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Welcome

Since November 2019, I am a Postdoc researcher at CNRS - CAMS, Paris (France).
[Supervisor: Prof. Henri Berestycki]

Previously, I was a Postdoc researcher at INdAM in the Departement of Mathematics and Geosciences at University of Trieste (Italy) [Supervisor: Prof. Pierpaolo Omari] and a Postdoc researcher at CMUP, Porto (Portugal) [Supervisor: Prof. Isabel S. Labouriau]. I was also a Visiting researcher at CMAF-CIO, University of Lisbon (Portugal).

My CV

Research

Highlights & Events
Works in Progress
Preprints

Topics

My research is driven in the area of nonlinear analysis concerning the theory of ordinary and partial differential equations and chaotic dynamical systems. I investigate solutions of boundary value problems associated with semilinear and quasilinear elliptic differential equations and wavefronts type solutions for reaction-diffusion equations. I am interested in population biology applications, and I deal with dynamical systems theory, topological and variational methods, and bifurcation techniques.

Keywords


Fundings

  • Postdoc Fellowship from FSMP. Research project: "Reaction-Diffusion Equations in Population Genetics: a study of the influence of geographical barriers on traveling waves and non-constant stationary solutions".
  • Postdoc Fellowship from INdAM (Istituto Nazionale di Alta Matematica). Research Project: "Problems in Population Dynamics: from Linear to Nonlinear Diffusion".

Publications

Recent Publications
Positive solutions of indefinite logistic growth models with flux-saturated diffusion.

Nonlinear Analysis, Theory, Methods and Applications, 2020, with Pierpaolo Omari.

This paper analyzes the quasilinear elliptic boundary value problem driven by the mean curvature operator \[\label{P}-\mathrm{div}\left(\nabla u/\sqrt{1+|\nabla u|^2}\right)=\lambda a(x)f(u) \quad \text{in }\Omega,\qquad u=0 \quad\text{on } \partial\Omega,\] with the aim of understanding the effects of a flux-saturated diffusion in logistic growth models featuring spatial heterogeneities. Here, \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with a regular boundary \(\partial \Omega,\) \(\lambda>0\) represents a diffusivity parameter, \(a\) is a continuous weight which may change sign in \(\Omega,\) and \(f\colon[0,L]\to\mathbb{R},\) with \(L>0\) a given constant, is a continuous function satisfying \(f(0)=f(L)=0\) and \(f(s)>0\) for every \(s\in ]0,L[.\) Depending on the behavior of \(f\) at zero, three qualitatively different bifurcation diagrams appear by varying \(\lambda\). Typically, the solutions we find are regular as long as \(\lambda\) is small, while as a consequence of the saturation of the flux they may develop singularities when \(\lambda\) becomes larger. A rather unexpected multiplicity phenomenon is also detected, even for the simplest logistic model, \(f(s) = s(L-s)\) and \(a\equiv 1\), having no similarity with the case of linear diffusion based on the Fick-Fourier's law.

Positive solutions of superlinear indefinite prescribed mean curvature problems.

Communications in Contemporary Mathematics, 2020, with Pierpaolo Omari.

This paper analyzes the superlinear indefinite prescribed mean curvature problem \[-\mathrm{div}\left({\nabla u}/{\sqrt{1+|\nabla u|^2}}\right)=\lambda a(x)h(u) \quad \text{in }\Omega,\qquad u=0 \quad\text{on } \partial\Omega,\] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with a regular boundary \(\partial \Omega\), \(h\in C^0(\mathbb{R})\) satisfies \(h(s) \sim s^{p}\), as \(s\to0^+,\) \(p>1\) being an exponent with \(p< \frac{N+2}{N-2}\) if \(N\geq 3\), \(\lambda> 0\) represents a parameter, and \(a\in C^0(\overline \Omega)\) is a sign-changing function. The main result establishes the existence of positive regular solutions when \(\lambda\) is sufficiently large, providing as well some information on the structure of the solution set. The existence of positive bounded variation solutions for \(\lambda\) small is further discussed assuming that \(h\) satisfies \(h(s) \sim s^{q}\) as \(s\to +\infty,\) \(q>0\) being such that \(q< \frac{1}{N-1}\) if \(N\geq 2\); thus, in dimension \(N\ge 2\), the function \(h\) is not superlinear at \(+\infty,\) although its potential \(H(s) = \int_0^sh(t)\, \mathrm{d}t\) is. Imposing such different degrees of homogeneity of \(h\) at \(0\) and at \(+\infty\) is dictated by the specific features of the mean curvature operator.

We deal with the periodic boundary value problem associated with the parameter-dependent second-order nonlinear differential equation \[u''+cu' + \bigr{(} \lambda a^{+}(x) - \mu a^{-}(x) \bigr{)} g(u) = 0,\] where \(\lambda,\mu>0\) are parameters, \(c\in\mathbb{R},\) \(a(x)\) is a locally integrable $P$-periodic sign-changing weight function, and \(g\colon[0,1]\to\mathbb{R}\) is a continuous function such that \(g(0)=g(1)=0,\) \(g(u)>0\) for all \(u\in]0,1[,\) with superlinear growth at zero. A typical example for \(g(u),\) that is of interest in population genetics, is the logistic-type nonlinearity \(g(u)=u^{2}(1-u).\) Using a topological degree approach, we provide high multiplicity results by exploiting the nodal behaviour of \(a(x).\) More precisely, when $m$ is the number of intervals of positivity of \(a(x)\) in a \(P\)-periodicity interval, we prove the existence of \(3^{m}-1\) non-constant positive \(P\)-periodic solutions, whenever the parameters \(\lambda\) and \(\mu\) 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 behaviour, coded by (possibly non-periodic) bi-infinite sequences of \(3\) symbols.

Chaos in periodically forced reversible vector fields.

Journal of Singularities, 2020, with Isabel S. Labouriau.

We discuss the appearance of chaos in time-periodic perturbations of reversible vector fields in the plane. We use the normal forms of codimension~\(1\) reversible vector fields and discuss the ways a time-dependent 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.


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Contact

To get in touch with me

Work Adress

École des Hautes Études en Sciences Sociales (EHESS)
Centre d'Analyse et de Mathématique Sociales (CAMS), CNRS
54 boulevard Raspail, 75006, Paris, France