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Abstract: Classical Liénard equations are two-dimensional vector fields, on the phase plane or on the Liénard plane, related to scalar differential equations . In this paper, we consider f to be a polynomial of degree , with l a fixed but arbitrary natural number. The related Liénard equation is of degree 2l. We prove that the number of limit cycles of such an equation is uniformly bounded, if we restrict f to some compact set of polynomials of degree exactly . The main problem consists in studying the large amplitude limit cycles, of which we show that there are at most l. [Copyright &y& Elsevier]

Abstract: The paper deals with two-dimensional slow-fast systems and more specifically with multi-layer canard cycles. These are canard cycles passing through n layers of fast orbits, with . The canard cycles are subject to n generic breaking mechanisms and we study the limit cycles that can be perturbed from the generic canard cycles of codimension n. We prove that this study can be reduced to the investigation of the fixed points of iterated translated power functions. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to study the reflection–transmission of diffractive geometrical optic rays described by semi-linear symmetric hyperbolic systems such as the Maxwell–Lorentz equations with the anharmonic model of polarization. The framework is that of P. Donnat''s thesis [P. Donnat, Quelques contributions mathématiques en optique non linéaire, chapters 1 and 2, thèse, 1996] and V. Lescarret [V. Lescarret, Wave transmission in dispersive media, M3AS 17 (4) (2007) 485–535]: we consider an infinite WKB expansion of the wave over long times/distances and because of the boundary, we decompose each profile into a hyperbolic (purely oscillating) part and elliptic (evanescent) part as in M. William [M. William, Boundary layers and glancing blow-up in nonlinear geometric optics, Ann. Sci. École Norm. Sup. 33 (2000) 132–209]. Then to get the usual sublinear growth on the hyperbolic part of the profiles, for every corrector, we consider , the space of bounded functions decomposing into a sum of pure transports and a “quasi compactly” supported part. We make a detailed analysis on the nonlinear interactions on which leads us to make a restriction on the set of resonant phases. We finally give a convergence result which justifies the use of “quasi compactly” supported profiles. [Copyright &y& Elsevier]

Abstract: This paper is concerned with Schrödinger equations whose principal operators are elliptic. Under certain degenerate condition we show the estimate of an oscillatory integral related to the solution operator in free case, and then employ fractionally integrated groups to obtain the estimate of solutions for the initial data belonging to a dense subspace of in the case of integrable potentials, which improves the corresponding result in [M. Balabane, H.A. Emami-Rad, estimates for Schrödinger evolution equations, Trans. Amer. Math. Soc. 292 (1985) 357–373]. [Copyright &y& Elsevier]

Abstract: The present paper proves the existence and the asymptotic stability of a stationary solution to the initial boundary value problem for a quantum hydrodynamic model of semiconductors over a one-dimensional bounded domain. We also discuss on a singular limit from this model to a classical hydrodynamic model without quantum effects. Precisely, we prove that a solution for the quantum model converges to that for the hydrodynamic model as the Planck constant tends to zero. Here we adopt a non-linear boundary condition which means quantum effect vanishes on the boundary. In the previous researches, the existence and the asymptotic stability of a stationary solution are proved under the assumption that a doping profile is flat, which makes the stationary solution also flat. However, the typical doping profile in actual devices does not satisfy this assumption. Thus, we prove the above theorems without this flatness assumption. Firstly, the existence of the stationary solution is proved by the Leray–Schauder fixed-point theorem. Secondly, we show the asymptotic stability theorem by using an elementary energy method, where the equation for an energy form plays an essential role. Finally, the classical limit is considered by using the energy method again. [Copyright &y& Elsevier]

Abstract: Consider a family of planar systems having a center at the origin and assume that for they have an isochronous center. Firstly, we give an explicit formula for the first order term in ε of the derivative of the period function. We apply this formula to prove that, up to first order in ε, at most one critical period bifurcates from the periodic orbits of isochronous quadratic systems when we perturb them inside the class of quadratic reversible centers. Moreover necessary and sufficient conditions for the existence of this critical period are explicitly given. From the tools developed in this paper we also provide a new characterization of planar isochronous centers. [Copyright &y& Elsevier]

