Special Session 14: New perspectives in the qualitative study of nonlinear differential equations and dynamical systems

Which diffusion strategy is best for a prey and a predator moving with Fractional Laplacians?

Elisa Affili Universite de Rouen Normandie France Co-Author(s):    Serena Dipierro, Giovanni Giacomin, Enrico Valdinoci

Abstract: We consider a predator and a prey moving with Fractional Laplacians, letting the predator and the prey free to choose the exponent characterising the distribution of the jump lengths. Which exponent should the predator use to maximise its chance to catch the prey? And which one should the prey select to minimise the possibility of meeting the predator? In this talk, we study a funtional that describes the chance of meeting taking into account both strategies. We focus in particular on the case of a large initial distance and we find ranges for the parameter of fractional diffusion corresponding to the maximum (for the predator) and the minimum (for the prey) in dependence of the choice of the other species. These findings rely on very fine estimates of the fundamental solution of the fractional heat equation.

Continuation theorems for periodic systems with nonlinear time-dependent differential operators

Pierluigi Benevieri University of S\~ao Paulo Brazil Co-Author(s):    Guglielmo Feltrin

Abstract: The talk proposes some continuation theorems for the periodic problem \begin{equation*} \begin{cases} \, x_{i}` = g_{i}(t,x_{i+1}), &i=1,\ldots,n-1, \ \, x_{n}` = h(t,x_{1},\ldots,x_{n}), \ \, x_{i}(0)=x_{i}(T), &i=1,\ldots,n, \end{cases} \end{equation*} providing a unified framework that improves and extends earlier contributions by Jean Mawhin and collaborators to second-order differential problems governed by nonlinear time-dependent differential operators of the form \begin{equation*} \begin{cases} \, (\phi(t,x`))`=f(t,x,x`), \ \, x(0)=x(T),\quad x`(0)=x`(T). \end{cases} \end{equation*} The proof is based on the topological degree theory. This is a joint work with Guglielmo Feltrin, University of Udine, Italy.

Non-smooth critical point theory: from relativistic celestial mechanics to symmetry breaking in PDEs

Alberto Boscaggin University of Turin Italy Co-Author(s):    Walter Dambrosio and Duccio Papini; Francesca Colasuonno, Benedetta Noris, Federica Sani and Tobias Weth

Abstract: I will briefly discuss two problems on which I have been working in the last few years: the first is the periodic problem for systems of ODEs in relativistic mechanics, the second is the Dirichlet problem for supercritical elliptic PDEs. Although seemingly unrelated, these two problems have in common the fact that they are addressed through the non-smooth critical point theory developed since the seminal paper by Szulkin in 1986. I will try to highlight similarities and differences between the two applications. Based on joint works with Walter Dambrosio and Duccio Papini (systems of ODEs) and Francesca Colasuonno, Benedetta Noris, Federica Sani and Tobias Weth (elliptic PDEs).

Existence of a periodic solution for planar systems with possible finite-time blow-up

Alberto Cagnetta Universit\`a degli Studi di Udine Italy Co-Author(s):    Paolo Gidoni

Abstract: Periodic solutions for superlinear planar systems have been extensively studied, in particular in the Hamiltonian setting, for instance in the work of A. Boscaggin (2012). Such results typically rely on the global existence of solutions, a structural assumption whose role was clarified by P. Gidoni (2023) for a class of superlinear second-order equations where global existence may fail. In this paper we extend the results of Gidoni (2023) to general non-autonomous $T$-periodic planar systems of the form \begin{equation*} \dot{z} = F(t,z), \end{equation*} without assuming global forward existence of solutions. Under suitable asymptotic bounds on the rotation number of solutions, we prove the existence of at least one $T$-periodic solution. In the Hamiltonian case $F(t,z)=J\nabla_z H(t,z)$, we further investigate super-quadratic systems and provide explicit sufficient conditions ensuring that the rotational assumptions are satisfied. This is a joint work with Paolo Gidoni (University of Udine).

Optimizing vaccine allocation in an age-structured SIR model

Romain Ducasse LJLL, universite Paris Cite France Co-Author(s):    Luis Almeida, Elisa Paparelli

Abstract: The SIR model is a nonlinear differential system that describes the spread of a disease in a population. In this talk, I will present an optimization problem where the goal is to maximize the final state of a SIR model by controling a vaccination term : in other terms, we study how to optimize the allocation of vaccines during an epidemic in order to minimize the casualties. Our model is an heterogeneous nonlinear integral system with an age structure and with a control (the vaccination). Using adequate comparison priciples, we manage to identify in some cases an optimal vaccination strategy.

