06/09/2022, 11:30 — 12:15 — Abreu Faro Amphitheatre
Teresa Faria, Faculdade de Ciências, Universidade de Lisboa

Periodic solutions for systems of impulsive delay differential equations

For a family of periodic systems of differential equations with (possibly infinite) delay and nonlinear impulses, sufficient conditions for the existence of at least one positive periodic solution are established.

The main technique used here is the Krasnoselskii fixed point theorem on cones.

Although fixed points methods have been extensively employed to show the existence of positive periodic solutions to scalar delay differential equations (DDEs), the literature on $n$-dimensional impulsive DDEs is very scarce.

Our criteria are applied to some classes of mathematical biology models, such as Nicholson-type systems with patch structure.

This a joint work with R. Figueroa (University of Santiago de Compostela) [1].

1. Faria, R. Figueroa, Positive periodic solutions for systems of impulsive delay differential equations, Discrete Contin. Dyn. Syst. Ser. B (to appear).

See also

tfaria_IST_IME22_rev1.pdf

06/09/2022, 10:30 — 11:15 — Abreu Faro Amphitheatre
Anibal Rodríguez-Bernal, Universidad Complutense de Madrid

Maximum principle and asymptotic dynamics for some nonlocal diffusion equations in metric measure spaces

06/09/2022, 09:45 — 10:30 — Abreu Faro Amphitheatre
Sérgio Oliva, Universidade de São Paulo

A local/nonlocal diffusion model

In this talk, we present some qualitative properties for solutions to an evolution problem that combines local and nonlocal diffusion operators acting in two different subdomains. The coupling takes place at the interface between these two domains in such a way that the resulting evolution problem is the gradient flow of an energy functional. We prove existence and uniqueness results, as well as that the model preserves the total mass of the initial condition. We also study the asymptotic behavior of the solutions. Besides, we show a suitable way to recover the heat equation at the whole domain from taking the limit at the nonlocal rescaled kernel. Finally, we propose a brief discussion about the extension of the problem to higher dimensions.

See also

apresentacao_sergio_ist_4.pdf

06/09/2022, 09:00 — 09:45 — Abreu Faro Amphitheatre
Bernold Fiedler, Freie Universität Berlin

Three-dimensional Sturm global attractors: orders and progress

05/09/2022, 15:45 — 16:30 — Abreu Faro Amphitheatre
Rui Loja Fernandes, University of Illinois Urbana-Champaign

Non-formal deformation quantization

05/09/2022, 15:00 — 16:15 — Abreu Faro Amphitheatre
Pedro Girão, Instituto Superior Técnico, Universidade de Lisboa

Higher Order Linear Stability and Instability of Reissner-Nordström

05/09/2022, 12:15 — 13:00 — Abreu Faro Amphitheatre
Fábio Tal, Universidade de São Paulo

Rotational Chaos for annular dynamics

05/09/2022, 11:30 — 12:15 — Abreu Faro Amphitheatre
Hugo Tavares, Instituto Superior Técnico, Universidade de Lisboa

Yamabe systems, optimal partitions and nodal solutions to the Yamabe equation

We give conditions for the existence of regular optimal partitions, with an arbitrary number $\ell\geq 2$ of components, for the Yamabe equation on a closed Riemannian manifold $(M,g)$.

To this aim, we study a weakly coupled competitive elliptic system of $\ell$ equations, related to the Yamabe equation. We show that this system has a least energy solution with nontrivial components if $\dim M\geq 10$, $(M,g)$ is not locally conformally flat and satisfies an additional geometric assumption whenever $\dim M=10$. Moreover, we show that the limit profiles of the components of the solution separate spatially as the competition parameter goes to $-\infty$, giving rise to an optimal partition. We show that this partition exhausts the whole manifold, and we prove the regularity of both the interfaces and the limit profiles, together with a free boundary condition.

For $\ell=2$ the optimal partition obtained yields a least energy sign-changing solution to the Yamabe equation with precisely two nodal domains.

