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This contribution considers the containment control problem for linear multi‐agent systems (MAS) using continuum models in form of linear parabolic PDEs. For this, a networked controller is designed, that ensures the asymptotic convergence of finite‐dimensional agents into a containment area specified by dynamic leaders. This problem is traced back to an output regulation problem for a continuum model independent of the number of agents, which also takes disturbances into account. A prescribed formation is assigned for the MAS in the containment area by using Bézier curves. Using bilateral backstepping a tracking controller is designed, which allows to distribute the control effort between the two boundary agents. The states are estimated by an infinite‐dimensional disturbance observer. For this, both a folding observer with network communication and two boundary observers without network communication are introduced. A fully distributed solution of the containment control problem is obtained by employing a continuum signal model observer, which communicates the states of the leader and disturbance model to all agents. The desired communication topology is imposed after the design by a spatial discretization. A simulation example illustrates the effectiveness of the new method for the networked controller design. [ABSTRACT FROM AUTHOR]

We establish the asymptotic expansion of the fundamental solutions with precise error estimates for second-order parabolic operators ∂ t - div (A (x / ε , t / ε ℓ) ∇) , 0 < ε < 1 , 0 < ℓ < ∞ , in the case ℓ ≠ 2 , where the spatial and temporal variables oscillate on non-self-similar scales and do not homogenize simultaneously. To achieve the goal, we explore the direct quantitative two-scale expansions for the aforementioned operators, which should be of some independent interests in quantitative homogenization of parabolic operators involving multiple scales. In the self-similar case ℓ = 2 , similar results have been obtained in Geng and Shen (Anal PDE 13(1): 147–170, 2020). [ABSTRACT FROM AUTHOR]

We consider the following chemotaxis-haptotaxis system: { u t = ∇ ⋅ (D (u) ∇ u) − χ ∇ ⋅ (S (u) ∇ v) − ξ ∇ ⋅ (u ∇ w) , x ∈ Ω , t > 0 , v t = Δ v − v + u , x ∈ Ω , t > 0 , w t = − v w , x ∈ Ω , t > 0 , under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ R n , n ≥ 3 with smooth boundary. It is proved that for S (s) D (s) ≤ A (s + 1) α for α < 2 n and under suitable growth conditions on D , there exists a uniform-in-time bounded classical solution. Also, we prove that for radial domains, when the opposite inequality holds, the corresponding solutions blow-up in finite or infinite-time. We also provide the global-in-time existence and boundedness of solutions to the above system with small initial data when D (s) = 1 , S (s) = s. [ABSTRACT FROM AUTHOR]

We consider the stabilization of a class of linear evolution systems z ′ = A z + B v under the observation y = C z by means of a finite dimensional control v. The control is based on the design of a Luenberger observer which can be infinite or finite dimensional (of dimension large enough). In the infinite dimensional case, the operator A is supposed to generate an analytical semigroup with compact resolvent and the operators B and C are unbounded operators whereas in the finite dimensional case, A is assumed to be a self-adjoint operator with compact resolvent, B and C are supposed to be bounded operators. In both cases, we show that if (A, B) and (A, C) verify the Fattorini-Hautus Criterion, then we can construct an observer-based control v of finite dimension (greater or equal than largest geometric multiplicity of the unstable eigenvalues of A) such that the evolution problem is exponentially stable. As an application, we study the stabilization of the diffusion system. [ABSTRACT FROM AUTHOR]

In this article, we study the boundary local exact controllability to any steady state of a one-dimensional parabolic system with coupled nonlinear boundary conditions by means of only one control. The significant point is that the state components are interacting only at the boundary points with the assistance of some nonlinear terms. We consider two cases : either the control function is acting through a mixed nonlinear boundary condition on the first component or through a Neumann condition on the second component. The results are slightly different in the two cases.To study this problem, we first consider the associated linearized systems around the given steady state. The method of moments let us to prove its controllability and to obtain a suitable estimate of the control cost of the form $ Me^{M(T+\frac{1}{T})} $. To this end, we need to develop a precise spectral analysis of a non self-adjoint operator.Thanks to those preliminary results, we can use the source term method developed in [29], followed by the Banach fixed point argument, to obtain the small-time boundary local exact controllability to the steady state for the original system. [ABSTRACT FROM AUTHOR]

