• March 25 at 1 pm (Brasilia time zone), Maurizio Garrione  (Politecnico di Milano)

Title:  Wave fronts for reaction-diffusion models with nonlinear diffusive term: an overview of recent results

Abstract: We present some recent results regarding fronts for reaction-diffusion models governed by a nonlinear (curvature type) diffusion, discussing in particular their existence and the shape of the critical profile for vanishing diffusivity. 

• April 8 at 1 pm (Brasilia time zone), Emanuele Pastorino (Politecnico di Milano)

Title: Unpredictable behavior of a partially damped systems of PDEs modeling suspension bridges 

Abstract:  We consider a nonlinear nonlocal coupled system of beam-wave equations, governing the dynamics of a degenerate plate modeling the behavior of suspension bridges. In line with the physical observations, the beam equation is both damped and forced, whereas the wave equation is “isolated”. Since the resulting overall damping is degenerate, the full dynamical system is only partially dissipative. This leads to an unpredictable behavior for the solutions of the system, which prevents to forecast the general behavior of bridges, encompassing the instability of the unique stationary solution and the richness of the ω-limit set, which contains infinitely many periodic solutions. We also discuss some striking differences between the sole beam equation and the coupled system, which may be the cause of some not fully understood paradoxes and erroneous conclusions. The talk is based on joint work with Maurizio Garrione and Filippo Gazzola (Politecnico di Milano).

VIDEO (Inglês/English): Link

• April 22 at 1 pm (Brasilia time zone), Marcone Côrrea Pereira (IME - USP)

Title: Rough Boundaries in Thin Domains with Signorini Constraints

Abstract: In this presentation, we explore a reaction-diffusion problem within a thin planar domain, subject to a Signorini-type boundary condition that models boundary-catalyzed reactions. Motivated by applications in microfluidics and free-boundary problems, we consider scenarios where the boundary exhibits resonant roughness—meaning the amplitude and period of the surface irregularities are on the same scale as the domain’s thickness. Depending on the intensity of the reaction mechanism governed by the Signorini condition, we identify three distinct behavioral regimes. Notably, we highlight the critical regime in which the influence of the domain’s geometry and all physically relevant aspects of the process are fully captured. This is a collaborative work with Prof. Antonio Gaudiello (Università degli Studi della Campania Luigi Vanvitelli) and Jean Carlos Nakasato (IME-USP).

• May 6 at 1 pm (Brasilia time zone), Delfina Gomez (Universidad de Cantabria)

Title:  Asymptotic analysis for spectral problems in thin structures

Abstract: We treat a spectral problem for the Laplace operator defined in some multi-structures where the thickness of each component depends on a small parameter. We assume mixed boundary conditions. We address the asymptotic behavior of the eigenvalues and  the corresponding eigenfunctions of such  problems as the thickness  vanish. In particular, we describe the behavior of the eigenfunctions associated with the so-called low and high frequencies.

Based on joint works with Antonio Gaudiello and Maria-Eugenia Pérez-Martínez.

• May 26 (MONDAY) at 1 pm (Brasilia time zone), Manuel Villanueva-Pesqueira (Universidad de Comillas)

Title: Beyond Periodicity: Weakly Oscillating Thin Domains

Abstract: We study the asymptotic behavior of solutions to ellipticPDEs posed in thin domains with weakly oscillating boundaries, including scenarios that go beyond classical periodicity such as quasi-periodic or almost-periodic geometries. Using dimension reduction and homogenization techniques, we rigorously derive effective limit equations that capture the effects of the domain’s geometry. In one setting, we also incorporate boundary-concentrated reaction terms supported in a narrow, oscillating strip near the upper boundary.

VIDEO (Inglês/English): Link

• June 3 at 1 pm (Brasilia time zone), Christian Pötzsche (Universität Klagenfurt)

Title: Evans function, parity and nonautonomous bifurcation

Abstract: We provide a functional-analytical approach to the bifurcation theory of bounded entire solutions of differential equations. On the one hand, it is rather specific in the sense that it applies only to unstable solutions of at least planar and necessarily nonautonomous problems. On the other hand, it can provide a detailed insight into the set of bounded entire solutions. 