Abstract: In this paper we consider dynamical systems generated by a diffeomorphism F defined on an open subset of , and give conditions over F which imply that their dynamics can be understood by studying the flow of an associated differential equation, , also defined on . In particular the case where F has functionally independent first integrals is considered. In this case X is constructed by imposing that it shares with F the same set of first integrals and that the functional equation , , has some non-zero solution, μ. Several examples for are presented, most of them coming from several well-known difference equations. [Copyright &y& Elsevier]

Abstract: This paper examines the time-dependent double diffusive convection in Brinkman flow in a semi-infinite cylinder. Under appropriate initial and boundary conditions the authors establish exponential decay of solutions in “energy” norm with distance from the finite end of the cylinder. [Copyright &y& Elsevier]

Abstract: This paper presents a partial classification for type-changing symplectic Monge–Ampère partial differential equations (PDEs) that possess an infinite set of first-order intermediate PDEs. The normal forms will be quasi-linear evolution equations whose types change from hyperbolic to either parabolic or to zero. The zero points can be viewed as analogous to singular points in ordinary differential equations. In some cases, intermediate PDEs can be used to establish existence of solutions for ill-posed initial value problems. [Copyright &y& Elsevier]

Abstract: Consider in this paper a linear skew-product system where or , and is a topological dynamical system on a compact metrizable space W, and where satisfies the cocycle condition based on θ and is continuously differentiable in t if . We show that ‘semi λ-exponential dichotomy’ of implies ‘λ-exponential dichotomy.’ Precisely, if Θ has no Lyapunov exponent λ and is almost uniformly λ-contracting along the λ-stable direction and if is constant a.e., then Θ is almost λ-exponentially dichotomous. To prove this, we first use Liao''s spectrum theorem, which gives integral expression of the Lyapunov exponents, and then use the semi-uniform ergodic theorem by Sturman and Stark, which allows one to derive uniform estimates from nonuniform ones. As a consequence, we obtain the open-and-dense hyperbolicity of eventual -cocycles based on a uniquely ergodic endomorphism, and of -cocycles based on a uniquely ergodic equi-continuous endomorphism, respectively. On the other hand, in the sense of -topology we obtain the density of -cocycles having positive Lyapunov exponent based on a minimal subshift satisfying the Boshernitzan condition. [Copyright &y& Elsevier]

Abstract: We construct a single transonic shock wave pattern in an infinite nozzle asymptotically converging to a cylinder, which is close to a uniform transonic shock wave. In other words, suppose there is a uniform transonic shock wave in an infinite cylinder nozzle which can be constructed easily, if we perturbed the supersonic incoming flow and the infinite nozzle a little bit, we can obtain a transonic wave near the uniform one. As a consequence, we can show that the uniform transonic wave is stable with respect to the perturbation of the incoming flow and nozzle wall. Based on the theory of [G.Q. Chen, M. Feldman, Existence and stability of multi-dimensional transonic flows through an infinite nozzle of arbitrary cross-sections, Arch. Ration. Mech. Anal. 184 (2007) 185–242], the crucial parts of this paper are to derive the uniform Schauder estimates of the linear elliptic equation for the infinite nozzle asymptotically converging to a cylinder. [Copyright &y& Elsevier]

Abstract: In this paper we consider the stationary Navier–Stokes equations in a bounded domain with a variable viscosity. We prove that one can uniquely determine the viscosity function from the knowledge of boundary data. [Copyright &y& Elsevier]

Abstract: In this paper we study the limit cycles of the Liénard differential system of the form , or its equivalent system , . We provide sufficient conditions in order that the system exhibits at least n or exactly n limit cycles. [Copyright &y& Elsevier]

Abstract: We study the existence of radial ground state solutions for the problem , . It is known that this problem has infinitely many ground states when , while no solutions exist if . A question raised by Ni and Serrin in [W.-M. Ni, J. Serrin, Existence and non-existence theorems for ground states for quasilinear partial differential equations, Atti Convegni Lincei 77 (1985) 231–257] is whether or not ground state solutions exist for . In this paper we prove the existence of a large, finite number of ground states with fast decay as provided that q lies below but close enough to the critical exponent . These solutions develop a bubble-tower profile as q approaches the critical exponent. [Copyright &y& Elsevier]

