1/2,1/4 < +∞. Thus, au ∈ C1/4([0, T ]). We can conclude that au ∈ C1/2,1/4(QT ). Analogously, we can verify that |a′u(t1)− a′u(t2)| |t1 − t2|1/4 ≤M |Ω|∥u∥1/2,1/4 < +∞. We can conclude that a′u ∈ C1/2,1/4(QT ). □ Claim 1: The mapping Λ0 is well-defined. Indeed, since ŷ ∈ C1,1(QT ), by Lemma 5.3 we have that aŷ = a (∫ Ω ŷ(x′, t) dx′ ) ∈ C 1 2 , 1 4 (QT ). By Lemma 5.1 applied to the system (5.1), we have Λ0(ŷ) = y ∈ C2+ 1 2 ,1+ 1 4 (QT ) ⊂ C1,1(QT ) satisfying ∥Λ0(ŷ)∥2+ 1 2 ,1+ 1 4 := ∥y∥2+ 1 2 ,1+ 1 4 ≤ C(M,a0, a1, |Ω|, ŷ) ( ∥y0∥2+ 1 2 + ∥v∥ 1 2 , 1 4 ) ≤ 2C(M,a0, a1, |Ω|, ŷ)∥y0∥2+ 1 2 , where C(M,a0, a1, |Ω|, ŷ) := C(M,a0, a1, |Ω|, ∥ŷ∥1/2,1/4). Claim 2: The mapping Λ0 is continuous. Let us consider a sequence (ŷn) ⊂ C1,1(QT ) such that ŷn −→ ŷ in C1,1(QT ). Denoting Λ0(ŷn) = yn and Λ0(ŷ) = y, we have that the function yn − y satisfies (yn − y)t − aŷ(t)∆(yn − y) = ( aŷn (t)− aŷ(t) ) ∆yn in QT , (yn − y)(x, t) = 0 on ΣT , (yn − y)(x, 0) = 0 in Ω. (5.2) By Claim 1, since the functions ŷ, ŷn ∈ C1,1(QT ), we have that y, yn ∈ C2+ 1 2 ,1+ 1 4 (QT ). 26 M. R. NUÑEZ-CHÁVEZ EJDE-2025/96 Thus, we can verify that ∆yn,∆y ∈ C1/2,1/4(QT ). Now, by Lemma 5.3 we have that aŷn , aŷ ∈ C1/2,1/4(QT ). We conclude that ( aŷn − aŷ ) ∆yn ∈ C1/2,1/4(QT ). Applying Lemma 5.1 to system (5.2), we obtain that ∥Λ0(ŷn)− Λ0(ŷ)∥2+ 1 2 ,1+ 1 4 = ∥yn − y∥2+ 1 2 ,1+ 1 4 ≤ C(M,a0, a1, |Ω|, ŷ) ∥∥(aŷn − aŷ ) ∆yn ∥∥ 1/2,1/4 . Let us estimate the right side of the last inequality. Given (x1, t1), (x2, t2) ∈ QT with (x1, t1) ̸= (x2, t2), we have ∣∣(aŷn(t1)− aŷ(t1) ) ∆yn(x1, t1)− ( aŷn(t2)− aŷ(t2) ) ∆yn(x2, t2) ∣∣ |x1 − x2|1/2 + |t1 − t2|1/4 ≤ ∣∣aŷn(t1)− aŷ(t1) ∣∣∣∣∆yn(x1, t1)−∆yn(x2, t2) ∣∣ |x1 − x2|1/2 + |t1 − t2|1/4 + ∣∣(aŷn (t1)− aŷ(t1) ) − ( aŷn (t2)− aŷ(t2) )∣∣|∆yn(x2, t2)| |x1 − x2|1/2 + |t1 − t2|1/4 := L1 + L2. Let us examine the estimates for L1 and L2. For L1, we use Remark 5.2 and Lemma 5.3, thus L1 ≤ ∣∣∣aŷn (t1)− aŷ(t1) ∣∣∣ |∆yn(x1, t1)−∆yn(x2, t2)| |x1 − x2|1/2 + |t1 − t2|1/4 ≤M |Ω| ∥ŷn − ŷ∥1,1∥yn∥2+ 1 2 , 1 4 ≤ CM |Ω| ∥ŷn − ŷ∥1,1 ( ∥y0∥2+ 1 2 + ∥v∥1/2,1/4 ) ≤ 2CM |Ω| ∥ŷn − ŷ∥1,1∥y0∥2+ 1 2 . For L2, we consider two cases: If t1 = t2, then L2 = 0. On the other hand if t1 ̸= t2, we use Remark 5.2 and Lemma 5.3; thus L2 ≤ ∣∣a′θn,ŷn,ŷ (t1) ∫ Ω (ŷn − ŷ)(t1) dx ′ − a′θn,ŷn,ŷ (t2) ∫ Ω (ŷn − ŷ)(t2) dx ′ ∣∣ |t1 − t2|1/4 |∆yn(x2, t2)| ≤ ∣∣(a′θn,ŷn,ŷ (t1)− a′θn,ŷn,ŷ (t2) ) ∫ Ω (ŷn − ŷ)(x′, t1) dx ′ ∣∣ |t1 − t2|1/4 |∆yn(x2, t2)| + ∣∣a′θn,ŷn,ŷ (t2) ∫ Ω ( (ŷn − ŷ)(x′, t1)− (ŷn − ŷ)(x′, t2) ) dx′ ∣∣ |t1 − t2|1/4 |∆yn(x2, t2)| ≤ ∣∣a′θn,ŷn,ŷ (t1)− a′θn,ŷn,ŷ (t2) ∣∣ |t1 − t2|1/4 (∫ Ω |(ŷn − ŷ)(x′, t1)| dx′ ) |∆yn(x2, t2)| + |a′θn,ŷn,ŷ(t2)| (∫ Ω |(ŷn − ŷ)(x′, t1)− (ŷn − ŷ)(x′, t2)| |t1 − t2|1/4 dx′ ) |∆yn(x2, t2)| ≤M |Ω|C∥θnŷn + (1− θn)ŷ∥1,1 (∫ Ω ∥ŷn − ŷ∥0,0 dx′ ) ∥yn∥2,0 +M (∫ Ω ∥ŷn − ŷ∥1/2,1/4 dx′ ) ∥yn∥2,0 ≤M |Ω|C ( ∥ŷn∥1,1 + ∥ŷ∥1,1 )( |Ω|∥ŷn − ŷ∥1,1 ) ∥yn∥2+ 1 2 ,1+ 1 