### 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).

### 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.

### Contact

#### 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