
Luisa MALAGUTI
Professore Ordinario Dipartimento di Scienze e Metodi dell'Ingegneria

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2023
 The role of convection in the existence of wavefronts for biased movements
[Articolo su rivista]
Berti, Diego; Corli, Andrea; Malaguti, Luisa
abstract
We investigate a model, inspired by Johnston et al. (2017) to describe the
movement of a biological population which consists of isolated and grouped
organisms. We introduce biases in the movements and then obtain a scalar
reaction–diffusion equation that includes a convective term as a consequence
of the biases. We focus on the case the diffusivity makes the parabolic equation
of forward–backward–forward type and the reaction term models a strong Allee
effect, with the Allee parameter lying between the two internal zeros of the
diffusion. In such a case, the unbiased equation (i.e., without convection) possesses
no smooth travelingwave solutions; on the contrary, in the presence of
convection, we show that travelingwave solutions do exist for some significant
choices of the parameters.We also study the sign of their speeds, which provides
information on the long term behavior of the population, namely, its survival or
extinction.
2022
 Diffusion–convection reaction equations with signchanging diffusivity and bistable reaction term
[Articolo su rivista]
Berti, Diego; Corli, Andrea; Malaguti, Luisa
abstract
We consider a reaction–diffusion equation with a convection term in one space variable, where the diffusion changes sign from the positive to the negative and the reaction term is bistable. We study the existence of wavefront solutions, their uniqueness and regularity. The presence of convection reveals several new features of wavefronts: according to the mutual positions of the diffusivity and reaction, profiles can occur either for a single value of the speed or for a bounded interval of such values; uniqueness (up to shifts) is lost; moreover, plateaus of arbitrary length can appear; profiles can be singular where the diffusion vanishes.
2022
 Lpexact controllability of partial differential equations with nonlocal terms
[Articolo su rivista]
Malaguti, Luisa; Perrotta, Stefania; Taddei, Valentina
abstract
The paper deals with the exact controllability of partial differential equations by linear controls. The discussion takes place in infinite dimensional state spaces since these equations are considered in their abstract formulation as semilinear equations. The linear parts are densely defined and generate strongly continuous semigroups. The nonlinear terms may also include a nonlocal part. The solutions satisfy nonlocal properties, which are possibly nonlinear. The states belong to Banach spaces with a Schauder basis and the results exploit topological methods. The novelty of this investigation is in the use of an approximation solvability method which involves a sequence of controllability problems in finitedimensional spaces. The exact controllability of nonlocal solutions can be proved, with controls in Lp spaces, 1<∞. The results apply to the study of the exact controllability for the transport equation in arbitrary Euclidean spaces and for the equation of the nonlinear wave equation.
2022
 Wavefront solutions to reactionconvection equations with PeronaMalik diffusion
[Articolo su rivista]
Corli, A.; Malaguti, L.; Sovrano, E.
abstract
We study a nonlinear reactionconvection equation with a degenerate diffusion of PeronaMalik's type and a monostable reaction term. Under quite general assumptions, we show the presence of wavefront solutions and prove their main properties. In particular, such wavefronts exist for every speed in a closed halfline and we give estimates of the threshold speed. The wavefront profiles are also strictly monotone and their slopes are uniformly bounded by the critical values of the diffusion.
2021
 Saturated Fronts in Crowds Dynamics
[Articolo su rivista]
Campos, Juan; Corli, Andrea; Malaguti, Luisa
abstract
We consider a degenerate scalar parabolic equation, in one spatial dimension, of fluxsaturated type. The equation also contains a convective term. We study the existence and regularity of travelingwave solutions; in particular we show that they can be discontinuous. Uniqueness is recovered by requiring an entropy condition, and entropic solutions turn out to be the vanishingdiffusion limits of travelingwave solutions to the equation with an additional nondegenerate diffusion. Applications to crowds dynamics, which motivated the present research, are also provided
2021
 Wavefronts for degenerate diffusionconvection reaction equations with signchanging diffusivity
[Articolo su rivista]
Berti, Diego; Corli, Andrea; Malaguti, Luisa
abstract
We consider in this paper a diffusionconvection reaction equation in one space dimension. The main assumptions are about the reaction term, which is monostable, and the diffusivity, which changes sign once or even more than once; then, we deal with a forwardbackward parabolic equation. Our main results concern the existence of globally defined traveling waves, which connect two equilibria and cross both regions where the diffusivity is positive and regions where it is negative. We also investigate the monotony of the profiles and show the appearance of sharp behaviors at the points where the diffusivity degenerates. In particular, if such points are interior points, then the sharp behaviors are new and unusual.
2021
 Wavefronts in Traffic Flows and Crowds Dynamics
[Capitolo/Saggio]
Corli, Andrea; Malaguti, Luisa
abstract
In this paper we give an overview of some recent results concerning partial differential equations modeling collective movements, namely, vehicular traffic flows and crowds dynamics. The focus is on travelingwave solutions to degenerate parabolic equations in one space dimension, even if we briefly discuss models based on different equations. The case of networks is also taken into consideration. The parabolic degeneration opens the possibilities of several different behaviors of the travelingwave solutions, which are investigated in details.
2020
 NONLOCAL SOLUTIONS AND CONTROLLABILITY OF SCHRODINGER EVOLUTION EQUATION
[Articolo su rivista]
Malaguti, L; Yoshii, K
abstract
The paper deals with semilinear evolution equations in complex Hilbert spaces. Nonlocal associated Cauchy problems are studied and the existence and uniqueness of classical solutions is proved. The controllability is investigated too and the topological structure of the controllable set discussed. The results are applied to nonlinear Schrodinger evolution equations with time dependent potential. Several examples of nonlocal conditions are proposed. The evolution system associated to the linear part is not compact and the theory developed in OkazawaYoshii [21] for its study is used. The proofs involve the SchauderTychonoff fixed point theorem and no strong compactness is assumed on the nonlinear part.
2020
 Uniqueness and nonuniqueness of fronts for degenerate diffusionconvection reaction equations
[Articolo su rivista]
Berti, D.; Corli, A.; Malaguti, L.
abstract
We consider a scalar parabolic equation in one spatial dimension. The equation
is constituted by a convective term, a reaction term with one or two equilibria, and
a positive diffusivity which can however vanish. We prove the existence and several
properties of travelingwave solutions to such an equation. In particular, we provide a
sharp estimate for the minimal speed of the profiles and improve previous results about
the regularity of wavefronts. Moreover, we show the existence of an infinite number of
semiwavefronts with the same speed.