4 +M ( C|Ω|∥ŷn − ŷ∥1,1 ) ∥yn∥2+ 1 2 ,1+ 1 4 ≤MC ( |Ω|, ∥ŷ∥1,1 ) ∥ŷn − ŷ∥1,1∥yn∥2+ 1 2 ,1+ 1 4 EJDE-2025/96 CONTROLLABILITY FOR NON-LINEAR AND NON-LOCAL PARABOLIC PDES 27 ≤MC ( |Ω|, ∥ŷ∥1,1 ) ∥ŷn − ŷ∥1,1 ( ∥y0∥2+ 1 2 + ∥v∥1/2,1/4 ) ≤ 2MC ( |Ω|, ∥ŷ∥1,1 ) ∥ŷn − ŷ∥1,1∥y0∥2+ 1 2 . Since L1, L2 → 0, we conclude that yn −→ y in C2+ 1 2 ,1+ 1 4 (QT ). It follows that Λ0 is continuous in C1,1(QT ). Let K := { ŷ ∈ C1+ 1 2 ,1+ 1 4 (QT ) : ∥ŷ∥1+ 1 2 ,1+ 1 4 ≤ R } . Claim 3: The set K is compact. Indeed, let us first see that Λ0(K) ⊂ K. For any ŷ ∈ K, from system (5.1) we have ∥Λ0(ŷ)∥2+ 1 2 ,1+ 1 4 ≤ C (M,a0, a1, |Ω|, ŷ) ( ∥y0∥2+ 1 2 + ∥v∥ 1 2 , 1 4 ) ≤ 2 C(M,a0, a1, |Ω|, R)∥y0∥2+ 1 2 . If we consider ∥y0∥2+ 1 2 ≤ 1 2C(M,a0,a1,|Ω|,R) , we have that Λ0(ŷ) ∈ K. From de compact immersion C1+ 1 2 ,1+ 1 4 (QT ) ↪→ C1,1(QT ), we conclude that K is compact. Thus, from Claims 1–3, by Schauder Fixed Point Theorem, there exists a function y ∈ K such that Λ0(y) = y. This means that there exists a solution y ∈ C2+ 1 2 ,1+ 1 4 (QT ) of the system (1.1) satisfying ∥y∥2+ 1 2 ,1+ 1 4 ≤ C(M,a0, a1, |Ω|, R) ( ∥y0∥2+ 1 2 + ∥v∥1/2,1/4 ) . □ Proof of uniqueness. Let y1, y2 ∈ C2+ 1 2 ,1+ 1 4 (QT ) be two solutions of system (1.1), then we have that y := y1 − y2 ∈ C2+ 1 2 ,1+ 1 4 (QT ) is solution of the system yt − ay1 ∆y = (ay1 − ay2 )∆y2 in QT , y(x, t) = 0 on ΣT , y(x, 0) = 0 in Ω. (5.3) Thus, by the results shown in the existence part (specifically Claim 2), ∥y∥2+ 1 2 ,1+ 1 4 = ∥y1 − y2∥2+ 1 2 ,1+ 1 4 ≤ C(M,a0, a1, |Ω|, R)∥ (ay1 − ay2 )∆y2∥1/2,1/4 ≤ C(M,a0, a1, |Ω|, R) ( ∥y1 − y2∥1,1∥y1(0)∥2+ 1 2 ) ≤ ( C(M,a0, a1, |Ω|, R)∥y0∥2+ 1 2 ) ∥y∥1,1 ≤ ( C(M,a0, a1, |Ω|, R)∥y0∥2+ 1 2 ) ∥y∥2+ 1 2 ,1+ 1 4 . If we consider ∥y0∥2+ 1 2 < 1 C(M,a0,a1,|Ω|,R) , we conclude that y = 0 in QT . Thus y1 = y2 in QT . □ 5.2. Proof of Proposition 2.1. Recall that the system (2.2) is zt − α(t, w)∆z = −β(t, w) (∫ Ω z(x′, t) dx′ ) ∆y + uϱω in QT , z(x, t) = 0 on ΣT , z(x, 0) = z0(x) in Ω, (5.4) where w ∈ Z := C1,1(QT ) is fixed, and α(t, w) := a (∫ Ω ( w(x′, t) + y(x′, t) ) dx′ ) , β(t, w) := ∫ 1 0 a′ (∫ Ω ( λw(x′, t) + y(x′, t) ) dx′ ) dλ. 28 M. R. NUÑEZ-CHÁVEZ EJDE-2025/96 Remark 5.4. From the notation in Appendix 5.1, we have α(t, w) = aw+y(t), β(t, w) = ∫ 1 0 a′λw+y(t)dλ. Lemma 5.5. We have that α(·, w) ∈W 1,∞(QT ). Proof. By Remark 5.4 and condition (1.2) on the function a(·), for every t ∈ [0, T ] we have |α(t, w)| = ∣∣∣a(∫ Ω ( w(x′, t) + y(x′, t) ) dx′ )∣∣∣ ≤ a1. Therefore, α(·, w) ∈ L∞(QT ). Now, we will verify that αt(·, w) ∈ L∞(QT ). Indeed, since w ∈ Z := C1,1(QT ), we have that |αt(·, w)| = ∣∣∣a′(∫ Ω ( w(x′, t) + y(x′, t) ) dx′ )∣∣∣ ∣∣∣ ∫ Ω ( wt(x ′, t) + yt(x ′, t) ) dx′ ∣∣∣ ≤M ∫ Ω (|wt(x ′, t)|+ |yt(x′, t)|)dx′ ≤M(∥w∥Z + ∥yt∥∞,1,0,T ) < +∞. We conclude that α(·, w) ∈W 1,∞(QT ). □ Lemma 5.6. We have that β(·, w) ∈ L∞(QT ). Proof. By Remark 5.4 and condition (1.2) on the