We consider fully parabolic two‐species chemotaxis system with p$$ p $$‐Laplacian diffusion in a smooth bounded domain Ω⊂ℝn,n≥2$$ \Omega \subset {\mathrm{\mathbb{R}}}&#x0005E;n,n\ge 2 $$ with 1

When applying the classical multistep schemes for solving differential equations, one often faces the dilemma that smaller time steps are needed with higher-order schemes, making it impractical to use high-order schemes for stiff problems. We construct in this paper a new class of BDF and implicit-explicit schemes for parabolic type equations based on the Taylor expansions at time t n+\beta with \beta > 1 being a tunable parameter. These new schemes, with a suitable \beta, allow larger time steps at higher order for stiff problems than that which is allowed with a usual higherorder scheme. For parabolic type equations, we identify an explicit uniform multiplier for the new second- to fourth-order schemes and conduct rigorously stability and error analysis by using the energy argument. We also present ample numerical examples to validate our findings. [ABSTRACT FROM AUTHOR]

Initial-boundary value problems are considered for homogeneous parabolic systems with Dini-continuous coefficients and zero initial conditions in a semibounded plane domain with a nonsmooth lateral boundary admitting cusps, on which general boundary conditions with variable coefficients are given. A theorem on unique classical solvability of these problems in the space of functions that are continuous and bounded together with their first spatial derivatives in the closure of the domain is proved by applying the boundary integral equation method. A representation of the resulting solutions in the form of vector single-layer potentials is given. [ABSTRACT FROM AUTHOR]

In this paper, we study the null controllability of systems of $ n $-coupled parabolic equations with dynamic boundary conditions, where the coupling and control matrices $ A $ and $ B $ are constant in time and space. Being different to the case of static boundary conditions, we will show that the Kalman rank condition $ rank[B, AB,\dots, A^{n-1}B ] = n $ is a sufficient condition, we also show that it is necessary for the null controlability under an extra assumption on the boundary coupling. The null controlability result will be proved by proving Carleman and observability inequalities for the corresponding adjoint problem. [ABSTRACT FROM AUTHOR]

In 1985, V. Scheffer discussed partial regularity for what he called solutions to the 'Navier-Stokes inequality', which only satisfy the incompressibility condition as well as the local and global energy inequalities and the pressure equation which may be derived formally from the Navier-Stokes system. One may extend this notion to a system introduced by F.-H. Lin and C. Liu in 1995 to model the flow of nematic liquid crystals, which include the Navier-Stokes system when the 'director field' $ d $ is taken to be zero. The model includes a further parabolic system which implies an a priori maximum principle for $ d $, which is lost when one considers the analogous 'inequality'.In 2018, Q. Liu proved a partial regularity result for solutions to the Lin-Liu model in terms of the 'parabolic fractal dimension' $ \text{dim}_{ \text{pf}} $, relying on the boundedness of $ d $ coming from the maximum principle. Q. Liu proves $ { \text{dim}_{ \text{pf}}(\Sigma_{-} \cap \mathcal{K}) \leq \tfrac{95}{63}} $ for any compact $ \mathcal{K} $, where $ \Sigma_{-} $ is the set of space-time points near which the solution blows up forwards in time. For solutions to the corresponding 'inequality', we prove that, without any compensation for the lack of maximum principle, one has $ { \text{dim}_{ \text{pf}}(\Sigma_{-} \cap \mathcal{K}) \leq \tfrac {55}{13}} $. We also provide a range of criteria, including as just one example the boundedness of $ d $, any one of which would furthermore imply that solutions to the inequality also satisfy $ { \text{dim}_{ \text{pf}}(\Sigma_{-} \cap \mathcal{K}) \leq \tfrac{95}{63}} $. [ABSTRACT FROM AUTHOR]

In this paper we study the quantitative homogenization of second-order parabolic systems with locally periodic (in both space and time) coefficients. The O (ε) scale-invariant error estimate in L 2 (0 , T ; L 2 d d − 1 (Ω)) is established in C 1 , 1 cylinders under minimum smoothness conditions on the coefficients. This process relies on critical estimates of smoothing operators. We also develop a new construction of flux correctors in the parabolic manner and a sharp estimate for temporal boundary layers. [ABSTRACT FROM AUTHOR]

In this article, we consider anisotropic parabolic systems of pp-Laplace type. The model case is the parabolic pi{p}_{i}-Laplace system ut−∑i=1n∂∂xi(∣Diu∣pi−2Diu)=0{u}_{t}-\mathop{\sum }\limits_{i=1}^{n}\frac{\partial }{\partial {x}_{i}}({| {D}_{i}u| }^{{p}_{i}-2}{D}_{i}u)=0 with exponents pi≥2{p}_{i}\ge 2. Under the assumption that the exponents are not too far apart, i.e., the difference pmax−pmin{p}_{\max }-{p}_{\min } is sufficiently small, we establish a higher integrability result for weak solutions. This extends a result, which was only known for the elliptic setting, to the parabolic setting.