Abstract: We study a large time behavior of a solution to the initial boundary value problem for an isentropic and compressible viscous fluid in a one-dimensional half space. The unique existence and the asymptotic stability of a stationary solution are proved by S. Kawashima, S. Nishibata and P. Zhu for an outflow problem where the fluid blows out through the boundary. The main concern of the present paper is to investigate a convergence rate of a solution toward the stationary solution. For the supersonic flow at spatial infinity, we obtain an algebraic or an exponential decay rate. Precisely, if an initial perturbation decays with the algebraic or the exponential rate in the spatial asymptotic point, the solution converges to the corresponding stationary solution with the same rate in time as time tends to infinity. An algebraic convergence rate is also obtained for the transonic flow. These results are proved by the weighted energy method. [Copyright &y& Elsevier]

Abstract: In this paper we consider the one-dimensional p-Laplacian boundary value problem on time scales where is p-Laplacian operator, i.e., , . Some new results are obtained for the existence of at least single, twin or triple positive solutions of the above problem by using Krasnosel''skii''s fixed point theorem, new fixed point theorem due to Avery and Henderson and Leggett–Williams fixed point theorem. This is probably the first time the existence of positive solutions of one-dimensional p-Laplacian boundary value problems on time scales has been studied. [Copyright &y& Elsevier]

Abstract: In this paper a half space problem for the one-dimensional Boltzmann equation with specular reflective boundary condition is investigated. It is shown that the solution of the Boltzmann equation time-asymptotically converges to a global Maxwellian under some initial conditions. Furthermore, a time-decay rate is also obtained. [Copyright &y& Elsevier]

Abstract: In this paper, we establish a generalized Hölder''s or interpolation inequality for weighted spaces in which the weights are non-necessarily homogeneous. We apply it to the stabilization of some damped wave-like evolution equations. This allows obtaining explicit decay rates for smooth solutions for more general classes of damping operators. In particular, for models, we can give an explicit decay estimate for pointwise damping mechanisms supported on any strategic point. [Copyright &y& Elsevier]

Abstract: In this paper we study the existence, uniqueness and propagation of regularity to infinite order partial differential evolution equations. Our approach is essentially functional and brings interesting results even when we restrict ourselves to finite order equations. [Copyright &y& Elsevier]

Abstract: In this paper we prove the existence and multiplicity of homoclinic orbits for first order Hamiltonian systems of the form where is asymptotically linear at ∞ and is not assumed to be periodic. [Copyright &y& Elsevier]

Abstract: We establish the existence of unique smooth center manifolds for ordinary differential equations in Banach spaces, assuming that admits a nonuniform exponential trichotomy. This allows us to show the existence of unique smooth center manifolds for the nonuniformly partially hyperbolic trajectories. In addition, we prove that the center manifolds are as regular as the vector field. Our proof of the smoothness of the manifolds uses a single fixed point problem in an appropriate complete metric space. To the best of our knowledge we establish in this paper the first smooth center manifold theorem in the nonuniform setting. [Copyright &y& Elsevier]

Abstract: In this paper, we study a free boundary problem for compressible spherically symmetric Navier–Stokes equations without a solid core. Under certain assumptions imposed on the initial data, we obtain the global existence and uniqueness of the weak solution, give some uniform bounds (with respect to time) of the solution and show that it converges to a stationary one as time tends to infinity. Moreover, we obtain the stabilization rate estimates of exponential type in -norm and weighted -norm of the solution by constructing some Lyapunov functionals. The results show that such system is stable under the small perturbations, and could be applied to the astrophysics. [Copyright &y& Elsevier]

Abstract: This paper deals with the existence of curved front solution of a partial differential equation coming from a mathematical model of stroke. The equation is of reaction–diffusion type in a cylinder of radius R and of diffusion and absorption type outside of the cylinder. We prove the nonexistence of a travelling front when R is small enough and the existence if R is large enough using a recent energy method. We construct the travelling front as the limit in time of a solution with a well-chosen initial condition, in a travelling referential. [Copyright &y& Elsevier]

Abstract: In this paper we study unique continuation properties of solutions to higher (fifth) order nonlinear dispersive models. The aim is to show that if the difference of two solutions of the equations, , decays sufficiently fast at infinity at two different times, then . [Copyright &y& Elsevier]

Abstract: Consider the periodic solutions of autonomous Hamiltonian systems on the given compact energy hypersurface . If Σ is convex or star-shaped, there have been many remarkable contributions for existence and multiplicity of periodic solutions. It is a hard problem to discuss the multiplicity on general hypersurfaces of contact type. In this paper we prove a multiplicity result for periodic solutions on a special class of hypersurfaces of contact type more general than star-shaped ones. [Copyright &y& Elsevier]

