Nonlinear Meeting 2021
Zoom, March 2223, 2021
The NLM2021 is the 6th of a series of annual workshops focusing on nonlinear analysis and differential problems. This edition's common thread will cover theoretical topics on reactiondiffusion equations and dynamical systems with possible applications to biology or physics.
Invited speakers
Henri Berestycki EHESSCAMSCNRS Paris 
Christian Kuehn Technical University of Munich 
Elaine Crooks Swansea University 
Peter Poláčik University of Minnesota 
Frank Hilker University of Osnabrück 
Pedro J. Torres University of Granada 
Program
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Monday, March 22
15:30  Henri Berestycki Reactiondiffusion systems in epidemiology 

The classical SIR type equations of epidemiology that take into consideration
spatial spreading lead to reactiondiffusion systems. Such systems also arise in the study of collective behaviors such as riots.
In this lecture, I will first recall the classical setting and then discuss a general framework for this type of systems that we call
''Activity / Susceptibility'' systems. In the second part of my talk, I will present a new model that describes the influence of roads
on the spread of epidemics. If time permits I will also show applications of these ideas to data of the current Covid19 epidemics. 

16:30  Elaine Crooks Travelling waves and minimality exchange in smectic C^{*} liquid crystals 
We consider minimality conditions for the speed of monotone travelling waves
in a model of a sample of smectic C^{*} liquid crystal subject to a constant electric field, dealing with both isotropic and anisotropic cases.
Such conditions are important in understanding switching properties of a liquid crystal, and our focus is on understanding how the presence of anisotropy
can affect the speed and nature of switching. Through a study of travellingwave solutions of a quasilinear parabolic equation, we obtain an estimate
of the influence of anisotropy on the minimal speed, and sufficient conditions for linear and nonlinear minimal speed selection mechanisms to hold
in different parameter regimes. We also discuss sufficient conditions for socalled “minimality exchange” in a general class of parameterdependent
monostable reactiondiffusion equations with explicit travellingwave solutions, when the minimal wave speed switches from the linearly determined value
to the speed of the explicitly determined front as a parameter changes. 

Break 

17:40  Christian Kuehn Geometric Singular Perturbation Theory for FastSlow PDEs 
Systems with multiple time scales appear in a wide variety of
applications. Yet, their mathematical analysis is challenging already in
the context of ODEs, where about four decades were needed to develop a
more comprehensive theory based upon invariant manifolds,
desingularization, variational equations, and many other techniques.
Yet, for PDEs progress has been extremely slow due to many obstacles in
generalizing several ODE methods. In my talk, I shall report on two
recent advances for fastslow PDEs, namely the extension of slow
manifold theory for unbounded operators driving the slow variables, and
the design of a blowup method for PDEs, where normal hyperbolicity is
lost [12]. 

Tuesday, March 23
15:30  Pedro J. Torres Periodic solutions of the Lorentz force equation 

The Lorentz force equation governs the motion of a charged particle under the effect of an electromagnetic field. In spite of that it plays a fundamental role in Classical Electromagnetism together with Maxwell equations, the application of the most known methods in Nonlinear Analysis and Dynamical Systems is still under development. The purpose of this talk is to present some recent results on the existence of periodic solutions when the electromagnetic field is periodic in time, and to identify several open problems. 

16:30  Frank Hilker Mathematical models of a socialecological system: coupling lake pollution dynamics and human behavior 
From global warming over landuse change to pollution, the world is facing many environmental problems. While the causes are mostly
wellknown from a natural sciences point of view, the challenge is the implementation of possible solutions. Societal demands, human behavior, and economic aspects not only impact the environmental state, but are
reversely affected by the environment. Mathematical models can be helpful in better understanding mutual feedbacks in coupled humanenvironment systems. In this talk, I will introduce a system of
two nonlinear differential equations, one describing lake water pollution and the other one describing human behavior of discharging pollutants. The latter uses approaches from evolutionary game theory.
Stability analysis reveals up to four alternative attractors as well as limit cycle oscillations. Numerical bifurcation analysis suggests the existence of BogdanovTakens and saddlenode homoclinic bifurcations.
In addition, some very preliminary results will be shown to indicate potential effects of discontinuous policy instruments. 

Break 

17:40  Peter Poláčik Quasiperiodic partially localized solutions of nonlinear elliptic equations on the entire space 
In this joint project with Dario Valdebenito, we study positive partially localized solutions of elliptic equations on the entire space. "Partially localized" means that the solutions decay to zero in all but one variable; we examine their behavior in the remaining variable. Our goal is to prove the existence of solutions which are quasiperiodic in the nondecay variable. In the lecture, I will discuss our techniques and report on recent progress. 

Registration
Participation in the meeting is free upon registration.
The registration for the NLM2021 is open!
For more information, please write to gidoni[at]utia[dot]cas[dot]cz
.
Information
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to join this meeting the use of the desktop client or mobile app is required.
If you have not already installed it, you can download it at this link.
Past NLMs
The workshop continues the series of Nonlinear Meetings, which started in Turin (2015).
Visit the websites of the past editions:
Turin2015, Milan2016,
Udine2017, Turin2019,
Milan2020.