We deal with nonlinear systems of parabolic type satisfying componentwise structural conditions. The nonlinear terms are Carathéodory maps having controlled growth with respect to the solution and the gradient and the data are in anisotropic Lebesgue spaces. Under these assumptions we obtain essential boundedness of the weak solutions. [ABSTRACT FROM AUTHOR]

The distributed null controllability for coupled parabolic systems with non-diagonalizable diffusion matrices with a reduced number of controls has been studied in the case of constant matrices. On the other hand, boundary controllability issues and distributed controllability with non-constant coefficients for this kind of systems is not completely understood. In this paper, we analyze the boundary controllability properties of a class of coupled parabolic systems with non-diagonalizable diffusion matrices in the constant case and the distributed controllability of a $ 2\times 2 $ non-diagonalizable parabolic system with space-dependent coefficients. For the boundary controllability problem, our strategy relies on the moment method. For the distributed controllability problem, our findings provide positive and negative control results by using the Fattorini-Hautus test and a fictitious control strategy. [ABSTRACT FROM AUTHOR]

The first and second initial-boundary value problems for second-order parabolic systems with coefficients satisfying the Dini condition in a semibounded plane domain with a nonsmooth lateral boundary admitting cusps are considered. Theorems on the uniqueness of classical solutions of these problems in the class of functions that are continuous and bounded together with their first spatial derivatives in the closure of this domain are proved. [ABSTRACT FROM AUTHOR]

In this paper we investigate the global existence and asymptotical stability of solutions to a class of parabolic systems with homogeneous nonlinearity for both bounded and unbounded domains. First we prove both global existence and finite time blow-up of solutions of the system for different initial conditions by using the potential well method, and the asymptotic behavior of the solutions are also considered. On the other hand, we also obtain global existence and finite time blow-up of solutions for both Sobolev subcritical and critical cases. We use a method of comparing least energy levels with that of semitrivial solutions to overcome the difficulties here. [ABSTRACT FROM AUTHOR]

We prove existence of variational solutions for a class of nonlocal evolution equations whose prototype is the double phase equation ∂ t u + P. V. ∫ ℝ N | u (x , t) − u (y , t) | p − 2 (u (x , t) − u (y − t)) | x − y | N + p s + a (x , y) | u (x , t) − u (y , t) | q − 2 (u (x , t) − u (y − t)) | x − y | N + q s d y = 0. The approach of minimization of parameter-dependent convex functionals over space-time trajectories requires only appropriate convexity and coercivity assumptions on the nonlocal operator. As the parameter tends to zero, we recover variational solutions. Under further growth conditions, these variational solutions are global weak solutions. Further, this provides a direct minimization approach to approximation of nonlocal evolution equations. [ABSTRACT FROM AUTHOR]

This paper investigates the formation problem of an array of large-scale mobile sensor networks. A new framework for the dynamic of mobile sensors as a continuum described by the parabolic system with boundary disturbance is proposed. The communication topology of agents is a chain graph and fixed. Leader feedback laws which are designed in a manner to the boundary control of large-scale mobile sensor networks allow the mobile sensors to achieve the formation steadily. By referring to the Lyapunov functional method and employing a boundary control approach, a new protocol is established to deal with a formation problem for the large-scale mobile sensor networks. Finally, numerical examples are given to illustrate the usefulness of the results. [ABSTRACT FROM AUTHOR]

We consider the controllability of an abstract parabolic system by using switching controls. More precisely, we show that, under general hypotheses, if a parabolic system is null-controllable for any positive time with N controls, then it is also null-controllable with the property that at each time, only one of these controls is active. The main difference with previous results in the literature is that we can handle the case where the main operator of the system is not self-adjoint. We give several examples to illustrate our result: coupled heat equations with terms of orders 0 and 1, the Oseen system or the Boussinesq system. [ABSTRACT FROM AUTHOR]

