
ELISA SOVRANO
Ricercatore t.d. art. 24 c. 3 lett. B Dipartimento di Scienze e Metodi dell'Ingegneria

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2024
 Wavefronts for Generalized PeronaMalik Equations
[Capitolo/Saggio]
Corli, A.; Malaguti, L.; Sovrano, E.
abstract
We consider a generalization of PeronaMalik equation with reaction and convective terms. By assuming that the reaction is monostable, we prove the existence of regular wavefronts as well as some of their qualitative properties. It turns out that the admissible speeds for subcritical or critical wavefronts form a closed halfline; the threshold cannot be computed explicitly but an estimate is provided. Moreover, the wavefronts are strictly monotone and their slope is bounded by the critical values of the diffusion.
2023
 Stationary fronts and pulses for multistable equations with saturating diffusion
[Articolo su rivista]
Garrione, M.; Sovrano, E.
abstract
We deal with stationary solutions of a reactiondiffusion equation with fluxsaturated diffusion and multistable reaction term, in dependence on a positive parameter epsilon. Motivated by previous numerical results obtained by A. Kurganov and P. Rosenau (Nonlinearity, 2006), we investigate stationary solutions of front and pulsetype and discuss their qualitative features. We study the limit of such solutions for epsilon > 0, showing that, in spite of the wide variety of profiles that can be constructed, there is essentially a unique configuration in the limit for both stationary fronts and pulses. We finally discuss some variational features that include the case where the solutions having continuous energy may not be global minimizers of the associated action functional.
2022
 On the number of positive solutions to an indefinite parameterdependent neumann problem
[Articolo su rivista]
Feltrin, G.; Sovrano, E.; Tellini, A.
abstract
We study the secondorder boundary value problem ( u00 = aλ;μ(t) u2(1  u); t 2 (0; 1); u0(0) = 0; u0(1) = 0; where aλ;μ is a stepwise indefinite weight function, precisely a; in [0;] [ [1  σ; 1] and a in (σ; 1  σ), for some 2  0; 1 2 , with λ and μ positive real parameters. We investigate the topological structure of the set of positive solutions which lie in (0; 1) as λ and μ vary. Depending on λ and based on a phaseplane analysis and on timemapping estimates, our findings lead to three different (from the topological point of view) global bifurcation diagrams of the solutions in terms of the parameter μ. Finally, for the first time in the literature, a qualitative bifurcation diagram concerning the number of solutions in the (λ; μ)plane is depicted. The analyzed Neumann problem has an application in the analysis of stationary solutions to reactiondiffiusion equations in population genetics driven by migration and selection.
2022
 Stability, bifurcations and hydra effects in a stagestructured population model with threshold harvesting
[Articolo su rivista]
Liz, Eduardo; Sovrano, Elisa
abstract
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
 Extinction or coexistence in periodic kolmogorov systems of competitive type
[Articolo su rivista]
Coelho, I.; Rebelo, C.; Sovrano, E.
abstract
We study a periodic Kolmogorov system describing two species nonlinear competition. We discuss coexistence and extinction of one or both species, and describe the domain of attraction of nontrivial periodic solutions in the axes, under conditions that generalise Gopalsamy conditions. Finally, we apply our results to a model of microbial growth and to a model of phyto plankton competition under the effect of toxins.
2021
 Positive solutions of superlinear indefinite prescribed mean curvature problems
[Articolo su rivista]
Omari, P.; Sovrano, E.
abstract
This paper analyzes the superlinear indefinite prescribed mean curvature problem div u/1 + u2 = λa(x)h(u)in ω,u = 0on ℓω, where ω is a bounded domain in N with a regular boundary ℓω, h C0() satisfies h(s)  sp, as s → 0+, p > 1 being an exponent with p < N+2 N2 if N ≥ 3, λ > 0 represents a parameter, and a C0(ω¯) is a signchanging function. The main result establishes the existence of positive regular solutions when λ is sufficiently large, providing as well some information on the structure of the solution set. The existence of positive bounded variation solutions for λ small is further discussed assuming that h satisfies h(s)  sq as s → +∞, q > 0 being such that q < 1 N1 if N ≥ 2; thus, in dimension N ≥ 2, the function h is not superlinear at + ∞, although its potential H(s) =0sh(t)dt is. Imposing such different degrees of homogeneity of h at 0 and at + ∞ is dictated by the specific features of the mean curvature operator.
2020
 Chaos in periodically forced reversible vector fields
[Articolo su rivista]
Labouriau, I. S.; Sovrano, E.
abstract
We discuss the appearance of chaos in timeperiodic perturbations of reversible vector fields in the plane. We use the normal forms of codimension 1 reversible vector fields and discuss the ways a timedependent periodic forcing term of pulse form may be added to them to yield topological chaotic behaviour. Chaos here means that the resulting dynamics is semiconjugate to a shift in a finite alphabet. The results rely on the classification of reversible vector fields and on the theory of topological horseshoes. This work is part of a project of studying periodic forcing of symmetric vector fields.