function a(·), for every t ∈ [0, T ] we have |β(t, w)| ≤ ∫ 1 0 |a′λw+y(t1)| dλ ≤ ∫ 1 0 Mdλ =M < +∞. Therefore, β(·, w) ∈ L∞(QT ). □ We divide the proof of Proposition 2.1 into two parts. Part I: Proof of existence. Once again, we apply the fixed-point method. Denoting the Banach space Z̃ := L∞(0, T ;L1(Ω)), we define the mapping Λ̃0 : Z̃ → Z̃ by Λ̃0(ẑ) := z, where z(x, t) is the solution of the system zt − α(t, w)∆z = −β(t, w) (∫ Ω ẑ(x′, t) dx′ ) ∆y + uϱω in QT , z(x, t) = 0 on ΣT , z(x, 0) = z0(x) in Ω. (5.5) For each ẑ ∈ Z̃, let us denote Ψw,ẑ(t) = −β(t, w) ( ∫ Ω ẑ(x′, t)dx′ ) . Lemma 5.7. For every ẑ ∈ Z̃, we have that Ψw,ẑ∆y ∈ L2(QT ). Proof. By Lemma 5.6, we have ∥Ψw,ẑ∆y∥2L2(QT ) = ∫ T 0 |Ψw,ẑ(t)|2∥∆y(t)∥22 dt ≤ ∫ T 0 |β(t, w)|2 (∫ Ω |ẑ(x′, t)| dx′ )2 ∥∆y(t)∥22 dt ≤ ∫ T 0 M2 ∥ẑ(t)∥21 ∥y(t)∥2H2(Ω)dt ≤M2 ∥ẑ∥2 Z̃ ∫ T 0 ∥y(t)∥2H2(Ω)dt =M2 ∥ẑ∥2 Z̃ ∥y∥2L2(0,T ;H2(Ω)) < +∞. Therefore, Ψw,ẑ∆y ∈ L2(QT ). □ EJDE-2025/96 CONTROLLABILITY FOR NON-LINEAR AND NON-LOCAL PARABOLIC PDES 29 Claim 1: The mapping Λ̃0 is well-defined. Indeed, for ẑ ∈ Z̃, by Lemma 5.7 we have that Ψw,ẑ∆y ∈ L2(QT ). We also have that uϱω ∈ L2(QT ), thus Ψw,ẑ∆y + uϱω ∈ L2(QT ). We have the initial data z0 ∈ L2(Ω). Therefore, by Lemmas 5.5 and 5.7, we can apply classical results of existence and uniqueness to the system (5.5). Thus, we have that Λ̃0(ẑ) = z ∈ W̃ ⊂ Z̃, where W̃ := { z ∈ L2(0, T ;H1 0 (Ω)) ∩ L∞(QT ) : zt ∈ L2(0, T ;H−1(Ω)) } . Moreover, by Lemma 5.7 we have ∥Λ̃0(ẑ)∥W̃ = ∥z∥W̃ ≤ C(M,a0, a1, |Ω|, w, y) ( ∥z0∥2 + ∥u∥2,0,T + ∥∥Ψw,ẑ∆y ∥∥ 2,0,T ) ≤ C(M,a0, a1, |Ω|, w, y) ( ∥z0∥2 + ∥u∥2,0,T + ∥y∥L2(0,T ;H2(Ω)) ∥∥ẑ∥∥ Z̃ ) , (5.6) where C(M,a0, a1, |Ω|, w, y) := C(M,a0, a1, |Ω|, ∥w∥Z , ∥yt∥∞,0,T ) > 0. Thus, the mapping Λ̃0 is well-defined. Claim 2: The mapping Λ̃0 is continuous. Let us consider a sequence (ẑn) ⊂ Z̃ such that ẑn → ẑ in Z̃. Denoting by Λ̃0(ẑn) = zn and Λ̃0(ẑ) = z, we have that the function zn − z satisfies (zn − z)t − α(t, w)∆(z − zn) = Ψw,ẑn−ẑ∆y in QT , (zn − z)(x, t) = 0 on ΣT , (zn − z)(x, 0) = 0 in Ω. (5.7) Remark 5.8. In system (5.7), we use the linearity of the function Ψw,ẑ with respect to the variable ẑ. From the estimate (5.6) applied to the system (5.7), we have ∥Λ̃0(ẑn)− Λ̃0(ẑ)∥W̃ = ∥zn − z∥W̃ ≤ C(M,a0, a1, w, y)∥y∥L2(0,T ;H2(Ω)) ∥∥ẑn − ẑ ∥∥ Z̃ . Thus, Λ̃0(ẑn) → Λ̃0(ẑ) in Z̃. Therefore, the mapping Λ̃0 is continuous. Let K̃ := { ẑ ∈ W̃ : ∥ẑ∥W̃ ≤ R } . Claim 3: The set K̃ is compact for some R̃ > 0. Let us first see that Λ̃0(K̃) ⊂ K̃. For any ẑ ∈ K̃, from system (5.5) and (5.6) we have ∥Λ̃0(ẑ)∥W̃ ≤ C(M,a0, a1, |Ω|, w, y) ( ∥z0∥2 + ∥u∥2,0,T + ∥y∥L2(0,T ;H2(Ω)) ∥∥ẑ∥∥ Z̃ ) . (5.8) If we consider R̃ > 0 (sufficiently large) such that 3C̃(M,a0, a1, |Ω|, w, y)∥z0∥2 ≤ R̃, 3C̃(M,a0, a1, |Ω|, w, y)∥u∥2,0,T ≤ R̃, ∥ẑ∥W̃ ≤ R̃, 3C̃(M,a0, a1, |Ω|, w, y)∥y∥L2(0,T ;H2(Ω))∥ẑ∥Z̃ ≤ R̃, then, from (5.8), we obtain ∥Λ̃0(ẑ)∥W̃ ≤ R̃ 3 + R̃ 3 + R̃ 3 = R̃. This