A delay-induced nonlocal free boundary problem

Jian Fang Harbin Institute of Technology Peoples Rep of China Co-Author(s):    Yihong Du, Ningkui Sun

Abstract: Incorporating time delay and Stefan type free boundary condition into reaction-diffusion equation yields a nonlocal problem. Under the KPP type setting we establish a dichotomy on propagation or vanishing. When propagation happens, the spreading speed is shown to exist and it is determined nonlinearly by a delay-induced nonlocal elliptic problem in half line.

Limit cycle and asymptotic gait for a dynamic model of rectilinear locomotion

Paolo Gidoni University of Udine Italy Co-Author(s):

Abstract: Biological and bio-inspired locomotion is usually described by recognizing periodic patterns, or gaits, in the movement of limbs or other body parts. But is the evolution of the system actually periodic? Or more properly, relative-periodic, since, presumably, each cycle will propel the animal (or robot) a little bit forward? The answer is often no, due, for instance, to inertia or elasticity. However, we might expect the behaviour to converge asymptotically to a relative-periodic one. In this talk we will introduce this issue considering, as a case study, a dynamic model of rectilinear crawling locomotion. We study the existence of a global periodic attractor for the reduced dynamics of the model, corresponding to an asymptotically relative-periodic motion of the crawler. The main result is of Massera-type, namely we show that the existence of a bounded solution implies the existence of the global periodic attractor for the reduced dynamics. Additional conditions and a counterexample for the existence of a bounded solution (and therefore of the attractor) will be briefly discussed. We conclude surveying the issue for some related models.

Propagating terraces in periodic media

Thomas Giletti University Clermont-Auvergne France Co-Author(s):    Luca Rossi

Abstract: In this talk, we will be interested in the propagation of solutions of a multistable reaction-diffusion equation in spatially periodic media. We will prove the existence of a so-called propagating terrace, which is a finite sequence of fronts whose speeds are ordered. These correspond to a situation where successive transitions occur between intermediate steady states. We will see that propagating terraces dictate the large-time behavior of solutions of the Cauchy problem, but also that new difficulties arise from the interplay between the different directions of propagation.

Rotating spirals and bifurcation analysis for competition systems

Zaizheng Li Hebei Normal University Peoples Rep of China Co-Author(s):    Susanna Terracini

Abstract: In this talk, we mainly discuss the existence of rotating spirals for competition-diffusion systems and related bifurcation results. For the Neumann problem, we establish the existence of unstable rotating spirals by applying the multi-parameter bifurcation theorem. In addition, for the non-homogeneous Dirichlet problem, the Rothe fixed point theorem is employed to prove the existence of rotating spirals. And we will deliver some bifurcation analysis for the stationary problem. This talk is based on joint works with Prof. Susanna Terracini.

Global multiplicity results in a Moore-Nehari type problem

Eduardo Mu\~noz-Hern\`andez Complutense University of Madrid Spain Co-Author(s):    Juli\`an L\`opez-G\`omez and Fabio Zanolin

Abstract: In this talk, we present a superlinear problem of the Moore-Nehari type. If there is a general non-negative weight in front of the nonlinear term, existence of positive and nodal solutions is guaranteed by classical bifurcation methods. On the other hand, if we restrict ourselves to the analysis of positive solutions for a symmetric non-negative weight vanishing in the middle interval, i.e., to the classical setting of the Moore-Nehari problem, it is possible to prove multiplicity of positive solutions for both signs of the bifurcation parameter. This is a joint work with Professors Juli\`an L\`opez-G\`omez (Complutense University of Madrid) and Fabio Zanolin (University of Udine).

Bifurcation from periodic solutions of central force problems in the three-dimensional space

Duccio Papini University of Modena and Reggio Emilia Italy Co-Author(s):    Alberto Boscaggin, Guglielmo Feltrin

Abstract: The talk deals with electromagnetic perturbations of a central force problem. The considered differential operator includes, as special cases, the classical one as well as that of special relativity. We investigate whether non-circular periodic solutions of the unperturbed problem can be continued into periodic solutions for small perturbations, both for the fixed-period problem and, if the perturbation is time-independent, for the fixed-energy problem. The proof is based on an abstract bifurcation theorem of variational nature, which is applied to suitable Hamiltonian action functionals. In checking the required nondegeneracy conditions we take advantage of the existence of partial action-angle coordinates as provided by the Mishchenko--Fomenko theorem for superintegrable systems. Physically relevant problems to which our results can be applied are homogeneous central force problems in classical mechanics and the Kepler problem in special relativity.