2020
 High Multiplicity and Chaos for an Indefinite Problem Arising from Genetic Models
[Articolo su rivista]
Boscaggin, A.; Feltrin, G.; Sovrano, E.
abstract
We deal with the periodic boundary value problem associated with the parameterdependent secondorder nonlinear differential equation u ′′ + c u ′ + (λ a + (x)  μ a  (x)) g (u) = 0, u^{primeprime}+cu^{prime}+igl{(}lambda a^{+}(x)mu a^{}(x)igr{)}g(u)% =0, where λ, μ > 0 {lambda,mu>0} are parameters, c {cinmathbb{R}}, a (x) {a(x)} is a locally integrable Pperiodic signchanging weight function, and g: [ 0, 1 ] → {gcolon{[0,1]} omathbb{R}} is a continuous function such that g (0) = g (1) = 0 {g(0)=g(1)=0}, g (u) > 0 {g(u)>0} for all u ] 0, 1 [ {uin{]0,1[}}, with superlinear growth at zero. A typical example for g (u) {g(u)}, that is of interest in population genetics, is the logistictype nonlinearity g (u) = u 2 (1  u) {g(u)=u^{2}(1u)}. Using a topological degree approach, we provide high multiplicity results by exploiting the nodal behavior of a (x) {a(x)}. More precisely, when m is the number of intervals of positivity of a (x) {a(x)} in a Pperiodicity interval, we prove the existence of 3 m  1 {3^{m}1} nonconstant positive Pperiodic solutions, whenever the parameters λ and μ are positive and large enough. Such a result extends to the case of subharmonic solutions. Moreover, by an approximation argument, we show the existence of a family of globally defined solutions with a complex behavior, coded by (possibly nonperiodic) biinfinite sequences of three symbols.
2020
 How to Construct Complex Dynamics? A Note on a Topological Approach
[Articolo su rivista]
Sovrano, E.
abstract
We investigate the presence of complex behaviors for the solutions of two different dynamical systems: one is of discrete type and the other is continuous. We give evidence of "chaos" in the framework of topological horseshoes and show how different problems can be analyzed by the same procedure.
2020
 Positive solutions of indefinite logistic growth models with fluxsaturated diffusion
[Articolo su rivista]
Omari, P.; Sovrano, E.
abstract
This paper analyzes the quasilinear elliptic boundary value problem driven by the mean curvature operator −div∇u∕1+∇u2=λa(x)f(u)inΩ,u=0on∂Ω,with the aim of understanding the effects of a fluxsaturated diffusion in logistic growth models featuring spatial heterogeneities. Here, Ω is a bounded domain in RN with a regular boundary ∂Ω, λ>0 represents a diffusivity parameter, a is a continuous weight which may change sign in Ω, and f:[0,L]→R, with L>0 a given constant, is a continuous function satisfying f(0)=f(L)=0 and f(s)>0 for every s∈]0,L[. Depending on the behavior of f at zero, three qualitatively different bifurcation diagrams appear by varying λ. Typically, the solutions we find are regular as long as λ is small, while as a consequence of the saturation of the flux they may develop singularities when λ becomes larger. A rather unexpected multiplicity phenomenon is also detected, even for the simplest logistic model, f(s)=s(L−s) and a≡1, having no similarity with the case of linear diffusion based on the Fick–Fourier's law.
2019
 Periodic solutions to parameterdependent equations with a ϕ Laplacian type operator
[Articolo su rivista]
Feltrin, G.; Sovrano, E.; Zanolin, F.
abstract
We study the periodic boundary value problem associated with the ϕLaplacian equation of the form (ϕ(u′))′+f(u)u′+g(t,u)=s, where s is a real parameter, f and g are continuous functions, and g is Tperiodic in the variable t. The interest is in Ambrosetti–Prodi type alternatives which provide the existence of zero, one or two solutions depending on the choice of the parameter s. We investigate this problem for a broad family of nonlinearities, under nonuniform type conditions on g(t, u) as u→ ± ∞. We generalize, in a unified framework, various classical and recent results on parameterdependent nonlinear equations.
2018
 A negative answer to a conjecture arising in the study of selection–migration models in population genetics
[Articolo su rivista]
Sovrano, E.
abstract
We deal with the study of the evolution of the allelic frequencies, at a single locus, for a population distributed continuously over a bounded habitat. We consider evolution which occurs under the joint action of selection and arbitrary migration, that is independent of genotype, in absence of mutation and random drift. The focus is on a conjecture, that was raised up in literature of population genetics, about the possible uniqueness of polymorphic equilibria, which are known as clines, under particular circumstances. We study the number of these equilibria, making use of topological tools, and we give a negative answer to that question by means of two examples. Indeed, we provide numerical evidence of multiplicity of positive solutions for two different Neumann problems satisfying the requests of the conjecture.
2018
 AmbrosettiProdi Periodic Problem under Local Coercivity Conditions
[Articolo su rivista]
Sovrano, E.; Zanolin, F.
abstract
In this paper we focus on the periodic boundary value problem associated with the Liénard differential equation x ′′ + f ( x ) x ′ + g ( t , x ) = s, where s is a real parameter, f and g are continuous functions and g is Tperiodic in the variable t. The classical framework of Fabry, Mawhin and Nkashama, related to the AmbrosettiProdi periodic problem, is modified to include conditions without uniformity, in order to achieve the same multiplicity result under local coercivity conditions on g. Analogous results are also obtained for Neumann boundary conditions.