See also

HT_IST-IME.pdf

05/09/2022, 10:30 — 11:15 — Abreu Faro Amphitheatre
Nicholas Alikakos, University of Athens

Some Estimates for the Triple Junction Problem on the Plane

05/09/2022, 09:45 — 10:30 — Abreu Faro Amphitheatre
Giorgio Fusco, Università degli Studi dellʼAquila

On the fine structure of minimizers in the Allen-Cahn theory of phase transitions

Let $u^\epsilon$ be a minimizer of the Allen-Cahn energy subjected to Dirichlet condition: \[J_\Omega^\epsilon(u^\epsilon)=\min_{u\vert_{\partial\Omega}=v_0^\epsilon}\int_\Omega\Big(\frac{\epsilon}{2}\vert\nabla u\vert^2+\frac{1}{\epsilon}W(u)\Big)dx,\] where $\Omega\subset\mathbb{R}^n$ is a smooth domain, $\epsilon\gt 0$ a small parameter, $v_0^\epsilon:\partial\Omega\rightarrow\mathbb{R}^m$ the boundary datum and $W:\mathbb{R}^m\rightarrow\mathbb{R}$ a smooth nonnegative potential that vanishes on a finite set: \[A=\{a_1,\ldots,a_N\}=\{W=0\}.\] The zeros $a_1,\ldots,a_N$ of $W$ represent equally preferred phases.

For $\delta\gt 0$ small we study the diffuse interface \[\mathscr{I}^{\epsilon,\delta}=\{x\in\bar{\Omega}:\min_{a\in A}\vert u^\epsilon(x)-a\vert\gt \delta\}.\]

We give sufficient conditions on $\Omega$ and $v_0^\epsilon$ ensuring the connectivity of the diffuse interface.

Then we restrict to two space dimensions and show that one can associate a sort of spine to the diffuse interface: a connected network $\mathscr{G}$ which is contained in $\mathscr{I}^{\epsilon,\delta}$ and divides $\Omega$ in $N$ connected subsets corresponding to the $N$ phases. The network $\mathscr{G}$ has a minimality property: it minimizes an energy, a weighted length, that, under a further condition, yields sharp lower bounds for the energy of minimizers and allows for a quantitative description of $\mathscr{G}$.

See also

GF-Lisbon-22.pdf

09/09/2016, 16:20 — 17:00 — Amphitheatre Pa2, Mathematics Building
Carlos Rocha, Instituto Superior Técnico, Universidade de Lisboa

On a definition of Morse-Smale evolution processes

We consider a definition of Morse-Smale evolution process that extends the notion of Morse-Smale dynamical system to the nonautonomous framework. In particular we consider nonautonomous perturbations of autonomous systems. In this case our definition of Morse-Smale evolution process holds for perturbations of Morse-Smale autonomous systems with or without periodic orbits. We establish that small nonautonomous perturbations of autonomous Morse-Smale evolution processes derived from certain nonautonomous differential equations are Morse-Smale evolution processes. We apply our results to examples of scalar parabolic semilinear differential equations generating evolution processes and possessing periodic orbits.

This is a joint work with Radoslaw Czaja and Waldyr Oliva.

09/09/2016, 15:40 — 16:20 — Amphitheatre Pa2, Mathematics Building
Mahendra Panthee, Universidade Estadual de Campinas

Well-posedness for multicomponent Schrödinger-gKdV systems and stability of solitary waves with prescribed mass

In this talk we discuss the well-posedness issues of the associated initial value problem, the existence of nontrivial solutions with prescribed $L^2$-norm, and the stability of associated solitary waves for two classes of coupled nonlinear dispersive equations. The first model describes the nonlinear interaction between two Schrödinger type short waves and a generalized Korteweg-de Vries type long wave and the second one describes the nonlinear interaction of two generalized Korteweg-de Vries type long waves with a common Schrödinger type short wave. The results here extend many of the previously obtained results for two-component coupled Schrödinger-Korteweg-de Vries systems.

This is a joint work with Adan J. Fernandes and Santosh Bhattarai.

09/09/2016, 15:00 — 15:40 — Amphitheatre Pa2, Mathematics Building
Radoslaw Czaja, University of Silesia at Katowice

Attractors for Dynamical Systems with Impulses

In this talk I will formulate a suitable definition of a global attractor for impulsive dynamical systems, which model the evolution of a continuous process interrupted by abrupt changes of state.