Initial point of our endeavor is the Evans function, an established concept for the stability analysis of traveling wave solutions to evolutionary partial differential equations. We relate this to the parity [1], which is a crucial tool in the abstract bifurcation theory of nonlinear Fredholm operators. It is illustrated that isolated zeros of the Evans function imply that critical spectral intervals of the Sacker-Sell (dichotomy) spectrum split, while sign changes of the Evans function are sufficient for local and global bifurcations of whole continua of bounded entire solutions. Previously known criteria for such nonautonomous bifurcations result as special cases. This is a joint work with Iacopo Longo (Imperial College, London, UK) and Robert Skiba (Nicolaus Copernicus University, Toruń, Poland) 

References: 

[1] J. Pejsachowicz, P. Rabier. Degree theory for $C^1$ Fredholm mappings of index 0, Journal d'Analyse Math{\'e}matique 76(1), 289–319, 1998.

[2] C. Pötzsche. Nonautonomous bifurcation of bounded solutions: Crossing curve situations, Stoch. Dyn. 12(2), 1150017, 2012.

[3] B. Sandstede. Stability of travelling waves, in Handbook of Dynamical Systems 2 (ed. B. Fiedler), Elsevier Science, Amsterdam, 983--1055, 2002. 

VIDEO (Inglês/English): Link

• June 26 (THURSDAY) at 1 pm (Brasilia time zone), Francesco Ferraresso (University of Verona)

Title:  Biharmonic and elasticity eigenvalue problems in singularly perturbed domains

Abstract: PDF file

VIDEO (Inglês/English): Link

• March 12 at 2 p.m. (Brasilia time zone), Pedro Tavares Paes Lopes (USP)

Title: Semigrupos analíticos na classe de funções contínuas gerados por operadores pseudodiferenciais elípticos 

Abstract: Apresentaremos métodos para mostrar que um operador diferencial gera um semigrupo analítico na classe de funções contínuas. Em particular, mostraremos o argumento de Herbert Amann e como ele pode ser estendido para operadores pseudodiferenciais, que incluem aplicações em problemas que envolvem operadores fracionários e do tipo Dirichlet-to-Neumann. (Trabalho baseado na tese de Weymar Andres Astaiza Sulez) 

VIDEO (Portuguese/Português):  Soon.

• April 9 at 2 p.m., Pedro Marín Rubio (Universidad de Sevilla)

Title: Longtime behavior of a non-autonomous 3D Navier-Stokes-Voigt model with and without

Abstract: The 3d Navier-Stokes-Voigt model was proposed by Cao, Lunasin and Titi (cf. [Cao, Lunasin, Titi, Commun. Math. Sci. 2006]) as a regularization of the 3d-Navier-Stokes equations for the purpose of direct numerical simulations. It is a non classical diffusion equation, which allows well-posedness but no immediate smoothing of the solution. Indeed the semigroup behaves asymptotically compact, as damped hyperbolic systems. Long-time behavior has been analyzed by many authors in several situations. We survey some old results (cf. [J. García-Luengo, PMR, J. Real, Nonlinearity 2012]) in the case of a non-autonomous variation of the problem (but without delay) and, if possible, some recent results in the case of delay terms included (cf. [J. García-Luengo, PMR, EJQTDE 2024]). Existence and regularity of pullback attractors will be discussed.

This work has been done in collaboration with Julia García-Luengo (Universidad Politécnica de Madrid).

References:

J. García-Luengo, P. Marín-Rubio, J. Real, Pullback attractors for three-dimensional non-autonomous Navier-Stokes-Voigt equations, Nonlinearity 25 (2012), 1-26.

J. García-Luengo, P. Marín-Rubio, Existence and regularity of pullback attractors for a 3D non-autonomous Navier-Stokes-Voigt model with finite delay, Electronic Journal of Qualitative Theory of Differential Equations, 14 (2024), 1-35. 

VIDEO (English/Inglês): Link

• May 7 (Tuesday) at 2 p.m., Oleksiy Kapustyan (Taras Shevchenko National University of Kyiv)

Title: Attractors for infinite-dimensional impulsive systems

Abstract: The qualitative behavior of dissipative infinite-dimensional continuous dynamical systems is investigated by the methods of the global attractor theory. The main problem we face when we try to expand this theory to impulsive DS is the lack of continuous dependence on the initial data.  Our approach  is based on the notion of uniform attractor, commonly used for non-autonomous problems, in particular, for systems with impulses at fixed moments of time. Sufficient conditions for existence of uniform attractors and investigation their properties for semilinear PDEs with impulsive perturbations are proposed.

VIDEO (Inglês/English): Link

• May 21 (Tuesday) at 2 p.m., Gabriela Planas (UNICAMP)

Title: On the decay of solutions for some nonlinear systems in critical spaces

Abstract: In this talk, I will discuss the decay of solutions to the Navier-Stokes, the Navier-Stokes-Coriolis, and the nonlinear heat equations in critical spaces.