In this paper, by using the so-called dynamical approach we study the existence of nontrivial solutions for a class of Kirchhoff type systems. The presented result is a generalization of previous works Alves and Boudjeriou (Nonlinear Anal. 197:1–17, 2020; Rend. Circ. Mat. Palermo 71(2):611–632, 2022) where the existence of nontrivial solutions for some scalar nonlocal elliptic problems has been studied. [ABSTRACT FROM AUTHOR]

In this paper, we show the existence of continuous positive solutions of a class of nonlinear parabolic reaction diffusion systems with initial conditions using techniques of functional analysis and potential analysis. [ABSTRACT FROM AUTHOR]

We present a weighted L p L p -theory of parabolic systems on a half space R d + R + d. The leading coefficients are assumed to be only measurable in time t t and have small bounded mean oscillations (BMO) with respect to the spatial variables x x , and the lower order coefficients are allowed to blow up near the boundary. [ABSTRACT FROM AUTHOR]

This work is devoted to the stabilization of parabolic systems with a finite-dimensional control subjected to a constant delay. Our main result shows that the Fattorini-Hautus criterion yields the existence of such a feedback control, as in the case of stabilization without delay. The proof consists in splitting the system into a finite dimensional unstable part and a stable infinite-dimensional part and to apply the Artstein transformation on the finite-dimensional system to remove the delay in the control. Using our abstract result, we can prove new results for the stabilization of parabolic systems with constant delay: the N N -dimensional linear reaction-convection-diffusion equation with N ≥ 1 N ≥ 1 and the Oseen system. We end the article by showing that this theory can be used to stabilize nonlinear parabolic systems with input delay by proving the local feedback distributed stabilization of the Navier-Stokes system around a stationary state. [ABSTRACT FROM AUTHOR]

The second initial-boundary value problem for a second-order Petrovskii parabolic system with constant coefficients in a semibounded plane domain with a nonsmooth lateral boundary is considered. The uniqueness of a solution to this problem in the class is proved. The minimum condition on the boundary function under which the solution of the problem belongs to is investigated. A constructive solution is obtained by applying the boundary integral equation method. [ABSTRACT FROM AUTHOR]

The work deals with the easily verifiable necessary conditions of the preservation of the nonnegativity of the solutions of a system of parabolic equations in the case of the double scale anomalous diffusion when the fractional Laplacian is added to the negative Laplace operator raised to another fractional power in the space of two dimensions. Such necessary conditions are extremely important for the applied analysis society because they impose the necessary form of the system of equations that must be studied mathematically. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the problem of constrained controllability for distributed parabolic linear system evolving in spatial domain Ω using the Reverse Hilbert Uniqueness Method (RHUM approach) introduced by Lions in 1988. It consists in finding the control u that steers the system from an initial state y0 to a state between two prescribed functions. We give some definitions and properties concerning this concept and then we resolve the problem that relays on computing a control with minimum cost in the case of ω = Ω and in the regional case where ω is a part of Ω. [ABSTRACT FROM AUTHOR]

The main concern of this article is to investigate the boundary controllability of some 2 × 2 one-dimensional parabolic systems with both the interior and boundary couplings: The interior coupling is chosen to be linear with constant coefficient while the boundary one is considered by means of some Kirchhoff-type condition at one end of the domain. We consider here the Dirichlet boundary control acting only on one of the two state components at the other end of the domain. In particular, we show that the controllability properties change depending on which component of the system the control is being applied. Regarding this, we point out that the choices of the interior coupling coefficient and the Kirchhoff parameter play a crucial role to deduce the positive or negative controllability results. Further to this, we pursue a numerical study based on the well-known penalized HUM approach. We make some discretization for a general interior-boundary coupled parabolic system, mainly to incorporate the effects of the boundary couplings into the discrete setting. This allows us to illustrate our theoretical results as well as to experiment some more examples which fit under the general framework, for instance a similar system with a Neumann boundary control on either one of the two components. [ABSTRACT FROM AUTHOR]

In this work, we extend the analysis of the problem of switching controls proposed in [E. Zuazua, J. Eur. Math. Soc. (JEMS), 13 (2011), pp. 85-117]. The problem asks the following question: Assuming that one can control a system using two or more actuators, does there exist a control strategy such that at all times, only one actuator is active? We answer positively when the controlled system corresponds to an analytic semigroup spanned by a positive self-adjoint operator which is null-controllable in arbitrary small times. Similarly to [E. Zuazua, J. Eur. Math. Soc. (JEMS), 13 (2011), pp. 85-117], our proof relies on analyticity arguments and will also work in finite dimensional settings and under some further spectral assumptions when the operator spans an analytic semigroup but is not necessarily self-adjoint. [ABSTRACT FROM AUTHOR]