is, Λ̃0(ẑ) ∈ K̃. From the compact immersion W̃ ↪→ Z̃, we conclude that K̃ is compact with R̃ > 0. Thus, from Claims 1–3, by Schauder Fixed Point Theorem, there exists a function z ∈ K̃ such that Λ̃0(z) = z. This means that there exists a solution z ∈ W̃ of the system (1.1) satisfying ∥z∥W̃ ≤ C(M,a0, a1, |Ω|, w, y) ( ∥z0∥2 + ∥u∥2,0,T + ∥y∥L2(0,T ;H2(Ω))∥z∥Z̃ ) . (5.9) We can simplify the expression (5.9) by using the condition on the trajectory y, that is, assuming that ∥y∥2+ 1 2 ,1+ 1 4 ≪ 1. We define the function P (·) := C(M,a0, a1, |Ω|, w, ·), it is clear that P is positive and increasing. So, for some s << 1, we have that P (s)s < 1 2 . Indeed, • If P (1) < 1, taking s < 1 2 , then s < 1 and P (s) < P (1) < 1. Thus, P (s)s < 1 2 . 30 M. R. NUÑEZ-CHÁVEZ EJDE-2025/96 • If P (1) ≥ 1, taking s < 1 2P (1) , then s < 1 2 < 1 and P (s) < P (1). Thus, P (s)s < 1 2 . In particular, for s0 = ∥y∥2+ 1 2 ,1+ 1 4 , we have ∥y∥L2(0,T ;H2(Ω)) ≤ s0 < 1 2P (s0) ≤ 1 2P (∥yt∥∞,0,T ) = 1 2C(M,a0, a1, |Ω|, w, y) . So, ∥z∥W̃ ≤ C(M,a0, a1, |Ω|, w, y) ( ∥z0∥2 + ∥u∥2,0,T ) . (5.10) Part II: Proof of uniqueness. Let z1, z2 ∈ W̃ be two solutions of the system (5.4), then we have that z = z1 − z2 ∈ W̃ is solution of the system zt − α(t, w)∆z = Ψw,z∆y in QT , z(x, t) = 0 on ΣT , z(x, 0) = 0 in Ω. (5.11) Thus, by applying (5.10) to system (5.11), we obtain that ∥z∥W̃ ≤ 0. Therefore, we conclude that z = 0 in QT , or equivalently, z1 = z2 in QT . 5.3. Proof of Proposition 2.2. The proof follows the same reasoning as in Appendix 5.1 and 5.2. Recall that system (2.2) is zt − α(t, w)∆z = −β(t, w) (∫ Ω z(x′, t) dx′ ) ∆y + uϱω in QT , z(x, t) = 0 on ΣT , z(x, 0) = z0(x) in Ω, (5.12) where w ∈ Z := C1,1(QT ) is fixed, and α(t, w) := a (∫ Ω ( w(x′, t) + y(x′, t) ) dx′ ) , β(t, w) := ∫ 1 0 a′ (∫ Ω ( λw(x′, t) + y(x′, t) ) dx′ ) dλ. By Remark 5.4, we have α(t, w) = aw+y(t), β(t, w) = ∫ 1 0 a′λw+y(t)dλ. (5.13) Lemma 5.9. We have that α(·, w) ∈ C1/2,1/4(QT ). Proof. Since w, y ∈ C1,1(QT ), it follows that w + y ∈ C1,1(QT ). By Lemma 5.3 and (5.13), we have that α(·, w) = aw+y ∈ C1/2,1/4(QT ). Furthermore, ∥α(·, w)∥ 1 2 . 1 4 = ∥aw+y∥ 1 2 . 1 4 ≤M |Ω|C(∥w∥Z + ∥y∥1/2,1/4). □ Lemma 5.10. We have that β(·, w) ∈ C1/2,1/4(QT ). Proof. For each λ ∈ [0, 1], since w, y ∈ C1,1(QT ), it follows that λw + y ∈ C1,1(QT ). By Lemma 5.3, we have that a′λw+y ∈ C1/2,1/4(QT ). From (5.13), for any t1, t2 ∈ [0, T ] with t1 ̸= t2, we have |β(t1, w)− β(t2, w)| |t1 − t2|1/4 ≤ ∫ 1 0 |a′λw+y(t1)− a′λw+y(t2)| |t1 − t2|1/4 dλ ≤ ∫ 1 0 ∥a′λw+y∥1/2,1/4dλ = ∥a′λw+y∥1/2,1/4 ≤ C(M, |Ω|)∥λw + y∥1/2,1/4 < +∞. Thus, we conclude that ∥β(·, w)∥1/2,1/4 ≤ C(M, |Ω|, w, y). Therefore, β(·, w) ∈ C1/2,1/4(QT ). □ EJDE-2025/96 CONTROLLABILITY FOR NON-LINEAR AND NON-LOCAL PARABOLIC PDES 31 We will divide the proof of Proposition 2.2 into two parts. Part I: Proof of existence. Once again, we apply the