### 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
• Reaction-diffusion equations with transmission conditions, with Henri Berestycki.
• Stability, bifurcations and hydra effects in an age-structured population model with threshold harvesting, with Eduardo Liz.
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
Extinction or coexistence in periodic Kolmogorov systems of competitive type.

Discrete Contin. Dyn. Syst., 2021 (to appear), with Isabel S. Labouriau. and Carlota Rebelo.

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 phytoplankton competion under the effect of toxins.

On the number of positive solutions to an indefinite parameter-dependent Neumann problem.

Discrete Contin. Dyn. Syst., 2021 (to appear), with Guglielmo Feltrin and Andrea Tellini.

We study the second-order boundary value problem$\begin{cases}\, -u''=a_{\lambda,\mu}(t) \, u^{2}(1-u), & t\in(0,1), \\\, u'(0)=0, \quad u'(1)=0,\end{cases}$ where $$a_{\lambda,\mu}$$ is a step-wise indefinite weight function, precisely $$a_{\lambda,\mu}\equiv\lambda$$ in $$[0,\sigma]\cup[1-\sigma,1]$$ and $$a_{\lambda,\mu}\equiv-\mu$$ in $$(\sigma,1-\sigma)$$, for some $$\sigma\in\left(0,\frac{1}{2}\right)$$, with $$\lambda$$ and $$\mu$$ positive real parameters. We investigate the topological structure of the set of positive solutions which lie in $$(0,1)$$ as $$\lambda$$ and $$\mu$$ vary. Depending on $$\lambda$$ and based on a phase-plane analysis and on time-mapping estimates, our findings lead to three different (from the topological point of view) global bifurcation diagrams of the solutions in terms of the parameter $$\mu$$. Finally, for the first time in the literature, a qualitative bifurcation diagram concerning the number of solutions in the $$(\lambda,\mu)$$-plane is depicted. The analyzed Neumann problem has an application in the analysis of stationary solutions to reaction-diffusion equations in population genetics driven by migration and selection.

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.