DEG1 Christmas Meeting 2023

Seminar room of
Dipartimento di Matematica, Informatica e Geoscienze
University of Trieste

15:00-15:45 – Jean Mawhin (Université Catholique de Louvain)
Periodic solutions of nonlinear systems with generalized variable exponents operators
Abstract. A continuation theorem of Manásevich and the author for periodic solutions of quasilinear systems of the form \[({\mathcal S}(u'))' = f(t,u,u')\] is extended to the more general situation \[({\mathcal S}(t,u'))' = f(t,u,u'),\] where \({\mathcal S}(t,\cdot)\) satisfies some monotonicity and coercivity conditions. The extension covers the case of \(p(t)\)-Laplacians \[(|u'|^{p(t) - 1}u')' = f(t,u,u')\] with continuous \(p(t) > 1\). Some applications are given. This is a joint work with M. Garcia-Huidobro, R. Manásevich and S. Tanaka.

15:45-16:30 – Guglielmo Feltrin (Università degli Studi di Udine)
Exact multiplicity and non-degeneracy of positive solutions for indefinite superlinear problems
Abstract. We analyse the positive solutions to second-order nonlinear ordinary differential equations of the form \(u''+q(t)g(u)=0\), where \(q\) is a sign-changing weight and \(g\) is a superlinear function. First, we review some classical results and recent contributions. Next, by exploiting a shooting technique and the comparison theorem, we present exact multiplicity and non-degeneracy results for positive solutions. Numerical examples and some related conjectures are also discussed. The talk is based on joint work with Christophe Troestler (University of Mons, Belgium).

17:00-17:45 – Fabio Zanolin (Università degli Studi di Udine)
Subharmonic solutions in a model of a periodically driven laser
Abstract. We present a planar ODE system modelling a class of periodically driven lasers and prove the existence of infinitely many subharmonic solutions. It may be interesting to observe that similar geometry and dynamics occur in some, apparently unrelated, classical models for population dynamics. This and other related equations will be considered in the talk. This is a joint work with Lakshmi Burra.

Alessandro Fonda
Andrea Sfecci