Using the Fourier Splitting Method and using properties arising from the scale invariance, we obtain an algebraic upper bound for the decay rate of solutions.

• June 4 (Tuesday) at 2 p.m., Maria-Eugenia Pérez-Martínez (Universidad de Cantabria)

Title: Thresholds in high contrast boundary homogenization problems on sieve-like structures.

Abstract: We analyse the asymptotic behaviour of solutions of boundary homogenization problems for the elasticity system on sieve-like structures with rapidly alternating boundary conditions of Winkler-Robin type containing reaction parameters in small regions. These boundary conditions can be nonlinear and the parameters can range from very large to very small conditioning the averaged behaviour of the solutions.  Depending on the relations between the parameters (period, reaction and sizes of the reaction regions), the homogenization of linear boundary conditions can imply the same asymptotic  behaviour. We provide a general framework for different operators and state the spectral convergence for associated spectral problems.   The talk is based on joint works  [1-3] and  some recent works in progress.

1.- D. Gómez,  S.A. Nazarov and M.-E. Pérez-Martínez:  Asymptotics for spectral problems with rapidly alternating boundary conditions on a strainer Winkler foundation. J. Elasticity. 142, 89-120, 2020.

2.- D. Gómez and M.-E Pérez-Martínez:  Boundary homogenization with large reaction terms on a strainer-type wall.  ZAMP,  73:234, p. 1-28, 2022.

3.- D. Gómez and M.-E Pérez-Martínez:  Averaged reaction for nonlinear boundary conditions on a grill-type Winkler foundation. MMA J., to appear, 2024.

• June 18 (Tuesday) at 2 p.m., Nataliia Goloshchapova (IME- USP)

Title: Introduction to Schrödinger models with point interactions: from linear to nonlinear case

Abstract: In this talk, we will introduce linear and nonlinear Schrödinger models with point interactions. 

First, we will explore 1D, 2D, and 3D Schrödinger operators with delta interactions at discrete points, focusing on their rigorous definitions and spectral properties. Historically, the seminal work on one-dimensional linear models with delta interactions was by Kronig and Penney in 1931, describing a nonrelativistic electron in a fixed crystal lattice. Later, Bethe, Peierls, and Thomas discussed three-dimensional models for nonrelativistic quantum particles interacting with "very short range" potentials.

Next, we will discuss the Nonlinear Schrödinger Equation (NLSE) with delta interaction on a line and on metric graphs, emphasizing their variational and stability properties. The standard NLSE with point interaction has been proposed as an effective model for Bose-Einstein Condensates (BEC) with defects or impurities. Applications extend to graph-like structures, such as planar self-focusing waveguides and various fiber optics devices.

In the third part of the talk, we will examine the NLSE with point-concentrated nonlinearity (nonlinear delta potential), focusing on its variational properties. Interest in this model has grown due to its applications in solid-state and condensed matter physics, such as charge accumulation in semiconductor interfaces, nonlinear propagation in Kerr-type media with localized defects, and BECs in optical lattices with isolated defects created by focused laser beams.

Finally, we will highlight open problems and future directions in the field.

VIDEO (Português/Portuguese): Link

• July 2 (Tuesday) at 2 p.m., Sergey Tikhomirov (PUC- RIO)

Title: Depinning bifurcation of travelling waves in ergodic media

Abstract: We study bistable reaction-diffusion equations in heterogeneous media. In such systems it is known phenomena of pinning – the solution forms stationarily front while bistable nonlinearity is asymmetric. With the change of parameters of the system stationarily front may transform to a traveling wave. Such bifurcation is called depining. It is believed that for periodic media the nature of depinning is similar to saddle-node bifurcation.

We study the case when heterogeneous media is ergodic, such assumption includes periodic and quasiperiodic media. We provide a set of conceptual assumptions under which we can prove power-law asymptotics for the speed with parameters depending on the dimension of the ergodic measure. The approach is based on invariant manifolds of dynamical systems in infinite-dimensional space, bifurcations in presence of symmetry group and ergodic theory.

VIDEO (Inglês/English):  Link

• August 20th (exceptionally on Tuesday) at 2 p.m., Juliana Fernandes (UFRJ)

Title: The Nehari manifold for a degenerate logistic parabolic equation

Abstract: Our main goal is to exploit the interplay between variational methods and dynamical systems. In particular, we use the so-called Nehari manifold to provide a picture of the phase plane, divided into regions of convergence over time. For the related semilinear stationary elliptic equation, general questions on the existence, multiplicity, and nonexistence of positive solutions have been discussed. The existence problem for sign-changing solutions is somewhat more delicate, with only a few results available in the literature. We tackle this problem by using Ambrosetti and Rabinowitz Mountain Pass Theorem on Nehari.