Abstract: In this paper, we study the biharmonic equation in the half-space , with . We prove in theory, with , existence and uniqueness results. We consider data and give solutions which live in weighted Sobolev spaces. [Copyright &y& Elsevier]

Abstract: In this paper we present new stability and extensibility results for skew-product semiflows with a minimal base flow. In particular, we describe the structure of uniformly stable and uniformly asymptotically stable sets admitting backwards orbits and the structure of omega-limit sets. As an application, the occurrence of almost periodic and almost automorphic dynamics for monotone non-autonomous infinite delay functional differential equations is analyzed. [Copyright &y& Elsevier]

Abstract: In this paper we discuss mesoscopic models describing pattern formation mechanisms for a prototypical model of surface processes that involves multiple microscopic mechanisms. We focus on a mean field partial differential equation, which contains qualitatively microscopic information on particle–particle interactions and multiple particle dynamics, and we rigorously derive the macroscopic cluster evolution laws and transport structure. We show that the motion by mean curvature is given by , where k is the mean curvature, σ is the surface tension and μ is an effective mobility that depends on the presence of the multiple mechanisms and speeds up the cluster evolution. This is in contrast with the Allen–Cahn equation where . [Copyright &y& Elsevier]

Abstract: In this paper we consider an initial boundary-value problem related to the electrodynamics of vibrating elastic media. The aim is to prove an existence and uniqueness result for a model describing the nonlinear interactions of the electromagnetic and elastic waves. We assume that the motion of the continuum occurs at velocities that are much smaller than the propagation velocity of the electromagnetic waves through the elastic medium. The model under study consists of two coupled differential equations, one of them is the hyperbolic equation (an analog of the Lamé system) and another one is the parabolic equation (an analog of the diffusion Maxwell system). One stability result is proved too. [Copyright &y& Elsevier]

Abstract: In this paper we prove a result on lower semicontinuity of pullback attractors for dynamical systems given by semilinear differential equations in a Banach space. The situation considered is such that the perturbed dynamical system is non-autonomous whereas the limiting dynamical system is autonomous and has an attractor given as union of unstable manifold of hyperbolic equilibrium points. Starting with a semilinear autonomous equation with a hyperbolic equilibrium solution and introducing a very small non-autonomous perturbation we prove the existence of a hyperbolic global solution for the perturbed equation near this equilibrium. Then we prove that the local unstable and stable manifolds associated to them are given as graphs (roughness of dichotomy plays a fundamental role here). Moreover, we prove the continuity of this local unstable and stable manifolds with respect to the perturbation. With that result we conclude the lower semicontinuity of pullback attractors. [Copyright &y& Elsevier]

Abstract: In this paper, one-dimensional (1D) nonlinear Schrödinger equation with the periodic boundary condition is considered. It is proved that for each given constant potential m and each prescribed integer , the equation admits a Whitney smooth family of small amplitude, time quasi-periodic solutions with N Diophantine frequencies. The proof is based on a partial Birkhoff normal form reduction and an improved KAM method. [Copyright &y& Elsevier]

Abstract: In this paper we study the number of critical points that the period function of a center of a classical Liénard equation can have. Centers of classical Liénard equations are related to scalar differential equations , with f an odd polynomial, let us say of degree . We show that the existence of a finite upperbound on the number of critical periods, only depending on the value of ℓ, can be reduced to the study of slow–fast Liénard equations close to their limiting layer equations. We show that near the central system of degree the number of critical periods is at most . We show the occurrence of slow–fast Liénard systems exhibiting critical periods, elucidating a qualitative process behind the occurrence of critical periods. It all provides evidence for conjecturing that is a sharp upperbound on the number of critical periods. We also show that the number of critical periods, multiplicity taken into account, is always even. [Copyright &y& Elsevier]

Abstract: In this paper, we give a global well-posedness result for the two-dimensional Boussinesq system with partial viscosity, when the initial data and . [Copyright &y& Elsevier]