2019
 Controllability in Dynamics of Diffusion Processes with Nonlocal Conditions
[Articolo su rivista]
Malaguti, Luisa; Rykaczewski, Krzysztof; Taddei, Valentina
abstract
The paper deals with semilinear evolution equations in Banach spaces. By means of linear control terms, the controllability problem is investigated and the solutions satisfy suitable nonlocal conditions. The Cauchy multipoint condition and the mean value condition are included in the present discussion. The final configuration is always achieved with a control with minimum norm. The results make use of fixed point techniques; two different approaches are proposed, depending on the use of norm or weak topology in the state space. The discussion is completed with some applications to dynamics of diffusion processes.
2019
 Exact controllability of infinite dimensional systems with controls of minimal norm
[Articolo su rivista]
Malaguti, Luisa; Perrotta, Stefania; Taddei, Valentina
abstract
The paper deals with the exact controllability of a semilinear system in a separable Hilbert space. A bounded linear part is considered and a linear control introduced. The state space is compactly embedded in a Banach space and the nonlinear term is continuous in its state variable in the norm of the Banach space. An infinite sequence of finite dimen sional controllability problems is introduced and the solution is obtained by a limiting procedure. To the best of our knowledge, the method is new in controllability theory. An application to an integrodifferential system in euclidean spaces completes the discussion.
2019
 Nonlocal solutions of parabolic equations with strongly elliptic differential operators
[Articolo su rivista]
Benedetti, Irene; Malaguti, Luisa; Taddei, Valentina
abstract
The paper deals with second order parabolic equations on bounded domains with Dirichlet conditions in arbitrary Euclidean spaces. Their interest comes from being models for describing reaction–diffusion processes in several frameworks. A linear diffusion term in divergence form is included which generates a strongly elliptic differential operator. A further linear part, of integral type, is present which accounts of nonlocal diffusion behaviours. The main result provides a unifying method for studying the existence and localization of solutions satisfying nonlocal associated boundary conditions. The Cauchy multipoint and the mean value conditions are included in this investigation. The problem is transformed into its abstract setting and the proofs are based on the homotopic invariance of the Leray–Schauder topological degree. A bounding function (i.e. Lyapunovlike function) theory is developed, which is new in this infinite dimensional context. It allows that the associated vector fields have no fixed points on the boundary of their domains and then it makes possible the use of a degree argument.
2019
 Viscous profiles in models of collective movement with negative diffusivity
[Articolo su rivista]
Corli, Andrea; Malaguti, Luisa
abstract
In this paper, we consider an advection–diffusion equation, in one space dimension, whose diffusivity can be negative. Such equations arise in particular in the modeling of vehicular traffic flows or crowds dynamics, where a negative diffusivity simulates aggregation phenomena. We focus on travelingwave solutions that connect two states whose diffusivity has different signs; under some geometric conditions, we prove the existence, uniqueness (in a suitable class of solutions avoiding plateaus) and sharpness of the corresponding profiles. Such results are then extended to the case of end states where the diffusivity is positive, but it becomes negative in some interval between them. Also the vanishing viscosity limit is considered. At last, we provide and discuss several examples of diffusivities that change sign and show that our conditions are satisfied for a large class of them in correspondence of real data.
2017
 An approximation solvability method for nonlocal differential problems in Hilbert spaces
[Articolo su rivista]
Benedetti, Irene; Loi, Nguyen V.; Malaguti, Luisa; Obukhovskii, Valeri
abstract
A new approach is developed for the solvability of nonlocal problems in Hilbert spaces associated to nonlinear differential equations. It is based on a joint combination of the degree theory with the approximation solvability method and the bounding functions technique. No compactness or condensivity condition on the nonlinearities is assumed. Some applications of the abstract result to the study of nonlocal problems for integrodifferential equations and systems of integrodifferential equations are then showed. A generalization of the result by using nonsmooth bounding functions is given.
2017
 Nonlocal diffusion second order partial differential equations
[Articolo su rivista]
Benedetti, Irene; Loi, Nguyen Van; Malaguti, Luisa; Taddei, Valentina
abstract
The paper deals with a second order integropartial differential equation in RnRn with a nonlocal, degenerate diffusion term. Nonlocal conditions, such as the Cauchy multipoint and the weighted mean value problem, are investigated. The existence of periodic solutions is also studied. The dynamic is transformed into an abstract setting and the results come from an approximation solvability method. It combines a Schauder degree argument with an Hartmantype inequality and it involves a ScorzaDragoni type result. The compact embedding of a suitable Sobolev space in the corresponding Lebesgue space is the unique amount of compactness which is needed in this discussion. The solutions are located in bounded sets and they are limits of functions with values in finitely dimensional spaces.
2017
 Sharp profiles in models of collective movement.
[Articolo su rivista]
Corli, Andrea; DI RUVO, Lorenzo; Malaguti, Luisa
abstract
We consider a parabolic partial differential equation that can be understood as a simple model for crowds flows. Our main assumption is that the diffusivity and the source/sink term vanish at the same point; the nonhomogeneous term is different from zero at any other point and so the equation is not monostable. We investigate the existence, regularity and monotone properties of semiwavefront solutions as well as their convergence to wavefront solutions.
2017
 Traveling waves for degenerate diffusive equations on networks
[Articolo su rivista]
Corli, Andrea; Ruvo, Lorenzo di; MALAGUTI, Luisa; Rosini, Massimiliano D.
abstract
In this paper we consider a scalar parabolic equation on a star graph; the model is quite general but what we have in mind is the description of traffic flows at a crossroad. In particular, we do not necessarily require the continuity of the unknown function at the node of the graph and, moreover, the diffusivity can be degenerate. Our main result concerns a necessary and sufficient algebraic condition for the existence of traveling waves in the graph. We also study in great detail some examples corresponding to quadratic and logarithmic flux functions, for different diffusivities, to which our results apply.