This is a joint work with L. Maia.

VIDEO (Inglês/English) : Link 

• September 18th, at 2 p.m., German Jesus Lozada Cruz (UNESP)

Title: Continuity of attractors for parabolic problems in domains with a small hole

Abstract: In this talk we discuss the continuity of global attractors for a class of semilinear parabolic problems with Dirichlet boundary conditions in domains with a small hole whose size is proportional to a small positive parameter ε.

             The talk will be based on a joint work with Elaine Tavares-Lima (IBICE/UNESP).

References

[1] Arrieta, J. M.; Carvalho, A. N. Spectral convergence and nonlinear dynamics of reaction-diffusion equations under perturbations of the domain, J. Differential Equa-

tions, 199 (2004), 143–178.

[2] Borisov, D. I.; Mukhametrakhimova, A. I. The norm resolvent convergence for elliptic operators in multi-dimensional domains with small holes, J. Math. Sci.(N.Y.)              232 (2018), 283–298.

[3] Tavares-Lima, E. A.; Lozada-Cruz, G. J., Dynamics of parabolic equations in domains with a small hole I. Continuity of the set of equilibria. Submitted por publication, 2024.

[4] Tavares-Lima, E. A.; Lozada-Cruz, G. J., Dynamics of parabolic equations in domains with a small hole II. Continuity of the attractors. Journal of Differential Equations 395 (2024) 308–332.

          VIDEO (Português/Portuguese): Link

• October 2nd, at 2 p.m., Everaldo de Mello Bonotto (ICMC - USP)

Title: On periodic solutions of abstract generalized ODEs and applications 

Abstract: It is known that generalized ODEs encompass other types of equations such as impulsive differential equations as well as functional differential equations. In this talk, we present the theory of periodic solutions in the framework of generalized ODEs in Banach spaces. We exhibit necessary and sufficient conditions for a solution of a generalized ODE to be periodic. Moreover, we develop the Floquet theory of homogeneous linear generalized ODEs and, as a consequence, we present a characterization of fundamental matrices for the finite dimensional case. As an illustration, we apply the main results to a class of impulsive differential equations.

VIDEO (Português/Portuguese): Link

• October 16th, at 2p.m., Yulia Petrova (PUC- Rio)

Title: Propagating terrace in a semi-discrete model of Incompressible Porous Medium (IPM) equation

Abstract: In the talk we will discuss gravitational fingering phenomenon - the unstable displacement of miscible liquids in porous media with the speed determined by Darcy's law. Laboratory and numerical experiments show the linear growth of the mixing zone, and we are interested in determining the exact speed of propagation of fingers. We consider two models: incompressible porous medium equation (IPM) and Transverse Flow Equilibrium (TFE). The existing theoretical upper bounds for the growth rate of the mixing zone (see e.g. Otto-Menon'2005) are higher than the observed speed from the numerical simulations. We believe that one of the possible mechanisms of slowing down the fingers' growth is due to convection in the transversal direction, and explain it by introducing a semi-discrete model of IPM and TFE.

The model consists of a system of advection-reaction-diffusion equations on concentration, velocity and pressure in several vertical tubes (real lines) and interflow between them. In the simplest setting of two tubes we show the structure of gravitational fingers - the profile of propagation is characterized by two consecutive travelling waves which we call a terrace. We prove the existence of such a propagating terrace for the parameters corresponding to small distances between the tubes. The main tool in the proof is the reduction of IPM model to TFE model using geometric singular perturbation theory. 

The talk is based on joint work with S. Tikhomirov and Ya. Efendiev (arXiv:2401.05981, to appear in SIMA)

VIDEO (Inglês/English): Link

• October 30th,  at. 2 p.m., Piotr Kalita (Jagiellonian University)

Title: Stability of phase portrait for a gradient ODE with memory

Abstract: We consider the problem governed by the gradient ODE $x'=\nabla F(x)$ in $\mathbb{R}^d$ on which we assume that it has a finite number of hyperbolic equilibria whose stable and unstable manifolds intersect transversally. The problem is perturbed by the memory term $x'=\nabla F(x) + \varepsilon\int_{-\infty}^t M(t-s)x(s)\, ds$ where $\varepsilon>0$ and $M(s)$ decay exponentially. The key result is that the structure of connections between the equilibria of the unperturbed problem is exactly preserved in the infinite dimensional dynamical system for a small $\varepsilon > 0$. The talk is based on a preprint https://arxiv.org/abs/2406.00910 which is a joint work with Piotr Zgliczyński.