2018
 Ambrosettiprodi type result to a neumann problem via a topological approach
[Articolo su rivista]
Sovrano, E.
abstract
We prove an AmbrosettiProdi type result for a Neumann problem associated to the equation u + f(x, u(x)) = µ when the nonlinearity has the following form: f(x, u):= a(x)g(u) − p(x). The assumptions considered generalize the classical one, f(x, u) → +∞ as u → +∞, without requiring any uniformity condition in x. The multiplicity result which characterizes these kind of problems will be proved by means of the shooting method.
2018
 An indefinite nonlinear problem in population dynamics: High multiplicity of positive solutions
[Articolo su rivista]
Feltrin, G.; Sovrano, E.
abstract
Reactiondiffusion equations have several applications in the feld of population dynamics and some of them are characterized by the presence of a factor which describes different types of food sources in a heterogeneous habitat. In this context, to study persistence or extinction of populations it is relevant to perform a search for nontrivial steady states. Our paper focuses on a onedimensional model given by a parameterdependent equation of the form u ′ +(λa+(t)μa (t)g(u) = 0, whereg: [0, 1] → R is a continuousfunction such thatg(0) = g(1) = 0,g(s) > 0 for every 0 < 1 and lim → g(s)/s = 0, and the weight a(t)has two positive humps separated by a negative one. In this manner, we consider bounded habitats which include two favorable food sources and an unfavorable one. We deal with various boundary conditions, including the Dirichlet and Neumann ones, and we prove the existence of eight positive solutions when → and μ are positive and suffciently large. Throughout the paper, numerical simulations are exploited to discuss the results and to explore some open problems.
2018
 Three positive solutions to an indefinite Neumann problem: A shooting method
[Articolo su rivista]
Feltrin, G.; Sovrano, E.
abstract
We deal with the Neumann boundary value problem u′′+(λa+(t)−μa−(t))g(u)=0,0<1,∀t∈[0,T],u′(0)=u′(T)=0,where the weight term has two positive humps separated by a negative one and g:[0,1]→R is a continuous function such that g(0)=g(1)=0, g(s)>0 for 0<1 and lims→0javax.xml.bind.JAXBElement@501dbb92g(s)∕s=0. We prove the existence of three solutions when λ and μ are positive and sufficiently large.
2017
 A periodic problem for first order differential equations with locally coercive nonlinearities
[Articolo su rivista]
Sovrano, E.; Zanolin, F.
abstract
In this paper we study the periodic boundary value problem associated with a first order ODE of the form x' + g(t, x) = s where s is a real parameter and g is a continuous function, Tperiodic in the variable t. We prove an AmbrosettiProdi type result in which the classical uniformity condition on g(t, x) at infinity is considerably relaxed. The Carathéodory case is also discussed.
2017
 Indefinite weight nonlinear problems with Neumann boundary conditions
[Articolo su rivista]
Sovrano, E.; Zanolin, F.
abstract
We present a multiplicity result of positive solutions for the Neumann problem associated with a second order nonlinear differential equation of the following form u″+a(t)g(u)=0, where the weight function a(t) has indefinite sign. The only assumption we make for the nonlinear term g(u) is that its primitive G(u) presents some oscillations at infinity, expressed by the condition involving lim_G(u)/u2=0
2017
 The AmbrosettiProdi periodic problem: Different routes to complex dynamics
[Articolo su rivista]
Sovrano, E.; Zanolin, F
abstract
2016
 About Chaotic Dynamics in the Twisted Horseshoe Map
[Articolo su rivista]
Sovrano, E.
abstract
The twisted horseshoe map was developed in order to study a class of density dependent Leslie population models with two age classes. From the beginning, scientists have tried to prove that this map presents chaotic dynamics. Some demonstrations that have appeared in mathematical literature present some difficulties or delicate issues. In this paper, we give a simple and rigorous proof based on a different approach. We also highlight the possibility of getting chaotic dynamics for a broader class of maps.
2015
 Remarks on Dirichlet problems with sublinear growth at infinity
[Articolo su rivista]
Sovrano, E.; Zanolin, F.
abstract
We present some existence and multiplicity results for positive solutions to the Dirichlet problem associated with δu + λa(x)g(u) = 0; under suitable conditions on the nonlinearity g(u) and the weight function a(x): The assumptions considered are related to classical theorems about positive solutions to a sublinear elliptic equation due to BrezisOswald and BrownHess.
2014
 Dolcher fixed point theorem and its connections with recent developments on compressive/expansive maps
[Articolo su rivista]
Sovrano, E.; Zanolin, F.
abstract
In 1948 Mario Dolcher proposed an expansive version of the Brouwer fixed point theorem for planar maps. In this article we reconsider Dolcher's result in connection with some properties, such as covering relations, which appear in the study of chaotic dynamics.