2016
 Nonsmooth feedback controls of nonlocal dispersal models
[Articolo su rivista]
Malaguti, Luisa; Rubbioni, Paola
abstract
The paper deals with a nonlocal diffusion equation which is a model for biological invasion and disease spread. A nonsmooth feedback control term is included and the existence of controlled dynamics is proved, satisfying different kinds of nonlocal condition. Jump discontinuities appear in the process. The existence of optimal control strategies is also shown, under suitably regular control functionals. The investigation makes use of techniques of multivalued analysis and is based on the degree theory for condensing operators in Hilbert spaces.
2016
 Semilinear delay evolution equations with measures subjected to nonlocal initial conditions
[Articolo su rivista]
Benedetti, I.; Malaguti, Luisa; Taddei, Valentina; Vrabie, I. I.
abstract
We prove a global existence result for bounded solutions to a class of abstract semilinear delay evolution equations with measures subjected to nonlocal initial data of the form: du(t)={Au(t)+f(t,u t )}dt+dg(t) with t∈R+ and u(t)=h(u)(t) for t∈[−τ,0], with τ≥0. The operator A:D(A)⊆X→X is the infinitesimal generator of a C0 semigroup, f:R+ ×R([−τ,0];X)→X is continuous, g∈BVloc (R+ ;X) and h:Rb (R + ;X)→R([−τ,0];X) is nonexpansive.
2016
 Semiwavefront solutions in models of collective movements with densitydependent diffusivity
[Articolo su rivista]
Corli, Andrea; Malaguti, Luisa
abstract
This paper deals with a nonhomogeneous scalar parabolic equation with possibly degenerate diffusion term; the process has only one stationary state. The equation can be interpreted as modeling collective movements (crowd dynamics, for instance). We show the existence of semiwavefront solutions for every wave speed; their properties are investigated. Proofs exploit comparisontype techniques and are carried out in the case of one spatial variable; the extension to the general case is straightforward.
2015
 Hartmantype conditions for multivalued Dirichlet problem in abstract spaces
[Relazione in Atti di Convegno]
Andres, Jan; Malaguti, Luisa; Pavlačková, Martina
abstract
The classical Hartman’s Theorem in for the solvability of the vector Dirichlet problem will be generalized and extended in several directions. We will consider its multivalued versions for Marchaud and upperCarath´eodory righthand sides with only certain amount of compactness in Banach spaces. Advanced topological methods are combined with a bound sets technique. Besides the existence, the localization of solutions can be obtained in this way.
2015
 Nonlocal problems in Hilbert spaces
[Relazione in Atti di Convegno]
Benedetti, Irene; Malaguti, Luisa; Taddei, Valentina
abstract
An existence result for differential inclusions in a separable Hilbert space is furnished. A wide family of nonlocal boundary value problems is treated, including periodic, antiperiodic, mean value and multipoint conditions. The study is based on an approximation solvability method. Advanced topological methods are used as well as a Scorza Dragonitype result for multivalued maps. The conclusions are original also in the singlevalued setting. An application to a nonlocal dispersal model is given.
2014
 Nonlocal Problems for Differential Inclusions in Hilbert Spaces
[Articolo su rivista]
Irene, Benedetti; Nguyen Van, Loi; Malaguti, Luisa
abstract
An existence theorem for differential inclusions in Hilbert spaces with nonlocal conditions is proved. Periodic, antiperiodic, mean value and multipoint conditions are included in this study. The investigation is based on a combination of the approximation solvability method with Hartmantype inequalities. A feedback control problem associated to a first order partial differential equation completes this discussion.
2014
 ScorzaDragoni approach to Dirichlet
problem in Banach spaces
[Articolo su rivista]
Jan, Andres; Malaguti, Luisa; Martina, Pavlacková
abstract
Hartmantype conditions are presented for the solvability of a multivalued Dirichlet
problem in a Banach space by means of topological degree arguments, bounding
functions, and a ScorzaDragoni approximation technique. The required transversality
conditions are strictly localized on the boundaries of given bound sets. The main
existence and localization result is applied to a partial integrodifferential equation
involving possible discontinuities in state variables. Two illustrative examples are
supplied. The comparison with classical singlevalued results in this field is also made.
2013
 Dirichlet problem in Banach spaces: the bound sets approach
[Articolo su rivista]
Jan, Andres; Malaguti, Luisa; Martina, Pavlackova
abstract
The existence and localization result is obtained for a multivalued Dirichlet problem in
a Banach space. The upperCarathéodory and Marchaud righthand sides are treated
separately because in the latter case, the transversality conditions derived by means
of bounding functions can be strictly localized on the boundaries of bound sets.
2013
 Nonlocal semilinear evolution equations without strong compactness: theory and applications
[Articolo su rivista]
Irene, Benedetti; Malaguti, Luisa; Taddei, Valentina
abstract
A semilinear multivalued evolution equation is considered in a reflexive Banach space. The nonlinear term has convex, closed, bounded values and a weakly sequentially closed graph when restricted to its second argument. No strong
compactness is assumed, neither on the evolution operator generated by the linear part, or on the nonlinear term. A wide family of nonlocal associated boundary value problems is investigated by means of a fixed point technique. Applications are given to an optimal feedback control problem, to a nonlinear hyperbolic integrodifferential
equation arising in agestructure population models, and to a multipoint boundary value problem associated to a parabolic partial differential equation.
2012
 A ScorzaDragoni approach to secondorder boundary value problems in abstract spaces
[Articolo su rivista]
J., Andres; Malaguti, Luisa; M., Pavlačková
abstract
The existence and localization of strong (Carathéodory) solutions is proved for a secondorder Floquet problem in a Banachspace. The result is obtained by combining a continuation principle together with a bounding (Liapunovlike) functions approach. Theapplication of the Scorza–Dragoni type technique allows us to use strictly localized transversality conditions.
2012
 Erratum and addendum to "Twopoint b.v.p. for multivalued equations with weakly regular r.h.s."
[Articolo su rivista]
I., Benedetti; Malaguti, Luisa; Taddei, Valentina
abstract
In this paper, we define a topological index for compact multivalued maps in convex metrizable subsets of a locally convex topological vector space in order to correct the proofs of Theorems 4.1 and 4.2 in BenedettiMalagutiTaddei, Nonlinear Anal. 74 (2011) 3657–3670.