fixed-point method. We define the mapping Λ̂0 : Z → Z by Λ̂0(ẑ) := z, where z(x, t) is the solution of the system zt − α(t, w)∆z = −β(t, w) (∫ Ω ẑ(x′, t) dx′ ) ∆y + uϱω in QT , z(x, t) = 0 on ΣT , z(x, 0) = z0(x) in Ω. (5.14) For each ẑ ∈ Z, by Appendix 5.2 we have that Ψw,ẑ(t) = −β(t, w) ( ∫ Ω ẑ(x′, t)dx′ ) . Lemma 5.11. For every ẑ ∈ Z, we have that Ψw,ẑ ∈ C1/2,1/4(QT ). Proof. Taking t1, t2 ∈ [0, T ] with t1 ̸= t2. By Remark 1.1 and Lemma 5.10, we have |Ψw,ẑ(t1)−Ψw,ẑ(t2)| |t1 − t2|1/4 = ∣∣β(t1, w)( ∫Ω ẑ(x′, t1) dx′)− β(t2, w) ( ∫ Ω ẑ(x′, t2) dx ′)∣∣ |t1 − t2|1/4 ≤ |β(t1, w)| ∫ Ω |ẑ(x′, t1)− ẑ(x′, t2)| |t1 − t2|1/4 dx′ + |β(t1, w)− β(t2, w)| |t1 − t2|1/4 ∫ Ω |ẑ(x′, t2)| dx′ ≤M∥ẑ∥1/2,1/4 + ∥β(·, w)∥1/2,1/4 (∫ Ω ∥ẑ∥0,0 dx′ ) ≤M∥ẑ∥Z + ∥β(·, w)∥1/2,1/4|Ω|∥ẑ∥Z < +∞. Thus, we conclude that ∥Ψw,ẑ∥1/2,1/4 ≤ C(M, |Ω|, w, y)∥ẑ∥Z . Therefore, Ψw,ẑ ∈ C1/2,1/4(QT ). □ Lemma 5.12. For every ẑ ∈ Z, we have that Ψw,ẑ∆y ∈ C1/2,1/4(QT ). Proof. Given (x1, t1), (x2, t2) ∈ QT with (x1, t1) ̸= (x2, t2), we have |Ψw,ẑ(t1)∆y(x1, t1)−Ψw,ẑ(t2)∆y(x2, t2)| |x1 − x2|1/2 + |t1 − t2|1/4 ≤ |Ψw,ẑ(t1)||∆y(x1, t1)−∆y(x2, t2)| |x1 − x2|1/2 + |t1 − t2|1/4 + |Ψw,ẑ(t1)−Ψw,ẑ(t2)||∆y(x2, t2)| |x1 − x2|1/2 + |t1 − t2|1/4 := Ψ1 +Ψ2. Let us examine the estimates for Ψ1 and Ψ2. For Ψ1, we use hypothesis about the function a(·) (1.2) and Remark 1.1, thus Ψ1 = |β(t1, w)| (∫ Ω |ẑ(x′, t1)| dx′ ) |∆y(x1, t1)−∆y(x2, t2)| |x1 − x2|1/2 + |t1 − t2|1/4 ≤M (∫ Ω ∥ẑ∥0,0 dx′ ) ∥∆y∥1/2,1/4 ≤M |Ω| ∥ẑ∥Z ∥y∥2+ 1 2 ,1+ 1 4 < +∞. For Ψ2, we will consider two cases: • If t1 = t2, then Ψ2 = 0. • If t1 ̸= t2, we use Lemma 5.11, thus Ψ2 ≤ |Ψw,ẑ(t1)−Ψw,ẑ(t2)| |t1 − t2|1/4 |∆y(x2, t2)| ≤ ∥Ψw,ẑ∥1/2,1/4∥y∥2,0 ≤ ∥Ψw,ẑ∥1/2,1/4∥y∥2+ 1 2 ,1+ 1 4 < +∞. We conclude that ∥Ψw,ẑ∆y∥1/2,1/4 ≤ C(M, |Ω|, w)∥y∥2+ 1 2 ,1+ 1 4 ∥ẑ∥Z . Therefore, Ψw,ẑ∆y ∈ C1/2,1/4(QT ). □ 32 M. R. NUÑEZ-CHÁVEZ EJDE-2025/96 Claim 1: The mapping Λ̂0 is well-defined. For ẑ ∈ Z, by Lemma 5.12 we have that Ψw,ẑ∆y ∈ C1/2,1/4(QT ). We also have that vϱω ∈ C1/2,1/4(QT ), thus Ψw,ẑ∆y + vϱω ∈ C 1 2 , 1 4 (QT ). We have the initial data z0 ∈ C1/2,1/4(QT ). By Lemma 5.1 applied to system (5.14), we obtain Λ̂0(ẑ) = z ∈ C2+ 1 2 ,1+ 1 4 (QT ) ⊂ Z, satisfying ∥Λ̂0(ẑ)∥2+ 1 2 ,1+ 1 4 = ∥z∥2+ 1 2 ,1+ 1 4 ≤ C(M,a0, a1, |Ω|, w, y) ( ∥z0∥2+ 1 2 + ∥v∥1/2,1/4 + ∥∥Ψw,ẑ∆y ∥∥ 1/2,1/4 ) ≤ C(M,a0, a1, |Ω|, w, y) ( ∥z0∥2+ 1 2 + ∥v∥1/2,1/4 + ∥y∥2+ 1 2 ,1+ 1 4 ∥∥ẑ∥∥ Z ) , where C(M,a0, a1, |Ω|, w, y) := C(M,a0, a1, |Ω|, ∥w∥Z , ∥y∥1/2,1/4). Thus, the mapping Λ̂0 is well- defined. Claim 2: The mapping Λ̂0 is continuous. Indeed, let us consider a sequence (ẑn) ⊂ Z such that ẑn → ẑ in Z. Denoting Λ̂0(ẑn) = zn and Λ̂0(ẑ) = z, we have that the function zn − z satisfies (zn − z)t − α(t, w)∆(z − zn) = Ψw,ẑn−ẑ∆y in QT , (zn − z)(x, t) = 0 on ΣT , (zn − z)(x, 0) = 0 in Ω. (5.15) In system (5.15), we use the linearity of the function Ψw,ẑ with respect to the variable ẑ. From