An impulsive dynamical system (IDS) consists of a continuous semigroup $\{\pi(t) : t\geq 0\}$ on a metric space $X$, a nonempty closed subset $M$ of $X$ called an impulsive set, which is “transversal” to the flow of the semigroup, and a continuous function $I : M \to X$ called an impulsive function. Whenever a trajectory for the semigroup $\pi$ hits the set $M$ the impulsive function $I$ redirects it to a new state, defining an impulsive trajectory. Assuming that all impulsive trajectories exist for all times $t\geq 0$, we obtain a possibly discontinuous semigroup $\{\tilde{\pi}(t):t\geq 0\}$.

To describe the long-time behavior of $\tilde{\pi}($ we introduce the notion of a global attractor $\mathcal{A}\subset X$, which is precompact, $\mathcal{A} = \overline{\mathcal{A}}\setminus M$, $\tilde{\pi}$-invariant and $\tilde{\pi}$-attracting all bounded subsets of $X$. Such sets are more suitable for impulsive dynamical systems and better describe their dynamics than compact global attractors known for continuous semigroups.

I will present several properties for this class of precompact global attractors and a theorem on existence of such attractors. The theory has applications to chosen ordinary and partial differential equations with impulsive functions. This is a joint work with E. M. Bonotto, A. N. Carvalho and R. Collegari from the University of São Paulo in São Carlos and M. C. Bortolan from the Federal University of Santa Catarina, Brazil ([1, 2]).

References

1. E. M. Bonotto, M. C. Bortolan, A. N. Carvalho, R. Czaja, Global attractors for impulsive dynamical systems - a precompact approach, Journal of Differential Equations, 259 (2015), 2602{2625, doi:10.1016/j.jde.2015.03.033.
2. E. M. Bonotto, M. C. Bortolan, R. Collegari, R. Czaja, Semicontinuity of attractors for impulsive dynamical systems, to appear in Journal of Differential Equations, doi:10.1016/j.jde.2016.06.024.

09/09/2016, 12:00 — 12:40 — Amphitheatre Pa2, Mathematics Building
Ademir Pastor, Universidade Estadual de Campinas

Scattering for a 3D Coupled Nonlinear Schrödinger System

We consider the three-dimensional cubic nonlinear Schrödinger system \[ \begin{cases} i \partial_t u + \triangle u + (|u|^2+\beta |v|^2)u = 0 , \\ i \partial_t v + \triangle v + (|v|^2+\beta |u|^2)v = 0 . \end{cases} \] Let $(P;Q)$ be any ground state solution of the above Schrödinger system. We show that for any initial data $(u_0,v_0)$ in $H^1(\mathbb{R}^3) \times H^1(\mathbb{R}^3)$ satisfying $M(u_0,v_0)A(u_0,v_0) < M(P,Q)A(P,Q)$ and $M(u_0,v_0)E(u_0,v_0) < M(P,Q)E(P,Q)$, where $M(u,v)$ and $E(u,v)$ are the mass and energy (invariant quantities) associated to the system, the corresponding solution is global in $H^1(\mathbb{R}^3) \times H^1(\mathbb{R}^3)$ and scatters. Our approach is in the same spirit of Duyckaerts-Holmer-Roudenko, where the authors considered the 3D cubic nonlinear Schrödinger equation. Joint work with L.G. Farah (UFMG).

09/09/2016, 11:20 — 12:00 — Amphitheatre Pa2, Mathematics Building
Ronaldo Garcia, IME, Universidade Federal de Goiás

Partially Umbilic Singularities of Hypersurfaces of $\mathbb{R}^4$ and of Plane Fields of $\mathbb{R}^3$

In this talk will be established the geometric structure of the lines of principal curvature of a hypersurface immersed in $\mathbb{R}^4$ in a neighborhood of the set $\mathcal{S}$ of its principal curvature singularities, consisting of the points at which at least two principal curvatures are equal. Under generic conditions defined by appropriate transversality hypotheses it is proved that $\mathcal{S}$ is the union of regular smooth curves $\mathcal{S}_{12}$ and $\mathcal{S}_{23}$, consisting of partially umbilic points, where only two principal curvatures coincide. This curve is partitioned into regular arcs consisting of points of Darbouxian types $D_1$, $D_2$, $D_3$, with common boundary at isolated semi-Darbouxian transition points of types $D_{12}$ and $D_{23}$.