2012
 Guidinglike functions for semilinear evolution equations
with retarded nonlinearities
[Articolo su rivista]
S., Cecchini; Malaguti, Luisa
abstract
The paper deals with a semilinear evolution equation in a reflexive and separable Banach space. The nonlinear term is multivalued, upper Caratheodory and it depends on a retarded argument. The existence of global almost exact, i.e. classical, solutions is investigated. The results are based on a continuation principle for condensing multifields and the required transversalities derive from the introduction of suitable generalized guiding functions. As a consequence, the equation has a bounded globally viable set. The results are new also in the lack of retard and in the single valued case. A brief discussion of a nonlocal diffusion model completes this investigation.
2012
 Semilinear evolution equations in abstract spaces and
applications
[Articolo su rivista]
I., Benedetti; Malaguti, Luisa; Taddei, Valentina
abstract
The existence of mild solutions is obtained, for a semilinear
multivalued equation in a reflexive Banach space. Weakly compact valued nonlinear terms are considered, combined with strongly continuous
evolution operators generated by the linear part. A continuation
principle or a fixed point theorem are used, according to the various
regularity and growth conditions assumed. Applications to the study of parabolic and hyperbolic partial differential equations are given.
2011
 Boundary value problem for differential inclusions in fréchet spaces with multiple solutions of the homogeneous problem
[Articolo su rivista]
I., Benedetti; Malaguti, Luisa; Taddei, Valentina
abstract
The paper deals with the multivalued boundary value problemx' Є A(t, x)x + F(t, x) for a.a. t Є [a, b], Mx(a)+Nx(b) = 0 in a separable, reflexive Banach space E. The nonlinearity F is weakly upper semicontinuous in x. We prove the existenceof global solutions in the Sobolev space W1,p([a, b], E) with 1 < p < ∞ endowed with the weak topology. We consider the case of multiple solutions of the associated homogeneouslinearized problem. An example completes the discussion.
2011
 Continuous dependence in front propagation for convective reactiondiffusion models with aggregative movements
[Articolo su rivista]
Malaguti, Luisa; C., Marcelli; S., Matucci
abstract
The paper deals with a degenerate reactiondiffusion equation, including aggregative movements and convective terms. The model also incorporates a real parameter causing the change from a purely diffusive to a diffusiveaggregativeand to a purely aggregative regime. Existence and qualitative properties of traveling wave solutions are investigated and estimates of their threshold speeds are furnished. Further, the continuous dependence of the threshold wave speed andof the wave profiles on a real parameter is studied, both when the process maintains its diffusionaggregation nature and when it switches from it to another regime.
2011
 On secondorder boundary value problems in Banach spaces: a bound sets approach
[Articolo su rivista]
J., Andres; Malaguti, Luisa; M., Pavlačková
abstract
The existence and localization of strong (Carathéodory) solutions is obtained for a secondorder Floquet problem in a Banach space. The combination of applied degree arguments and bounding (Liapunovlike) functions allows some solutions to escape from a given set. The problems concern both semilinear differential equations and inclusions. The main theorem for upperCarathéodory inclusions is separately improved for Marchaudinclusions (i.e. for globally upper semicontinuous righthand sides) in the form of corollary. Three illustrative examples are supplied.
2011
 On the behavior of periodic solutions of planar autonomous Hamiltonian systems with multivalued periodic perturbations
[Articolo su rivista]
O., Makarenkov; Malaguti, Luisa; P., Nistri
abstract
Aim of the paper is to provide a method to analyze the behavior of Tperiodic solutions of a perturbed planar Hamiltonian system near a cycle x_0, of smallest period T, of the unperturbed system. The perturbation is represented by a Tperiodic multivalued map which vanishes as the parameter tends to zero. In several problems from nonsmooth mechanical systems this multivalued perturbation comes from the Filippovregularization of a nonlinear discontinuous Tperiodic term. Through the paper, assuming the existence of a Tperiodic solution of the parameterized problems, under the conditionthat x_0 is a nondegenerate cycle of the linearized unperturbed Hamiltonian system, we provide a formula for the distance between any point x_0(t) and the trajectories of the perturbed period solutions along a transversal direction to x_0(t).
2011
 Strictly localized bounding functions and Floquet boundary value problems
[Articolo su rivista]
S., Cecchini; Malaguti, Luisa; Taddei, Valentina
abstract
Semilinear multivalued equations are considered, in separable Banach spaces with the RadonNikodym property. An effective criterion for the existence of solutions to the associated Floquet boundary value problem is showed. Its proof is obtained combining a continuation principle with a Liapunovlike technique and a ScorzaDragoni type theorem. A strictly localized transversality condition is assumed. The employed method enables to localize the solution values in a not necessarily invariant set; it allows also to introduce nonlinearities with superlinear growth in the state variable.
2011
 Twopoint b.v.p. for multivalued equations with weakly regular r.h.s.
[Articolo su rivista]
I., Benedetti; Malaguti, Luisa; Taddei, Valentina
abstract
A twopoint boundary value problem associated to a semilinear multivalued evolution equation is investigated, in reflexive and separable Banach spaces. To this aim, an original method is proposed based on the use of weak topologies and on a suitable continuation principle in Fréchet spaces. Lyapunovlike functions are introduced, for proving the required transversality condition. The linear part can also depend on the state variable x and the discussion comprises the cases of a nonlinearity with sublinear growth in x or of a noncompact valued one. Some applications are given, to the study of periodic and Floquet boundary value problems of partial integrodifferential equations and inclusionsappearing in dispersal population models. Comparisons are included, with recent related achievements.
2010
 Asymptotic speed of propagation for Fishertype degenerate reactiondiffusionconvection equations
[Articolo su rivista]
Malaguti, Luisa; Ruggerini, Stefano
abstract
The paper deals with the initialvalue problem for the degenerate reactiondiffusionconvection equationu_t + h(u)u_x = (u^m)_xx + f(u), x Є R, t>0, m>1,with f, h continuous and f of Fishertype. By means of comparison type techniques, we prove that the equilibrium u ≡ 1 is an attractor for all solutions with a continuous, bounded, nonnegative initial condition u_0(x) = u(x, 0) ≠ 0. Whenu_0 is also compactly supported and satisfies 0 ≤ u0 ≤ 1, the convergence is such that an asymptotic estimate of the interface can be obtained. The employed techniques involve the theory of travellingwave solutions that we improve in thiscontext. The assumptions on f and h guarantee that the threshold speed wavefront is not stationary and we show that the asymptotic speed of the interface equals this minimal speed.