Liouville-type results for the Fisher-KPP equation: old and new

Luca Rossi Sapienza University of Rome Italy Co-Author(s):

Abstract: In this talk, we present a joint work with O. Tough on the existence and uniqueness of positive bounded stationary solutions to the Fisher-KPP equation in unbounded domains, with Dirichlet boundary conditions. Under strong KPP-type assumptions, we derive a necessary and sufficient condition for existence in dimensions $\leq 6$, expressed in terms of the generalized principal eigenvalue. We further show that, whenever it exists, the solution is unique. As an application, one infers that for branching Brownian motion, global survival implies local survival in low dimensions.

Large positive solutions for a class of 1-D diffusive logistic problems with general boundary conditions

Alejandro Sahuquillo Universidad Complutense de Madrid Spain Co-Author(s):    Juli\`{a}n L\`{o}pez-G\`{o}mez and Andrea Tellini

Abstract: The first goal of this paper is to establish the existence of a positive solution for the singular boundary value problem (1.1), where $\mathcal{B}$ is a general boundary operator of Dirichlet, Neumann or Robin type, either classical or non-classical, not necessarily of Sturm--Liouville type. Since $f(u):=au^p -\lambda u$, $u\geq 0$, is not increasing if $\lambda>0$, the uniqueness of the positive solution is not obvious, even when $a(x)$ is a positive constant. The second goal of this paper is to establish the uniqueness of the positive solution of (1.1) in that case. At a later stage, denoting by $L_\lambda$ the unique positive solution of (1.1) when $a(x)$ is a positive constant, we will characterize the point-wise behavior of $L_\lambda$ as $\lambda \to \pm \infty$. Finally, we establish the uniqueness of the positive solution of (1.1) when $a(x)$ is non-increasing in $[0,R]$, $\lambda \geq 0$, and $\beta < 0$ if $-u`(0)+\beta u(0)=0$. This is a joint work with Juli\`{a}n L\`{o}pez-G\`{o}mez and Andrea Tellini.

Reaction--diffusion systems with degenerate diffusivity: wavefronts and their qualitative properties

Elisa Sovrano University of Modena and Reggio Emilia Italy Co-Author(s):

Abstract: We consider a system of two coupled reaction--diffusion equations in which one component is governed by doubly degenerate diffusion. These systems do not preserve the total mass, separating them from standard scalar reaction-diffusion equations. We investigate the existence and qualitative properties of wavefront solutions, namely profiles that propagate with constant speed and are represented by a pair of strictly monotone functions. Using a combination of shooting techniques and fixed-point arguments, we derive conditions ensuring the existence of wavefronts and obtain estimates for threshold speeds. We also address the regularity of the wavefront profiles and discuss the possible occurrence of sharp wavefronts arising from diffusion degeneracy. The model arises in the study of the spatio-temporal evolution of bacterial colonies growing on nutrient-rich agar substrates. Joint work with L. Malaguti, V. Taddei (University of Modena and Reggio Emilia), and E. Mu\~noz-Hern\`andez (Complutense University of Madrid).

Morse index and symmetry-breaking bifurcation of positive solutions for the one-dimensional Liouville type equation with a step weight

Satoshi Tanaka Tohoku University Japan Co-Author(s):    Satoshi Tanaka

Abstract: We consider the boundary value problem $u'' + \lambda h(x,\alpha) e^u = 0$ for $x \in (-1,1)$; $u(-1) = u(1) = 0$, where $\lambda$ is a positive parameter, $\alpha\in(0,1)$, $h(x,\alpha)=0$ for $x\in(-\alpha,\alpha)$, and $h(x,\alpha)=1$ for $\alpha \le |x| \le 1$. We compute the Morse index of positive even solutions, and then we prove the existence of an unbounded connected set of positive non-even solutions emanating from a symmetry-breaking bifurcation point. This is a joint work with Kanako Manabe (JG Corporation).

Nodal solutions for the Minkowski mean curvature operator: multiplicity and large-$\\mu$ asymptotics

Ricardo Ziegele Universit\\`{a} degli studi di Torino Italy Co-Author(s):    Alberto Boscaggin, Francesca Colasuonno

Abstract: We study the Dirichlet problem for the mean curvature operator in Minkowski space on a bounded domain $\Omega \subset \mathbb{R}^n$. For this problem, we study the interaction between two parameters $(\lambda,\mu)$ and ther effects on the multiplicity of nodal solutions. In the general setting, using non-smooth variational methods, we prove the existence of a ground-state solution and a linking solution. In the radial case, we prove existence and multiplicity of nodal solutions. Finally, we characterize the limiting profile of these solutions in both settings as $\mu \to +\infty$. This is a joint work with Alberto Boscaggin (Universit\`a di Torino) and Francesca Colasuonno (Universit\`a di Torino).