VIDEO (Inglês/English): Link

• November 27th, at 2 p.m., Daniel Alberto Morales Ramirez (PhD IME-USP)

Title: Pullback attractors for a semilinear parabolic equation with nonlinear Neumann boundary conditions and time-varying domains

Abstract: We are interested in studying a semilinear parabolic problem with nonlinear Neumann boundary conditions and domains varying with the time parameter. We will prove the existence and uniqueness of solutions to this equation and prove the existence of pullback attractors.

VIDEO (Português/Portuguese): Link

• December 6 at 14h, Matheus Bortolan (UFSC)

Title: Weak global attractor for the 3D-Navier-Stokes 

Abstract: In this work we construct a set that attracts a class of solutions of the 3D-Navier-Stokes equations in the weak topology of L2(𝛀). This set is also weakly compact and, considering this class of solutions, is invariant. We call this set the ``weak global attractor'' for the 3D-Navier-Stokes equations. To construct this set and this specific class of solutions we use the unique global solutions of a collection of ``globally modified Navier-Stokes equations''. 

This is a joint work with Alexandre Carvalho (ICMC-USP), Pedro Marín-Rubio (Universidad de Sevilla) e José Valero (Universitas Miguel Hernández). 

Video (Portuguese): Link

• November 22 at 14h, José Valero (Universidad Miguel Hernández de Elche)

Title: Convergence of solutions and attractors for reaction-diffusion equations governed by a fractional Laplacian

Abstract: We study a nonlocal reaction-di§usion equation governed by a fractional Laplacian. First, we consider the case when the domain is bounded. We prove the strong convergence of solutions of the equation governed by the fractional Laplacian to the solutions of the classical equation governed by the standard Laplacian, when the fractional parameter grows to 1. Then, in the autonomous situation, we establish the upper semicontinuity of global attractors with respect to the attractor of the limit problem. Second, we study the situation when the domain is the whole space ℝn. In this case, we prove the convergence of solutions with respect to the weak topology. 

Video (English): Link

• November 8 at 14h, Marcio A. Jorge da Silva (UEL)

Title: Kolmogorov ε-Entropy Analysis of Attractors in Extensible Beams with Inherent Polynomial Dynamics.

Abstract: The main objective of this talk is to present a model for hyperbolic evolution equations that pertain to extensible beams with a nonlocal, potentially degenerate damping coefficient of the Balakrishnan-Taylor type. The peculiar damping structure prevents any form of exponential behavior within the weak phase space over time, while allowing for an inherent polynomial range of stability as proved for the corresponding Lyapunov functional. Consequently, we give a preliminary examination of the long-term behavior of the associated dynamical system by analyzing the Kolmogorov ε-entropy of the family of compact attractors.

Video (Portuguese): Link

• October 25 at 14h, Angela Jiménez Casas (Universidad Pontificia Comillas)

Title: Damped wave equation

Abstract: We show how solutions of a wave equation with distributed damping near the boundary converge to solutions of a wave equation with boundary feedback damping.

Sufficient conditions are given for the convergence of solutions to occur in the natural energy space.

• October 11 at 14h, Flank David Morais Bezerra (UFPB)

Title: Problemas de Cauchy de ordem superior e potências fracionárias 

Abstract: Nesta palestra apresentaremos alguns resultados relacionados à existência e regularidade de soluções para alguns problemas de Cauchy de ordem superior com uma abordagem via potências fracionárias de operadores lineares.

Video (Portuguese):  Link

• September 27 at 14h, Mirelson Martins Freitas (UFPA)

Title: Global Attractors and Synchronization of Coupled Critical Lamé Systems

Abstract: We investigate the global attractors and synchronization phenomenon of a coupled Lamé system defined on a smooth bounded domain Ω ⊂ R3 with nonlinear forces of critical growth. Firstly, we prove the existence, finiteness of fractal dimension, and smoothness of global attractors. Considering κ as the coupling coefficient, we prove the upper semicontinuity of the attractors as κ → ∞. Finally, we study the asymptotic synchronization for the coupled Lamé system.