Lemma 5.1 applied to the system (5.15), we have that ∥Λ̂0(ẑn)− Λ̂0(ẑ)∥2+ 1 2 ,1+ 1 4 = ∥zn − z∥2+ 1 2 ,1+ 1 4 ≤ C(M,a0, a1, w, y)∥Ψw,ẑn−ẑ∆y∥1/2,1/4. Using Lemma 5.12, we conclude that ∥Λ̂0(ẑn)− Λ̂0(ẑ)∥2+ 1 2 ,1+ 1 4 ≤ C(M,a0, a1, |Ω|, w, y)∥y∥2+ 1 2 ,1+ 1 4 ∥ẑn − ẑ∥Z . Thus, Λ̂0(ẑn) → Λ̂0(ẑ) em Z. Therefore, the mapping Λ̂0 is continuous. Let K̂ := { ẑ ∈ C2+ 1 2 ,1+ 1 4 (QT ) : ∥ẑ∥2+ 1 2 ,1+ 1 4 ≤ R } . Claim 3: The set K̂ is compact for some R > 0. First we show that Λ̂0(K̂) ⊂ K̂. For any ẑ ∈ K̂, from system (5.14) we have that ∥Λ̂0(ẑ)∥2+ 1 2 ,1+ 1 4 ≤ C(M,a0, a1, |Ω|, w, y) ( ∥z0∥2+ 1 2 + ∥u∥1/2,1/4 + ∥y∥2+ 1 2 ,1+ 1 4 ∥∥ẑ∥∥ Z ) . If we consider R̂ > 0 (sufficiently large) such that 3C(M,a0, a1, |Ω|, w, y)∥z0∥2+ 1 2 ≤ R̂, 3C(M,a0, a1, |Ω|, w, y)∥v∥1/2,1/4 ≤ R̂, 3C(M,a0, a1, |Ω|, w, y)∥y∥2+ 1 2 ,1+ 1 4 ∥ẑ∥Z ≤ R̂. Thus, we have Λ̂0(ẑ) ∈ K̂. From the compact immersion C2+ 1 2 ,1+ 1 4 (QT ) ↪→ Z, we conclude that K̂ is compact with R̂ > 0. Thus, from Claims 1–3, by Schauder Fixed Point Theorem, there exists a function z ∈ K̂ such that Λ̂0(z) = z. This means that there exists a solution z ∈ C2+ 1 2 ,1+ 1 4 (QT ) of the system (1.1) satisfying ∥z∥2+ 1 2 ,1+ 1 4 ≤ C(M,a0, a1, |Ω|, w, y) ( ∥z0∥2+ 1 2 + ∥u∥1/2,1/4 + ∥y∥2+ 1 2 ,1+ 1 4 ∥∥ẑ∥∥ Z ) . (5.16) We can simplify the expression (5.16) by using the condition on the trajectory y, that is, assuming that ∥y∥2+ 1 2 ,1+ 1 4 << 1. EJDE-2025/96 CONTROLLABILITY FOR NON-LINEAR AND NON-LOCAL PARABOLIC PDES 33 We define the function P̃ (s) := C(M,a0, a1, |Ω|, w, s) for all s > 0, it is clear that P̃ is positive and increasing. So, for some s << 1, we have that P̃ (s)s < 1 2 . Indeed, • If P̃ (1) < 1, taking s < 1 2 , then s < 1 and P̃ (s) < P̃ (1) < 1. Thus, P̃ (s)s < 1 2 . • If P̃ (1) ≥ 1, taking s < 1 2P̃ (1) , then s < 1 2 < 1 and P̃ (s) < P̃ (1). Thus, P̃ (s)s < 1 2 . In particular, for s0 = ∥y∥2+ 1 2 ,1+ 1 4 , C(M,a0, a1, |Ω|, w, y)∥y∥2+ 1 2 ,1+ 1 4 = P̃ (∥y∥1/2,1/4)s0 ≤ P̃ (s0)s0 < 1 2 . So, ∥z∥2+ 1 2 ,1+ 1 4 ≤ C(M,a0, a1, w, y) ( ∥z0∥2+ 1 2 + ∥u∥1/2,1/4 ) . (5.17) Part II: Proof of uniqueness. Let z1, z2 ∈ C2+ 1 2 ,1+ 1 4 (QT ) be two solutions of the system (5.12), then we have that z = z1 − z2 ∈ C2+ 1 2 ,1+ 1 4 (QT ) is solution of the system zt − α(t, w)∆z = Ψw,z∆y in QT , z(x, t) = 0 on ΣT , z(x, 0) = 0 in Ω, (5.18) Thus, by applying (5.17) to system (5.18), we obtain that ∥z∥2+ 1 2 ,1+ 1 4 ≤ 0. Therefore, we conclude that z = 0 in QT , or equivalently, z1 = z2 in QT . 