2010
 Continuous dependence in front propagation of convective reactiondiffusion equations
[Articolo su rivista]
Malaguti, Luisa; C., Marcelli; S., Matucci
abstract
Continuous dependence of the threshold wave speed and of thetravelling wave profiles for reactiondiffusionconvection equationsis here studied with respect to the diffusion, reaction and convection terms.
2010
 Semilinear differential inclusions via weak topologies
[Articolo su rivista]
I., Benedetti; Malaguti, Luisa; Taddei, Valentina
abstract
The paper deals with the multivalued initial value problem x'(t) Є A(t, x)x+ F (t, x) for a.a. t in[a, b], x(a) = x_0 in a separable, reflexive Banach space E. The nonlinearity F is weakly upper semicontinuous in x and the investigation includes the case when both A and F have asuperlinear growth in x. We prove the existence of local and global classical solutions in the Sobolev space W1,p ([a, b], E) with 1 < p < ∞. Introducing a suitably defined Lyapunovlikefunction, we are able to investigate the topological structure of the solution set. Our main tool is a continuation principle in Frechét spaces and we prove the required pushingcondition in two different ways. Some examples complete the discussion.
2009
 Bound sets approach to boundary value problems for vector secondorder differential inclusions
[Articolo su rivista]
J., Andres; M., Kozusníková; Malaguti, Luisa
abstract
A continuation principle is established for the solvability of vector secondorder boundary value problems associated with upperCarathéodory differential inclusions. For Floquet secondorder problems, this principle is combined with a bound sets approach. The viability result is also obtained in this way.
2009
 On boundary value problems in Banach spaces
[Articolo su rivista]
J., Andres; Malaguti, Luisa; Taddei, Valentina
abstract
The paper deals with boundary value problemsassociated to firstorder differential inclusions in Banach spaces. The solvability is investigated in the (strong) Carathèodory sense on compact intervals. To this aim, we develop a general method that relies on degree arguments. This method is still combined with a bound sets technique for checking the behavior of trajectories in the neighborhood of a suitable parametric set of candidate solutions. On this basis, we obtain effective criteria for the existence of solutions of Floquet problems. The existence of entirely bounded solutions is also established by means of a sequence of solutions on compact increasing intervals.
2009
 On the Floquet problem for secondorder Marchaud differential systems
[Articolo su rivista]
J., Andres; M., Kozusníková; Malaguti, Luisa
abstract
Solutions in a given set of the Floquet boundary value problem are investigated for secondorder Marchaud systems. The methods used involve a fixed point index techniquedeveloped by ourselves earlier with a bound sets approach. Since the related bounding (Liapunovlike) functions are strictly localized on the boundaries of parameter sets of candidate solutions, some trajectories are allowed to escape from these sets. The main existence and localization theorem is illustrated by two examples for periodic and antiperiodic problems.
2009
 Periodic Solutions of Semilinear Multivalued and Functional Evolution Equations in Banach Spaces
[Articolo su rivista]
Cecchini, Simone; Malaguti, Luisa
abstract
This paper deals with the semilinear multivalued evolution equationx'(t) + A(t)x(t) Є F(t, x(t)), t Є [a, b] and x Є E,in an arbitrary Banach space E.The linear operators {A(t) : t Є [a, b]} are densely defined on a common domain in E and generate a strongly continuous evolution system. We discuss the existence of mild periodic solutions, also in the case when the nonlinear term F depends on aretarded argument. We also show that in both cases the solutions set is compact. The proofs are based on topological arguments and make use of the theory of condensing multimaps.
2009
 Strictly localized bounding functions for vector secondorder boundary value problems
[Articolo su rivista]
J., Andres; Malaguti, Luisa; M., Pavlačková
abstract
The solvability of the secondorder Floquet problem in a given set is established by means of C2bounding functions for vector upperCarathéodory systems. The applied ScorzaDragoni type technique allows us to impose related conditions strictly on the boundaries of bound sets. An illustrating example is supplied for a dry friction problem.
2007
 A bounding functions approach to multivalued boundary value problems
[Articolo su rivista]
J., Andres; Malaguti, Luisa; Taddei, Valentina
abstract
The solvability of Floquet boundary value problems is investigated for upperCaratheodory differential inclusions by means of strictly localized C2bounding functions. The existence of an entirely bounded solution is obtained in a sequential way. Our criteria can be regarded as a multivalued extension of recent results of Mawhin and Thompson concerning periodic and bounded solutions of Caratheodory differential equations. A simple illustrating example is supplied.
2007
 Aggregative movement and front propagation for bistable population models
[Articolo su rivista]
P. K., Maini; Malaguti, Luisa; C., Marcelli; S., Matucci
abstract
Front propagation for aggregationdiffusionreaction equationsis investigated, with a bistable reactionterm and a diffusion coeffcient with changing sign, modeling aggregatingdiffusing processes. We provide necessary and suffcient conditions for the existence of traveling wave solutions and classify them according to how or if they attain their equilibria at finite times. We also show that the dynamics can exhibit the phenomena of finite speed of propagation and/or finite speed of saturation.
2007
 Trends in Differential Equations and Dynamical Systems
[Esposizione]
Gavioli, Andrea; Malaguti, Luisa; Villarini, Massimo
abstract
The workshop took place at Modena, from November 29th to 30th.The main speakers were J. Andres, from Palachy University (Olomouc, SK), P. K. Maini, from the University of Oxford (UK), V. Obukhovskii, from Voronezh State University (Russia), and other speakers from Italy (R. Johnson, F. Papalini, M. Tarallo). Furthermore, several talks were given by young researchers. The topics of the meeting covered many different areas in the field of differential equations and related problems. In particular, here are some of the exposed subjects: frontpropagation in reactiondiffusion equations, which often arise from biological models, SturmLiouville operators, perturbation theory for Hamiltonian systems, impulsive control systems, boundary value problems and the theory of bound sets, delay equations, differential inclusions.