Video (Portuguese): Link

• September 13 at 14h, Arthur Cavalcante Cunha (UFBA)

Title: Existence and stability of pullback exponential attractors for a nonautonomous semilinear evolution equation of second order 

Abstract: We consider a nonautonomous semilinear evolution problem of second order and investigate the existence and stability of a family of pullback exponential attractors for our problem under suitable growth and dissipativity conditions. Moreover, we also prove the upper and lower semicontinuity of this family of pullback exponential attractors. Finally, we guarantee the existence of the pullback attractor in an appropriate space, prove its upper semicontinuity and, lastly, obtain a regularity result of this pullback attractor.

V.T. Azevedo, E.M. Bonotto, A.C. Cunha, M.J.D. Nascimento, Existence and stability of pullback exponential attractors for a nonautonomous semilinear evolution equation of second order, Journal of Differential Equations v.365, p. 521-559, 2023.

Video (Portuguese) : Link

• August 16 at 14h, Alexandre Nolasco de Carvalho (ICMC - USP)

Title: Bifurcation and hyperbolicity for a nonlocal quasilinear parabolic problem 

Abstract: In this talk, we study the scalar one-dimensional nonlocal quasilinear problem of the form ut=a(||u_x||2)uxx+𝜈 f(u), with Dirichlet boundary conditions on the interval [0,𝜋], where a: R+→[m,M]⊂ (0,∞) and f: R→R are continuous functions that satisfy suitable additional conditions. We give a complete characterization of the bifurcations and hyperbolicity for the corresponding equilibria. With respect to bifurcation, the existing result requires that the function a(.) be non-decreasing and shows that bifurcations are pitchfork supercritical bifurcations from zero. We extend these results to the case of a general smooth nonlocal diffusion function $a(.)$ and show that bifurcations may be pitchfork or saddle-node, both subcritical or supercritical. Concerning hyperbolicity, we specify necessary and sufficient conditions for its occurrence. We also explore some examples to exhibit the variety of possibilities, depending on the choice of the function a(.), that may occur as the parameter 𝜈  varies.

Video (English):  Link

• July 12 at 14h, Sofia B. S. D. Castro (Universidade do Porto)

Title: Switching dynamics near heteroclinic networks

Abstract: This talk discusses the dynamics near a heteroclinic network, that is, near a connected union of two or more heteroclinic cycles. Near a node/equilibrium which belongs to more than one cycle, trajectories can follow one of the outgoing connections at this node and therefore, follow different cycles. Switching dynamics refers to the fact that trajectories can ‘switch’ from following one of the cycles to following a different one at such a node. Switching dynamics can be simple or very complex depending on the possible choices that are available. I shall show that, near a heteroclinic network such that the Jacobian matrix at each node has only real eigenvalues, only finite switching exist. This means that not all arbitrary combinations of paths along the network can be followed by nearby trajectories. This result does not mean that for such networks the dynamics are uninteresting, as some game theory problems illustrate.

Video: Link

• June  28 at 14h, Leandro del Pezzo (Universidad de Buenos Aires)

Title: The fundamental solution of the fractional $p-$Laplacian 

Abstract: In this talk, we find the fundamental solution of the fractional p-laplacian and use them to prove two different Liouville-type theorems. The non-existence Liouville-type results are for p-superharmonic  functions and for the fractional p-laplacian operator plus a nonlinearity. This is a joint work with Alexander Quaas.

Video: Link

• June 14 at 14h, Philippe Souplet (Université Sorbonne Paris Nord)

Title: Diffusive Hamilton-Jacobi equations and their singularities 

Abstract: We consider the diffusive Hamilton-Jacobi equation $u_t-\Delta u=|\nabla u|^p$ with homogeneous Dirichlet boundary conditions, which plays an important role in stochastic optimal control theory and in certain models of surface growth (KPZ). Despite its simplicity, in the superquadratic case p>2 it displays a variety of interesting and surprising behaviors. We will discuss two classes of phenomena: - Gradient blow-up (GBU) on the boundary: time rate, single-point GBU, space and time-space profiles, Liouville type theorems and their applications; - Continuation after GBU as a global viscosity solution with loss and recovery of boundary conditions. In particular, we will present the complete classification of solutions in one space dimension, which describes the losses and recoveries of boundary conditions at multiple times, as well as all the possible GBU and recovery rates. This talk is based on a series of joint works in collaboration with A. Attouchi, R. Filippucci, Y. Li, N. Mizoguchi, A. Porretta, P. Pucci, Q. Zhang.