Also a similar result is obtained for principal lines associated to plane fields of $\mathbb{R}^3$.

09/09/2016, 10:20 — 11:00 — Amphitheatre Pa2, Mathematics Building
Miguel Abreu, Instituto Superior Técnico, Universidade de Lisboa

On the mean Euler characteristic of Gorenstein toric contact manifolds

In this talk I will prove that the mean Euler characteristic of a Gorenstein toric contact manifold is equal to half the normalized volume of the corresponding toric diagram. I will also give some immediate applications of this result. This is joint work with Leonardo Macarini.

09/09/2016, 09:40 — 10:20 — Amphitheatre Pa2, Mathematics Building
Sérgio Oliva, IME, Universidade de São Paulo

Human mobility in epidemic models and nonlocal diffusions

Following Brockmann, where human mobility is introduced in a simple SIR model, we get a Reaction Diffusion equation with fractional power diffusion. The first interesting mathematical and epidemiological question is how to characterize the existence of positive equilibrium in these equations. We also present a correlation network between occurrences of reported cases of dengue between cities in the state of Rio de Janeiro, Brazil.

09/09/2016, 09:00 — 09:40 — Amphitheatre Pa2, Mathematics Building
Bernold Fiedler, Freie Universität Berlin

Sturm global attractors: the three-dimensional balls

Sturm attractors $\mathcal{A}$ are global attractors of dissipative PDEs \[ u_t = u_{xx}+f(x,u,u_x), \] for $0 \lt x \lt 1$, , say with Neumann boundary conditions and hyperbolic equilibria $v$. The collection of unstable manifolds $c_v=W^u(v)$ forms a regular complex of cells $c_v \in \mathcal{C}$, in the sense of algebraic topology. Caveat: this is true for Sturm attractors, but totally wrong in general! We call $\mathcal{C}$ the Sturm complex.

We describe all $3$-ball Sturm complexes, i.e., all $\mathcal{C}$ which arise from the closure of a single $3$-cell $c_0$. We relate our description to the associated Fusco-Rocha meanders, and to the braid $x\mapsto(x,v(x),v_x(x))$ of equilibrium spaghetti. In particular we explain why, against all intuition, the top and bottom equilibria must be chosen as adjacent corners, in the octahedral $3$-cell complex, rather than antipodally opposite.

All results hold, equally, for the Jacobi systems studied by Waldyr M. Oliva. This is joint work with Carlos Rocha, and contains artwork by Anna Karnauhova.

See also http://dynamics.mi.fu-berlin.de/.

08/09/2016, 17:00 — 17:40 — Amphitheatre Pa2, Mathematics Building
Philippo Lappicy, Freie Universität Berlin

Einstein Constraints: A Dynamical Approach

The Einstein constraint equations describe the space of initial data for the evolution equations, dictating how space should curve within spacetime. Under certain assumptions, the constraints reduce to a single quasilinear parabolic equation on the sphere with various singularities, and nonlinearity being the prescribed scalar curvature of space. We focus on self-similar solutions of Schwarzschild type. Those describe, for example, the initial data of black holes. We give a detailed study of the axially symmetric solutions, since the domain is now one dimensional and nodal properties can be used to describe certain asymptotics of the rescaled self-similar solutions. In particular, we mention examples for certain prescribed scalar curvatures.

08/09/2016, 16:20 — 17:00 — Amphitheatre Pa2, Mathematics Building
Xavier Carvajal, IM, Universidade Federal do Rio de Janeiro

Local well-posedness and dissipative limit of high dimensional KdV-type equations

Considered in this work is an $n$-dimensional dissipative version of the Korteweg-deVries equation. Our goal here is to investigate the well-posedness issue for the associated initial value problem in the anisotropic Sobolev spaces. We also study the limit behavior of this equation when the dissipative effects are reduced.