2006
 Diffusionaggregation processes with monostable reaction terms
[Articolo su rivista]
P. K., Maini; Malaguti, Luisa; C., Marcelli; S., Matucci
abstract
This paper analyses front propagation of the aggregationdiffusionreaction equation with a monostable reaction term and a diffusion coefficient which changes sign from positive to negative values. This model equation accounts for simultaneous diffusive and aggregative behaviors of a population dynamic depending on the population density. The existence of infinitely many traveling wave solutions is proven. These fronts are parameterized by their wave speed and monotonically connect the stationary states 0 and 1. In the degenerate case sharp profiles appear, corresponding to the minimum wave speed. They also have new behaviors, in addition to those already observed in diffusive models, since they can be right compactly supported, left compactly supported, or both. The dynamics can exhibit, respectively, the phenomena of finite speed of propagation, finite speed of saturation, or both.
2005
 Finite speed of propagation in monostable degenerate reactiondiffusionconvection equations
[Articolo su rivista]
Malaguti, Luisa; C., Marcelli
abstract
We study the existence and properties of travelling wave solutions of the FisherKPP reactiondiffusionconvection equation u_t + h(u)u_x = [D(u)u_x]_x + g(u), where the diffusivity D(u) is simply or doubly degenerate. Both the cases when D(0) and D(1) are possibly zero real values or infinity, are treated. We discuss the effects, due to the presence of a convective term, concerning the property of finite speed of propagation. Moreover, in the doubly degenerate case we show the appearance of new types of profiles and provide their classification according to sharp relations between the nonlinear terms of the model. An application is also presented, concerning the evolution of a bacterial colony.
2005
 Fixed point analysis for nonoscillatory solutions of quasi linear ordinary differential equations
[Articolo su rivista]
Malaguti, Luisa; Taddei, Valentina
abstract
The paper deals with a quasilinear ordinarydifferential equation when the nonlinearity is not necessarily monotone in its second argument. We find necessary and sufficient conditions for the existence of unbounded nonoscillatory solutions. By means of a fixed point technique we investigate their growth, proving the coexistence of solutions with different asymptotic behaviors. In some special cases we are able to show the exact asymptotic growth of these solutions. We apply previous analysis for studying the nonoscillatory problem. Several examples are included.
2005
 On transitional solutions of second order nonlinear differential equations
[Articolo su rivista]
Malaguti, Luisa; C., Marcelli; N., Partsvania
abstract
The paper deals with the existence of bounded solutions of the nonlinear diferential equation u''=f(t,u,u') satisfying suitable conditions at infinity. New existence results are obtained, which generalize and unify previous quoted investigations. The main technique for proving all the results derives from the comparisontype theory introduced by Kiguradze and Shekhter.
2005
 The effects of convective processes on front propagation in various reactiondiffusion equations
[Relazione in Atti di Convegno]
Malaguti, Luisa; Marcelli, C.; Matucci, S.
abstract
We present a brief survay on our recent results concerning the existence and properties of travelling wave solutions to reactiondiffusionconvection equations with different types of reaction terms. In particular, we show that in certain cases the presence of an additional transport term causes the disappearance of wavefronts.
2005
 Trends in Differential Equations and Dynamical Systems
[Esposizione]
Gavioli, Andrea; Malaguti, Luisa; Villarini, Massimo
abstract
The workshop took place at Reggio Emilia, from September 29th to 30th. The main speakers were S. Kamin from TelAviv University, TelAviv (Israel) and L. Sanchez from Lisbon University (Lisbon, Portugal) and other speakers from Italy (A. Agrachev and S. Terracini). Furthermore, several talks were given by young researchers. The topics of the meeting covered many different areas in the field of differential equations and control problems.
2004
 Bounded solutions and wavefronts for discrete dynamics
[Articolo su rivista]
Malaguti, Luisa; P., Rehak; Taddei, Valentina
abstract
This paper deals with the secondorder nonlinear difference equation Delta(r(k)Deltau(k)) + q(k)g(u(k+1)) = 0 where {r(k)} and {q(k)} are positive real sequences defined on N, and the nonlinearity g : R > R is nonnegative and nontrivial. Sufficient and necessary conditions are given, for the existence of bounded solutions starting from a fixed initial condition u(0). The same dynamic, with f instead of g such that uf (u) > 0 for u not equal 0, was recently extensively investigated. On the contrary, our nonlinearity g is of a small appearance in the discrete case. Its introduction is motivated by the analysis of wavefront profiles in biological and chemical models. The paper emphasizes the many different dynamical behaviors caused by such a g with respect to the equation involving function f. Some applications in the study of wavefronts complete this work.
2004
 Front propagation in bistable reactiondiffusionadvection equations
[Articolo su rivista]
Malaguti, Luisa; C., Marcelli; S., Matucci
abstract
The paper deals with the existence and properties of frontpropagation between the stationary states 0 and 1 of the reactiondiffusionadvection equation with a bistable reaction term G and a strictly positive diffusive process. We show that the additional transport term h can cause the disappearance of such wavefronts and prove that their existence depends both on the local behavior of G and h near the unstable equilibrium and on a suitable sign condition on h in [0, 1]. We also provide an estimate of the wave speed, which can be negative unlike what happens to the mere reactiondiffusion dynamic occurring when h ≡ 0.
2003
 A unifying approach to travelling wavefronts for reactiondiffusion equations arising from genetics and combustion models
[Articolo su rivista]
Malaguti, Luisa; C., Marcelli; S., Matucci
abstract
We present a new approach for investigating the existence of travelling wave solutions of the reactiondiffusion equation with density dependent diffusion. Such an apporach allows to treat contemporarily different types of reaction terms which frequently appear in genetics and combustion models.
2003
 Bounded solutions of Carathéodory differential inclusions: a bound sets approach
[Articolo su rivista]
J., Andres; Malaguti, Luisa; Taddei, Valentina
abstract
A bound sets technique is developed for Floquet problems to Carathèodory differential inclusions. It relies on the construction of either continuous or locally Lipschitzian Lyapunovlike bounding functions. Proceeding sequentially, the existence of bounded trajectories is then obtained. Nontrivial examples are supplied to illustrate our approach.