Abstract: This article is motivated by the fact that very little is known about variational inequalities of general principal differential operators with critical growth. The concentration compactness principle of P.L. Lions [P.L. Lions, The concentration compactness principle in the calculus of variation. The limit case I, Rev. Mat. Iberoamericana 1 (1) (1985) 145–201; P.L. Lions, The concentration compactness principle in the calculus of variation. The limit case II, Rev. Mat. Iberoamericana 1 (2) (1985) 45–121] is a widely applied technique in the analysis of Palais–Smale sequences. For critical growth problems involving principal differential operators Laplacian or p-Laplacian, much has been accomplished in recent years, whereas very little has been done for problems involving more general main differential operators since a nonlinearity is observed between the corresponding functional and measure μ introduced in the concentration compactness method. In this paper, we investigate a Leray–Lions type operator and behaviors of its sequence. [Copyright &y& Elsevier]

Abstract: We analyze the dynamics of a reaction–diffusion equation with homogeneous Neumann boundary conditions in a dumbbell domain. We provide an appropriate functional setting to treat this problem and, as a first step, we show in this paper the continuity of the set of equilibria and of its linear unstable manifolds. [Copyright &y& Elsevier]

Abstract: This paper is devoted to regularity results and geometric properties of the singular set of the parabolic obstacle problem with variable right-hand side. Making use of a monotonicity formula for singular points, we prove the uniqueness of blow-up limits at singular points. These results apply to parabolic obstacle problem with variable coefficients. [Copyright &y& Elsevier]

Abstract: In this paper we deal with discontinuous vector fields on and we prove that the analysis of their local behavior around a typical singularity can be treated via singular perturbation. The regularization process developed by Sotomayor and Teixeira is crucial for the development of this work. [Copyright &y& Elsevier]

Abstract: We let Ω be a smooth bounded domain of and a sequence of functions such that in . We consider a sequence of functions such that in Ω for all . We address in this paper the question of the asymptotic behavior of the ''s when . The corresponding problem in dimension 2 was considered by Brézis and Merle, and Li and Shafrir (among others), where a blow-up phenomenon was described and where a quantization of this blow-up was proved. Surprisingly, as shown by Adimurthi, Struwe and the author in [Adimurthi, F. Robert and M. Struwe, Concentration phenomena for Liouville equations in dimension four, J. Eur. Math. Soc., in press, available on http://www-math.unice.fr/~frobert], a similar quantization phenomenon does not hold for this fourth-order problem. Assuming that the ''s are radially symmetrical, we push further the analysis of the mentioned work. We prove that there are exactly three types of blow-up and we describe each type in a very detailed way. [Copyright &y& Elsevier]

Abstract: In this paper, some sufficient conditions for the existence of solutions to the boundary value problems of a class of second order difference equation are obtained by using the critical point theory. [Copyright &y& Elsevier]

Abstract: This paper is concerned with Hamiltonian systems of linear differential equations with periodic coefficients under a small perturbation. It is well known that Krein''s formula determines the behavior of definite multipliers on the unit circle and is quite useful in studying the (strong) stability of Hamiltonian system. Our aim is to give a simple formula that determines the behavior of indefinite multipliers with two multiplicity, which is generic case. The result does not require analyticity and is proved directly. Applying this formula, we obtain instability criteria for solutions with periodic structure in nonlinear dissipative systems such as the Swift–Hohenberg equation and reaction–diffusion systems of activator–inhibitor type. [Copyright &y& Elsevier]

Abstract: In this paper we consider the singular value decomposition (SVD) of a fundamental matrix solution in order to approximate the Lyapunov and exponential dichotomy spectra of a given system. One of our main results is to prove that SVD techniques are sound approaches for systems with stable and distinct Lyapunov exponents. We also show how the information which emerges with the SVD techniques can be used to obtain information on the growth directions associated to given spectral intervals. [Copyright &y& Elsevier]

Abstract: In this paper we consider nonlinear boundary value problems whose simplest model is the following: where is a summable function in Ω (bounded open set in , ), , and . [Copyright &y& Elsevier]

Abstract: In this paper we prove the existence of a weak solution of the incompressible 2D Euler equations in the exterior of a reflection symmetric smooth bluff body with symmetric initial flow corresponding to vortex sheet type data whose vorticity is of distinguished sign on each side of the symmetry axis. This work extends the results proved for full plane flow by the authors in [M.C. Lopes Filho, H.J. Nussenzveig Lopes, Z. Xin, Existence of vortex sheets with reflection symmetry in two space dimensions, Arch. Ration. Mech. Anal. 158 (3) (2001) 235–257]. [Copyright &y& Elsevier]