We consider coupled parabolic systems with homogeneous boundary conditions. We establish a family of L q -Carleman inequalities, q ∈ [ 2 , ∞) and use them to obtain stability estimates in L q and L ∞ norms for the sources in terms of the solution in a subdomain. We apply these estimates to reaction–diffusion systems. [ABSTRACT FROM AUTHOR]

In the paper the positivity problem of the model of an one dimensional heat transfer process is addressed. Such a problem has not been considered yet. The considered thermal process is described by the fractional order state equation, derived from parabolic heat equation with homogenous Neumann boundary conditions and distributed control and observation. The internal and external positivity of the model depend on heater and sensor location as well as the size of the model. It is proved that the external positivity of the considered system can be achieved without internal positivity. Conditions of the internal and external positivity are proposed and proved. Theoretical considerations are supported by experiments. Experiments were done using the real system containing typical industrial components. The proposed results can be applied in real temperature measurements, for example in thermal cameras. • The positivity problem for the one dimensional thermal process is addressed. The considered process is described by the fractional order state space model. This problem has not been considered yet. • The positivity is determined by size and location of heater and sensors as well as the size of model. More generally, the positivity of the model can be achieved by choosing the right place for measurement. • The positivity does not depend on non integer orders of the model. This allows to use the proposed results to integer order systems too. • The crucial result is that the external positivity can be achieved without internal positivity by suitable location of heate and sensors. • The proposed external positivity condition is easy in use and it can be employed in real temperature measurements, for example in thermal cameras. [ABSTRACT FROM AUTHOR]

We consider systems of n parabolic equations coupled in zero or first order terms with scalar controls acting through a control matrix B. We are interested in stabilization with a control in feedback form. Our approach relies on the approximate controllability of the linearized system, which in turn is related to unique continuation property for the adjoint system. For the unique continuation we establish algebraic Kalman type conditions. [ABSTRACT FROM AUTHOR]

The paper aims to extend the notion of regional observability with constraints on the gradient (also called regional enlarged observability) to the semilinear parabolic case, in order to reconstruct the gradient of the initial data between two prescribed functions in an internal subregion ω of the evolution domain Ω. The approach is based on an extension of the Hilbert Uniqueness Method (HUM) combined with the fixed point techniques. The analysis leads to an algorithm which is successfully numerically implemented and illustrated with examples and simulations. [ABSTRACT FROM AUTHOR]

In this article we establish a partial Hölder continuity result for weak solutions of parabolic systems, where the nonlinear vector field A(⋅) satisfies a standard p-growth condition and a non-degenerate ellipticity condition with respect to the gradient variable, while in the space-time variable z = (x,t) it verifies a VMO-type condition. Thus, no continuity in the space-time variable is assumed. The proof is based on the method of A-caloric approximation, applied on suitably chosen intrinsic cylinders. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to investigate the boundary controllability of some coupled parabolic systems in the cascade form in the case where the boundary conditions are of Robin type. In particular, we prove that the associated controls satisfy suitable uniform bounds with respect to the Robin parameters, that let us show that they converge towards a Dirichlet control when the Robin parameters go to infinity. This is a justification of the popular penalisation method for dealing with Dirichlet boundary data in the framework of the controllability of coupled parabolic systems. [ABSTRACT FROM AUTHOR]

In L 2 (R d ; C n) , we consider a semigroup e - t A ε , t ⩾ 0 , generated by a matrix elliptic second-order differential operator A ε ⩾ 0 . Coefficients of A ε are periodic, depend on x / ε , and oscillate rapidly as ε → 0 . Approximations for e - t A ε were obtained by Suslina (Funktsional Analiz i ego Prilozhen 38(4):86–90, 2004) and Suslina (Math Model Nat Phenom 5(4):390–447, 2010) via the spectral method and by Zhikov and Pastukhova (Russ J Math Phys 13(2):224–237, 2006) via the shift method. In the present note, we give another short proof based on the contour integral representation for the semigroup and approximations for the resolvent with two-parametric error estimates obtained by Suslina (2015). [ABSTRACT FROM AUTHOR]