5.4. Proof of Lemma 2.6. For each w ∈ Z, we consider the system −φt − α(t, w)∆φ = 0 in QT φ(x, t) = 0 on ΣT φ(x, T ) = f(x) in Ω. (5.19) Then, we have the following Carleman inequality for system (5.19),∫∫ QT e− C0 T−t |φ|2 dx dt ≤ C̃0 ∫∫ ω1×(0,T ) e−2sσξ3|φ|2 dx dt, (5.20) where C̃0 := C̃0(Ω, a0, a1, ∥w∥Z , ∥y∥Z) is a positive constant. From the spectral theory applied to the solution of the system (5.19), we obtain φ(x, t) = +∞∑ j=1 e−λj ∫ T t α(s,w) ds(f, ϕj)ϕj(x), ∥φ∥22 = +∞∑ j=1 e−2λj ∫ T t α(s,w) ds|(f, ϕj)|2. The Carleman inequality (5.20) becomes∫ T 0 +∞∑ j=1 e−2λj ∫ T t α(s,w) ds− C0 T−t |(f, ϕj)|2dt ≤ C̃0 ∫∫ ω1×(0,T ) e−2sσξ3|φ|2 dx dt. Then +∞∑ j=1 [( ∫ T 0 e−2λja1(T−t)− C0 T−t dt ) |(f, ϕj)|2 ] ≤ C̃0 ∫∫ ω1×(0,T ) e−2sσξ3|φ|2 dx dt. (5.21) We will approximate the integral I = ∫ T 0 e−2λja1(T−t)− C0 T−t dt using the Laplace method. To do this, we make the following change of variables s = T − t, so the integral becomes I = ∫ T 0 e−2λja1s−C0 s ds. 34 M. R. NUÑEZ-CHÁVEZ EJDE-2025/96 We define g(s) = 2λja1s+ C0 s for all s ∈ [0, T ], then g′(s) = 0 ⇐⇒ 2λja1 − C0 s2 = 0 ⇐⇒ s = √ C0 2λja1 , g′′(s) = 2C0 s3 > 0. The point s0 = √ C0 2λja1 is a minimum for g(s). Applying the Taylor expansion (Taylor polynomial) to the function g(s) at the point s0, we obtain g(s) = g(s0) + g′(s0)(s− s0) + g′′(s0) 2 (s− s0) 2 +O((s− s0) 3). Performing the calculations of the terms of the polynomial: g(s0) = 2λ1a1s0 + C0 s0 = 2λja1 √ C0 2λja1 + C0√ C0 2λja1 = √ 2λja1C0 + √ 2λja1C0 = 2 √ 2λja1C0, g′(s0) = 0, g′′(s0) = 2C0(√ C0 2λja1 )3 = 2 √ (2λja1)3√ C0 . Thus, g(s) = 2 √ 2λja1C0 + √ (2λja1)3 C0 (s− s0) 2 +O((s− s0) 3). We can approximate the integral I as I = ∫ T 0 e−g(s) ds ≈ ∫ T 0 e − ( 2 √ 2λja1C0+ √ (2λja1)3 C0 (s−s0) 2 ) ds = e−2 √ 2λja1C0 ∫ T 0 e − √ (2λja1)3 C0 (s−s0) 2 ds. Let us calculate the integral on the right-hand side. To do this, by making the substitution s̃ = 4 √ (2λja1)3 C0 (s− s0), we obtain ∫ T 0 e − √ (2λja1)3 C0 (s−s0) 2 ds = 4 √ C0 (2λja1)3 ∫ 4 √ (2λja1)3 C0 (T−s0) − 4 √ (2λja1)3 C0 s0 e−s̃2ds̃. Since λja1 → +∞, we have I = ∫ T 0 e−2λja1(T−t)− C0 T−t dt = ∫ T 0 e−g(s) ds ≈ e−2 √ 2λja1C0 4 √ C0 (2λja1)3 ∫ +∞ −∞ e−s̃2ds̃, I ≈ ( π2C0 (2λja1)3 )1/4 e−2 √ 2C0λja1 . Thus, we deduce that ∫ T 0 e−2λja1(T−t)− C0 T−t dt ≥ C̃1e −2R0 √ λj , where R0 := √ 2C0a1. Substituting the last inequality into the estimate (5.21), we can conclude the proof of Lemma 2.6. EJDE-2025/96 CONTROLLABILITY FOR NON-LINEAR AND NON-LOCAL PARABOLIC PDES 35 References [1] N. Chang, M. Chipot; Nonlinear nonlocal evolution problems, Revista Real Academia de Ciencias Serie A Matemáticas (RACSAM), 97 (3) (2003), 393–415. [2] V. Costa, J. Ĺımaco, A. R. Lopes, L. Prouvée; On the Controllability of a Free-Boundary Problem for 1D Heat Equation with Local and Nonlocal Nonlinearities, Applied Mathematics & Optimization, 87 (2023), 11. [3] E. Fernández-Cara, H. R. Clark, J. Ĺımaco,L. A. Medeiros; Theoretical and numerical local null controllability for a parabolic system with local and nonlocal nonlinearities, Applied Mathematics and Computation, 223, 2013, 483–505. [4] E. Fernández-Cara, J. Ĺımaco, S. Menezes; Null controllability for a parabolic equation with nonlocal nonlin- earities, Systems & Control Letters, 61 (2012), 107–111. [5] E. Fernández-Cara, J. Ĺımaco, D. Nina-Huaman, M. R. Nuñez-Chávez, Exact controllability to the trajectories for parabolic PDEs with nonlocal nonlinearities, Mathematics of Control, Signals and Letters, 31 (2019), 415–431. [6] E. Fernández-Cara, Q. Liu, E. Zuazua; Null controllability of linear heat and wave equations with nonlocal spatial terms, SIAM, Journal on Control and Optimization. 