2003
 Existence and multiplicity of heteroclinic solutions for a nonautonomous boundary eigenvalue problem
[Articolo su rivista]
Malaguti, L.; Marcelli, C.
abstract
In this paper we investigate the boundary eigenvalue problem x''b(c,t,x)x'+g(t,x)=0, x(∞)=0, x(+∞)=1 depending on the real parameter c. We take the function b continuous and positive and assume that g is bounded and becomes active and positive only when it exceeds a threshold value theta in (0,1). At the point theta we allow g to have a jump. Additional monotonicity properties are required, when needed. Our main discussion deals with the nonautonomous case. In this context we prove the existence of a continuum of values for which this problem is solvable and we estimate the interval of such admissible values. In the autonomous case, we show its solvability for at most one c*. In the special case when b=c+h(x) with h continuous, we also give a nonexistence result, for any real c. Our methods combine comparisontype arguments, both for first and second order dynamics, with a shooting technique. Some applications of the obtained results are included.
2003
 On a twopoint boundary value problem for the second order ordinary differential equations with singularities
[Articolo su rivista]
A., Lomtatidze; Malaguti, Luisa
abstract
The paper deals with the boundary value problemu''= f(t; u; u'), u(a+) = 0; u(b−) = 0, where f is a realvalued function defined on (a; b) × R× R and it is Caratheodory on (a +c ; b −c) × R× R for any small c>0.
2003
 Sharp profiles in degenerate and doublydegenerate FisherKPP equations
[Articolo su rivista]
Malaguti, Luisa; C., Marcelli
abstract
This paper investigates the effects of a degenerate diffusion term in reactiondiffusion models ut=[D(u)ux]x + g(u) with FisherKPP type g. Both in the case when D(0) = 0 and when D(0) = D(l) = 0, with D(u) > 0 elsewhere, we obtain a continuum of travelling wave solutions having wave speed c greater than a threshold value c* and we show the appearance of a sharptype profile when c = c*. These results solve recent conjectures formulated by SanchezGarduno and Maini (J. Differential Equations 117 (1995) 281) and Satnoianu et al. (Discrete Continuous Dyn. Systems (Series B) 1 (2000) 339).
2003
 The influence of convective effects on front propagation in certain diffusive models
[Relazione in Atti di Convegno]
Malaguti, Luisa; Marcelli, C.
abstract
We consider a reactiondiffusionconvection equation where the reaction term well describes those phenomena which activate only after a certain threshold value. We address our interest in the existence of travelling wave solutions (t.w.s.) between two equilibria and their corresponding wave sppeds. We show that, when the convective term H is, in some sense, weak, this model behaves as in the absence of convection and it admits a unique (up to space shifts) t.w.s. with a positive wave speed. On the contrary, when H prevails over the other terms, no t.w.s. exists.
2003
 Wave fronts in reactiondiffusion equations
[Relazione in Atti di Convegno]
Malaguti, Luisa; C., Marcelli; S., Matucci
abstract
A new approach, based on upper and lower solutions, was recently employed by the same authors to unify and generalize existing results for wave fronts in reactiondiffusion equations arising in combustion and genetic models. The statement of the problem and then main results are here explaned, and relations with existing results are analyzed. Some open problems and directions for futurre research are also indicated.
2002
 Heteroclinic Orbits in Plane Dynamical Systems
[Articolo su rivista]
Malaguti, Luisa; C., Marcelli
abstract
We consider general second order boundary value problems on the whole line of the type u''=h(t, u, u'), u(∞)=0, u(+∞)=1, for which we provide existence, nonexistence, multiplicity results. The solutions we find can be reviewed as heteroclinic orbits in the (u, u') plane dynamical system.
2002
 Travelling wavefronts in reactiondiffusion equations with convection effects and nonregular terms
[Articolo su rivista]
Malaguti, Luisa; C., Marcelli
abstract
This paper deals with the appearance of monotone bounded travelling wave solutions for a parabolic reactiondiffusion equation which frequently meets both in chemical and biological systems. In particular, we prove the existence of monotone front type solutions for any wave speed c greater than or equal to c* and give an estimate for the threshold value c*. Our model takes into account both of a density dependent diffusion term and of a nonlinear convection effect. Moreover, we do not require the main nonlinearity g to be a regular C1 function; in particular we are able to treat both the case when g'(0) = 0, giving rise to a degenerate equilibrium point in the phase plane, and the singular case when g'(0) = +∞. Our results generalize previous ones due to ARONSON and WEINBERGER [Adv. Math. 30 (1978), pp. 33  76], GIBBS and MURRAY (see MURRAY [Mathematical Biology, SpringerVerlag, Berlin, 1993]) and MCCABE, LEACH and NEEDHAM [SIAM J. Appl. Math. 59 (1998), pp. 870899]. Finally, we obtain our conclusions by means of a comparisontype technique which was introduced and developed in this framework in a recent paper by the same authors.
2001
 Floquet Boundary Value Problems for Differential Inclusions: a Bound Sets Approach
[Articolo su rivista]
J., Andres; Malaguti, Luisa; Taddei, Valentina
abstract
A technique is developed for the solvability of the Floquet boundary value problem associated to a differential inclusion. It is based on the usage of a not necessarily C1class of Liapunovlike bounding functions. Certain viability arguments are applied for this aim. Some illustrating examples are supplied.
2001
 The rational expectation dynamics of a model for the term structure and monetary policy
[Articolo su rivista]
Malaguti, Luisa; Torricelli, Costanza
abstract
In the present paper we set up a rational expectation model for the interaction between the expectation theory of the term structure, monetary policy and a timevarying premium. The results we obtain allow to conclude that the information content of the spread over the short/long rate depends onwhether or not the joint working of monetary policy and the timevaryingterm premium, which is peculiar to our model, exactly compensate the classicalimplication of the Expectation Theory. So our results can rationalise regression testsobtaining small or even negative values for the spread coefficient.