Abstract: We construct solutions for 2- and 3-D stochastic nonhomogeneous incompressible Navier–Stokes equations with general multiplicative noise. These equations model the velocity of a mixture of incompressible fluids of varying density, influenced by random external forces that involve feedback; that is, multiplicative noise. Weak solutions for the corresponding deterministic equations were first found by Kazhikhov [A.V. Kazhikhov, Solvability of the initial and boundary-value problem for the equations of motion of an inhomogeneous viscous incompressible fluid, Soviet Phys. Dokl. 19 (6) (1974) 331–332; English translation of the paper in: Dokl. Akad. Nauk SSSR 216 (6) (1974) 1240–1243]. A stochastic version with additive noise was solved by Yashima [H.F. Yashima, Equations de Navier–Stokes stochastiques non homogènes et applications, Thesis, Scuola Normale Superiore, Pisa, 1992]. The methods here extend the Loeb space techniques used to obtain the first general solutions of the stochastic Navier–Stokes equations with multiplicative noise in the homogeneous case [M. Capiński, N.J. Cutland, Stochastic Navier–Stokes equations, Applicandae Math. 25 (1991) 59–85]. The solutions display more regularity in the 2D case. The methods also give a simpler proof of the basic existence result of Kazhikhov. [Copyright &y& Elsevier]

Abstract: In this paper we study bifurcation from simple eigenvalues for systems of differential equations; we prove the existence of global bifurcating branches of solutions on which the Maslov index of suitable associated linear systems is preserved. [Copyright &y& Elsevier]

Abstract: Under a certain restriction, singular first-order linear partial differential equations of nilpotent type with two variables are divided into two classes. In the previous paper Part I, we dealt with the one class, and comprehended that there was a close affinity between the Borel summability of divergent solutions and global analytic continuation properties for coefficients. In this Part II, we give a similar consideration on the other class. More precise global estimates than those given in Part I for coefficients will be required to prove the Borel summability of divergent solutions. [Copyright &y& Elsevier]

Abstract: For finite-dimensional bifurcation problems, it is well known that it is possible to compute normal forms which possess nice symmetry properties. Oftentimes, these symmetries may allow for a partial decoupling of the normal form into a so-called “radial” part and an “angular” part. Analysis of the radial part usually gives an enormous amount of valuable information about the bifurcation and its unfoldings. In this paper, we are interested in the case where such bifurcations occur in retarded functional differential equations, and we revisit the realizability and restrictions problem for the class of radial equations by nonlinear delay-differential equations. Our analysis allows us to recover and considerably generalize recent results by Faria and Magalhães [T. Faria, L.T. Magalhães, Restrictions on the possible flows of scalar retarded functional differential equations in neighborhoods of singularities, J. Dynam. Differential Equations 8 (1996) 35–70] and by Buono and Bélair [P.-L. Buono, J. Bélair, Restrictions and unfolding of double Hopf bifurcation in functional differential equations, J. Differential Equations 189 (2003) 234–266]. [Copyright &y& Elsevier]

Abstract: In this paper we introduce the generalized eigenvalues of a quasilinear elliptic system of resonant type. We prove the existence of infinitely many continuous eigencurves, which are obtained by variational methods. For the one-dimensional problem, we obtain an hyperbolic type function defining a region which contains all the generalized eigenvalues (variational or not), and the proof is based on a suitable generalization of Lyapunov''s inequality for systems of ordinary differential equations. We also obtain a family of curves bounding by above the kth variational eigencurve. [Copyright &y& Elsevier]

Abstract: In this paper we are interested in establishing up-to boundary uniform estimates for the one phase singular perturbation problem involving a nonlinear singular/degenerate elliptic operator. Our main result states: if is a domain, for some and verifies where , and with some positive constants B and M, then there exists a constant independent of ε such that . [Copyright &y& Elsevier]

Abstract: This paper is devoted to the study of differential inclusions, particularly discontinuous perturbed sweeping processes in the infinite-dimensional setting. On the one hand, the sets involved are assumed to be prox-regular and to have a variation given by a function which is of bounded variation and right continuous. On the other hand, the perturbation satisfies a linear growth condition with respect to a fixed compact subset. Finally, the case where the sets move in an absolutely continuous way is recovered as a consequence. [Copyright &y& Elsevier]