54(4) (2016), 2009–2019. [7] A. V. Fursikov, O. Yu. Imanuvilov; Controllability of Evolution Equations, Lecture Notes Series, vol. 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996. [8] O. Yu. Imanuvilov, M. Yamamoto; Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations, Publ. Res. Inst. Math. Sci. 39 (2) 2003, 227–274. [9] O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Ural’ceva; Linear and Quasilinear Equations of Parabolic Type, Transl, Math. Monogr., 23, AMS, Providence, RI, 1968. [10] Xu Liu, Xu Zhang; Local controllability of multidimensional quasi-linear parabolic equations, SIAM Journal on Control and Optimization, 50 (4) (2012), 2046–2064. [11] J. Lohéac, E. Trelát, E. Zuazua; Minimal controllability time for the heat equation under unilateral state or control constraints, Mathematical Models Methods and Applications, 27 (2017), 1587–1644. [12] A. R. Lopes, J. Ĺımaco; Local null controllability for a parabolic equation with local and nonlocal nonlinearities in moving domains, Evolution Equations and Control Theory, 11, (2022), 749–779. [13] M. R. Nuñez-Chávez; Controllability under positive constraints for quasilinear parabolic PDEs, Mathematical Control and Related Fields, 12(2) (2021), 327–341. [14] D. Pighin, E. Zuazua; Controllability under positivity constraints of semilinear heat equations, Mathematical Control and Related Fields, 8(3-4) (2018), 935–964. [15] D. Pighin, E. Zuazua; Controllability under positivity constraints of multi-d wave equations, Trends in Control Theory and Partial Differential Equation, Springer INdAM Series, 32 2019, 195–232. [16] L. Prouvée, J. Ĺımaco; Local null controllability for a parabolic-elliptic system with local and nonlocal nonlin- earities, Electronic J. of Qualitative Theory of Differential Equations, 2029, 74, (2019), 1–31. [17] Z. Wu, J. Yin, Ch. Wang; Eliptic and parabolic equations, World Scientific, New York-London-Singapure- Beijing-Shangai-Hong Kong-Taipei-Chennai, 2006. Miguel R. Nuñez-Chávez UFMT- Federal University of Mato Grosso, ICET, Department of Mathematics, 78060-900, Cuiabá, MT, Brazil Email address: miguel.chavez@ufmt.br 1. Introduction 1.1. Important results 1.2. Outline 2. Proof of Theorem 1.5 2.1. Null controllability of the linearized system (2.2) 2.2. Null local controllability of the nonlinear system (2.1) 3. Proof of Theorem 1.7 4. Proof of Theorem 1.8 5. Appendix 5.1. Proof of Remark 1.2 5.2. Proof of Proposition 2.1 5.3. Proof of Proposition 2.2 5.4. Proof of Lemma 2.6 References