2000
 Existence of Bounded Trajectories Via Upper and Lower Solutions
[Articolo su rivista]
Malaguti, Luisa; C., Marcelli
abstract
The paper deals with the boundary value problem (on the whole line) u''f(u,u')+g(u)=0, u(∞)=0, u(+∞)=1, where g is a continuous nonnegative function with support [0, 1], and f is a continuous function. By means of a new approach, based on a combination of lower and uppersolutions methods and phaseplane techniques, we prove an existence result for the problem when f is superlinear in u'; by a similar technique, we also get a nonexistence one. As an application, we investigate the attractivity of the singular point (0,0) in the phaseplane (u, u'). Applications of these results in the field of fronttype solutions for reaction diffusion equations can be found in L. Malaguti, C. Marcelli, Math. Nachr. 242 (2002), 148—164
2000
 On a Nonlocal Boundary Value Problem for Second Order Nonlinear Singular Differential Equations
[Articolo su rivista]
A., Lomtatidze; Malaguti, Luisa
abstract
Criteria for the existence and uniqueness of a solution of a nonlocal second order boundary value problem are estabilished. These criteria apply, in particular, to the case when the nonlinearity has nonintegrable singularities.
2000
 Viable solutions of differential inclusions with memory in Banach spaces
[Articolo su rivista]
Gavioli, Andrea; Malaguti, Luisa
abstract
In this paper we study functional differential inclusions with memory and state constraints. We assume the state space to be a separable Banach space and prove existence results for an upper semicontinuous orientor field; we consider both the case of a globally measurable orientor field and the case of a Caratheodory one.
1999
 A comparisontype approach for travelling fronts
[Relazione in Atti di Convegno]
Malaguti, Luisa; C., Marcelli
abstract
We propose a new approach, based on differential inequalities, in order to study travelling wave solutions for a diffusion equation u_t=u_xx+g(u) with g having compact support and positive inside it. We are able to consider any arbitrary continuous nonlinear term g; in particular we do not need to assume a linear growth condition on g, even of local type.
1998
 Asymptotic properties of an ordinary differential equation via topological methods
[Articolo su rivista]
Malaguti, Luisa; Taddei, Valentina
abstract
The existence of bounded and unbounded solutions for a second order equation is obtained via topological methods
1997
 Monetary policy and the term structure of interest rates: a generalization of McCallum model
[Capitolo/Saggio]
Malaguti, Luisa; Torricelli, Costanza
abstract
McCallum (1994) sets up a Rational Expectation model for the interaction of monetary policy and the Expectation Theory of the term structure which rationalises some empirical failures of the latter and demonstrates the inappropriateness of usual regression tests for the information in the term structure. In the present paper, we generalize McCallum (1994) twoperiod model by introducing a different, financetheoretic characterization of the term premium. Our results still account for those empirical findings which are at odds with the Expectation Theory of the term structure and yet support validity of usual regressions performed to assess the information content of the term structure. Our results depend on the relative magnitude of the relevant parameters in the model and therefore final conclusions have to be left to empirical investigations.
1997
 The Interaction Between Monetary Policy and the Expectation Hypothesis of the Term Structure of Interest Rates in a NPeriod Rational Expectation Model
[Working paper]
Malaguti, L.; Torricelli, C.
abstract
1996
 Monetary policy and the term structure of interest rates
[Working paper]
Malaguti, L.; Torricelli, C.
abstract
1996
 Monotone Trajectories of Differential Inclusions in Banach Spaces
[Articolo su rivista]
Malaguti, Luisa
abstract
Existence results are obtained for monotone trajectories of nonlinear differential inclusions in a separable Banach space. They extend in two directions previous ones due to AubinCellina, Deimling and Haddad.
1996
 Optimal Solutions in a Growth Model with Irreversible Investments
[Articolo su rivista]
Malaguti, Luisa
abstract
We consider an economical model where a single commodity can be either consumed or invested. We assume that investments are irreversible and study the problem of maximizing a disconted utility functional on an unbounded time interval. Using existence theorems for infinitehorizon optimal control problmes, we prove the existence of strongly optimal solutions. The result obtained generalizes a previous one due to ArrowKurz.
1995
 Bounded Solutions For a Class of Second Order Nonlinear Differential Equations, Differential Equations and Dynamical Systems
[Articolo su rivista]
Malaguti, Luisa
abstract
An existence result of bounded solutions with assigned initial conditions for a second order differential equation is given. The techniques applied are then used to solve a boundary value problem suggested both by the theory of population genetics and by the study of combustion models.
1995
 Limit Properties for Solutions of a Class of Second Order Nonlinear Differential Equations
[Articolo su rivista]
Malaguti, Luisa
abstract
The paper deals wtih the boundedness and the asymptotic behaviour of the solutions of a second order differential equation which is nonlinear in the solution variable. The considered equation was suggested both by the theory of population genetics and by the study of combustion models.
1987
 Soluzioni periodiche dell’equazione di Liénard: biforcazione dall’infinito e non unicità
[Articolo su rivista]
Malaguti, Luisa
abstract
The paper deals with nonexistence and nonuniqueness results for periodic solutions of the Liénard equation.
1987
 Sull'ampiezza della soluzione periodica in un modello non lineare del ciclo economico
[Relazione in Atti di Convegno]
Malaguti, Luisa
abstract
Si studia un modello non lineare di ciclo economico del tipo moltiplicatoreaccelleratore nel quale le oscillazioni si ottengono dalle interazioni tra il reddito e l'investimento. Da un punto di vista matematico si tratta di una equazione differenziale non lineare del secondo ordine del tipo Lienard. L'uso della teoria classica su questa classe di equazioni consente di mostrare la presenza di una unica soluzione periodica, che risulta un ciclo limite stabile. Il modello contiene un parametro reale, che dipende dal fenomeno economico cosiddetto dell'accelleratore e l'indagine prosegue con uno studio dettagliato dell'ampiezza del ciclo al variare di questo parametro.
1985
 Su alcune congetture nel modello di Goodwin
[Relazione in Atti di Convegno]
Malaguti, Luisa
abstract
Il modello del sistema capitalistico elaborato da R.M. Goodwin nel 1967 si propone di cogliere le fluttuazioni dinamiche delle principali variabili macroeconomiche. Il ciclo viene, qui, visto come un effetto naturale delle contraddizioni interne al capitalismo. Per gli agenti economici, lavoratori e capitalisti, queste continue oscillazioni non sono vantaggiose ed e', quindi, ragionevole attendersi che essi si sforzino di contrastarle. Il punto stazionario verso il quale cercano di portare l'economia e', pero', diverso per le due classi sociali. Scopo del lavoro e' analizzare il comportamento asintotico del processo economico soggetto a questo fenomeno di conflittualita' mediante l'uso degli strumenti che sono propri della Teoria dei Giochi.