Video: Link 

• May 31 at 14h, Tomás Caraballo (Universidad de Sevilla) 

Title: Global versus random attractors: effects of noise on dynamical systems 

Abstract: The aim of this talk is to present some features concerning the effects of noise on the asymptotic behaviour of dynamical systems. It is well-known now the stabilizing and destabilizing effects which the appearance of different kinds of noise (e.g. Ito or Stratonovich) may have on the stationary solutions (equilibria) of deterministic dynamical systems. Now we will report some results on the appearance of exponentially stable stationary (in the stochastic sense) solutions  when some noise is added to the model, as well as, the analysis of the existence of random attractor when the deterministic model is not known to have (or does  not have) a global attractor. These results will show some kind of stabilization on global attractors instead of only  on equilibria.

Video: Link

• May 17 at 14h, Mariya Ptashnyk (Heriot-Watt University) 

Title: From individual-based models to continuum descriptions: Modelling and analysis of interactions between different populations.

Abstract: First we will show that the continuum counterpart of the discrete individual-based mechanical model that describes the dynamics of two contiguous cell populations is given by a free-boundary problem for the cell densities.  Then, in addition to interactions, we will consider the microscopic movement of cells and derive a fractional cross-diffusion system as the many-particle limit of a multi-species system of moderately interacting particles.

Video: Link

• May 03 at 14h, Richard Höfer (Universität Regensburg)

Title: Homogenization of the Navier-Stokes equations in perforated domains in the inviscid limit

Abstract: We revisit homogenization problems of fluid flows in perforated domains which have applications in porous media and particulate flows. We consider the solution $u_\eps$ to the Navier-Stokes equations in $\R^3$ perforated by small particles centered at $(\eps \Z)^3$ with no-slip boundary conditions at the particles. We study the behavior of $u_\eps$  for small $\eps$, depending on the diameter $\eps^\alpha$, $\alpha > 1$, of the particles and the viscosity $\eps^\gamma$, $\gamma > 0$, of the fluid. If the local Reynolds number on the length-scale of the particles 

Video: Link

• April 19 at 14h, Julián López-Gómez (Universidad Complutense de Madrid)

Title: Metasolutions 

Abstract: Metasolutions were designed to characterize the dynamics of some important classes of semilinear parabolic equations in Population Dynamics in the presence of spatial heterogeneities. In this talk we are going to give a general introduction to them.

Video: Link

• April 5 at 14h, Bernold Fiedler (Freie Universität Berlin) 

Title: Time reversible attractors of parabolic PDEs: a meandering tale of three noses 

Abstract: As a paradigm of irreversibility, we consider dissipative parabolic semilinear “heat equations” u_t = u_{xx} + f(x, u, u_x) (PDE) on the unit interval with Neumann boundary. By results of Angenent, Fusco, Henry, Matano, Rocha and myself, the global attractors are determined by the shooting meanders for the ODE boundary value problem 0 = v'' + f(x, v, v' ). The celebrated 1974 Chafee-Infante case of cubic f features meanders with two noses. As a natural next step, we investigate the heteroclinic connection graphs for meanders with three noses, i.e. innermost arcs to p, q, p + q nested meander arcs, respectively. Such meanders arise in the PDE context if, and only if, p = r(q + 1), for some integer r. Surprisingly, all their connection graphs are time reversible: a rather non-intuitive bijection of equilibria reverses all heteroclinic directions. We also explore the case q = r(p − 1), which leads to the absurdity of negative “unstable dimensions” ranging from 1-r to -1. Only after r-1 unstable double cone suspensions, such meanders first provide parabolic attractors again. Much to our surprise, their connection graphs then seem to coincide with their time-reversible cousins above. For general meanders however, i.e. for general co-prime p−1 and q+1, a full classification remains elusive – after almost 50 years. On and on, this is joint work with Carlos Rocha.

Video: Link

• Starting on 2022, UFRJ also takes part in our organizational agenda.

• December 14 at 14h00, Anibal Rodriguez-Bernal  (Universidad Complutense de Madrid) 

Title: Homogeneous spaces, operators and semigroups: optimal estimates,  spectral analysis and perturbations 

Abstract: Homogeneous operators are a class of operators in $\R^N$ that include the Laplacian and all its powers (including fractional ones) while homogeneous semigroups are the ones generated by homogeneous operators. Homogeneous spaces are a class of spaces that include, among others, Lebesgue, Lorentz and Morrey ones.