In this paper, the finite-time non-fragile boundary feedback control problem is investigated for a class of nonlinear parabolic systems, where both the multiplicative and additive controller gain variations are considered to describe the actuator parameter perturbation. Non-fragile boundary control strategies are designed with respect to two controller gain variations via collocated or non-collocated boundary measurement, respectively. In light of the finite-time stability and Lyapunov-based techniques, some sufficient conditions are presented in terms of linear matrix inequalities such that the resulting closed-loop system is well-posedness and practically finite-time stable. Finally, numerical examples are given to verify the effectiveness of the proposed design method. [ABSTRACT FROM AUTHOR]

We consider systems of parabolic equations coupled in zero and first order terms. We establish Lipschitz estimates in L q {L^{q}} -norms, 2 ≤ q ≤ ∞ {2\leq q\leq\infty} , for the source in terms of the solution in a subdomain. The main tool is a family of appropriate Carleman estimates with general weights, in Lebesgue spaces, for nonhomogeneous parabolic systems. [ABSTRACT FROM AUTHOR]

Rough polyharmonic splines (RPS) is a variational method which has recently been developed for linear divergence-form operators with arbitrary rough coefficients. RPS method does not rely on concepts of ergodicity or scale-separation, but on compactness properties of the solution space. In this paper, we extend RPS approach method for the optimal control problem governed by parabolic systems with rough L ∞ coefficients. RPS method is used for the spatial discretization, while the temporal discretization is performed by the finite difference method. As the iterative solution of the optimal control problem requires solving the state and co-state equations many times with different right hand sides, RPS method only requires one-time pre-computation on the fine scale and the following iterations can be done to coarse degrees of freedom. First of all, we extend an approximation method for the multiscale optimal control problem. Secondly, we obtain the error estimates of the multiscale optimal control problem. Finally, numerical experiments are presented to validate the theoretical analysis. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to investigate the controllability properties of semi-discrete in space coupled parabolic systems with less controls than equations, in dimension greater than 1. We are particularly interested in the boundary control case which is notably more intricate that the distributed control case, even though our analysis is more general. The main assumption we make on the geometry and on the evolution equation itself is that it can be put into a tensorized form. In such a case, following [5] and using an adapted version of the Lebeau-Robbiano construction, we are able to prove controllability results for those semi-discrete systems (provided that the structure of the coupling terms satisfies some necessary Kalman condition) with uniform bounds on the controls. To achieve this objective we actually propose an abstract result on ordinary differential equations with estimates on the control and the solution whose dependence upon the system parameters are carefully tracked. When applied to an ODE coming from the discretization in space of a parabolic system, we thus obtain uniform estimates with respect to the discretization parameters. [ABSTRACT FROM AUTHOR]

Singularities of plane into plane mappings described by parabolic two-component systems of quasi-linear partial differential equations of the first order are studied. Impediments arising in the application of the original Whitney's approach to such a case are discussed. Hierarchy of singularities is analyzed by the double-scaling expansion method for the simplest 2 -component Jordan system. It is shown that flex is the lowest singularity while higher singularities are given by (k + 1 , k + 2) curves which are of cusp type for k = 2 n + 1 , n = 1 , 2 , 3 , .... Regularization of these singularities by deformation of plane into plane mappings into surface S 2 + k (⊂ ℝ 2 + k) to plane is discussed. Applicability of the proposed approach to other parabolic type mappings is noted. We finally compare the results obtained for the parabolic case with non-generic gradient catastrophes for hyperbolic systems. [ABSTRACT FROM AUTHOR]

We consider quantitative estimates in the homogenization of second-order parabolic systems with periodic coefficients that oscillate on multiple spatial and temporal scales, ∂ t − div (A (x , t , x / ε 1 , ... , x / ε n , t / ε 1 ′ , ... , t / ε m ′) ∇) , where ε ℓ = ε α ℓ , ε k ′ = ε β k , ℓ = 1 ,... , n , k = 1 ,... , m , with 0 < α 1 <... < α n < ∞ and 0 < β 1 <... < β m < ∞. The convergence rate in the homogenization is derived in the L 2 space, and the large-scale interior and boundary Lipschitz estimates are also established. In the case n = m = 1 , such issues have been addressed by Geng and Shen (2020) [12] based on an interesting scale reduction technique developed therein. Our investigation relies on a quantitative reiterated homogenization theory. [ABSTRACT FROM AUTHOR]