When homogeneous operators or semigroups act on homogeneous spaces, homogeneity implies much more precise results than in the general case. For example, we will show in this talk that homogeneous semigroups must satisfy quite sharp estimates. Also, we sill show that the resolvent set of an homogeneous operator must be formed by semilines passing through zero in the complex plane and a sharp estimate is available on the resolvent operator on each semiline. These properties, in turn, give that for homogeneous operators, Hille-Yosida and Lumer-Phillips theorems for the generation of a semigroup are much easier to check. Also, conditions to check that an homogeneous operator is sectorial (and hence it generates an analytic semigroup) are much easier to met. These conditions become specially simple in the case of an homogeneous Hilbert space. We will also discuss the problem of perturbing an homogeneous operator by homogeneous operators of lower degree (non resonant case) or the same degree (resonant one). We will analyse the spectrum and resolvent of the perturbed operator and show in particular that any non resonant perturbation of a sectorial operator is still sectorial. We will also show smoothing estimates for the perturbed semigroup. Finally we apply these results to some linear diffusion problems, including fractional diffusion with Hardy type potentials. In this case we show that resonant and non resonant conditions correspond to some Hardy and Gagliardo-Nirenberg type inequalities respectively. The evolution problem in the non resonant and resonant cases is also analysed.

Video: Link

• November 30 at 14h00, Alexandre Rodrigues (ISEG - University of Lisbon and CMUP - University of Porto)

Title: Stability of heteroclinic cycles: a new approach

Abstract: In this informal talk, I discuss the stability of cycles within a heteroclinic network formed by different cycles, for a one-parameter model developed in the context of game theory. I describe an asymptotic technique to decide which cycle (within the network) is visible in numerics. The technique consists of reducing the relevant dynamics to a suitable one-dimensional map -- the projective map. Stability of the fixed points of the projective map determines the stability of the associated cycles. All concepts will be gently introduced and the talk will be accessible to non-specialists. This is a joint work with Telmo Peixe (ISEG, CEMAPRE).

Video: Link

• November 16 at 14h00, Renata Bunoiu (Université de Lorraine)  

Title: Homogenization for Problems with Sign-Changing Coefficients.

Abstract: This talk focuses on the presentation of some homogenization results for sign-changing problems. The problems studied are inspired by metamaterials, composite materials that can produce an index of refraction negative, and which have practical applications in optics, electromagnetism, acoustics. By adapting the T-coercivity method on the one hand and homogenization techniques on the other hand, we prove that the problems under study, as well as the homogenized problems, are well posed and we obtain results of convergence of the initial solutions towards the solutions of the homogenized problems. This is joint work with L. Chesnel, K. Ramdani, M. Rihani, C. Timofte.

• October 19 at 14h00, Maximilian Engel (Freie Universität Berlin) 

Title: On the pitchfork bifurcation for the Chafee-Infante equation with additive noise

Abstract: The talk will present recent results for pitchfork bifurcations in a stochastic reaction diffusion system perturbed by an infinite-dimensional Wiener process. It is well-known that the random attractor is a singleton, independently of the value of the bifurcation parameter; this phenomenon is often referred to as the "destruction'' of the bifurcation by the noise. However, we show that some remnant of the bifurcation persists for this SPDE model in the form of a positive finite-time Lyapunov exponent. Additionally, we prove finite-time expansion of volume with increasing dimension as the bifurcation parameter crosses further eigenvalues of the Laplacian.

Video: Link

• October 5 at 14h00, François Béguin (Université Sorbonne Paris Nord)

Title: Past-asymptotic behavior of spatially homogeneous spacetimes

Abstract: I will present a result roughly states that, if one picks at random a spatially homogeneous spacetime, then, with positive probability, the geometry of the spacelike slices of the picked spacetime will oscillate in a chaotic manner as one approaches the initial singularity. I will also : - explain some nice mathematical properties of the finite dimensional dynamical system which describes the evolution of the geometry of the spacelike slices, - discuss further questions on the past asymptotic geometry of spatially homogeneous spacetimes. The main result I will discuss was obtained in collaboration with my  student T. Dutilleul, building upon previous works by S. Liebscher, J. Härterig, K. Webster, M. Georgi, M. Reiterer and E. Trubowitz.

Video: Link

• September 21 at 14h00, Marcus Marrocos (Universidade Federal do Amazonas) 

Title: On eigenvalues of a class of second order elliptic operators on Riemannian manifold

Abstract: In this talk, we compute variations Hadamard type formulas for the eigenvalues of a class of elliptic operators L on a compact Riemannian manifold M. As an application we analyse the behavior of eigenvalues when the metric changes along Ricci flow in a closed Riemannian manifold, in particular, we prove that it increase under suitable hypothesis. Considering λ_k(Ω), the k-th eigenvalue of L with Dirichlet boundary condition in Ω, as a functional on the set of domains of fixed volume in M we obtain necessary and sufficient conditions for a domain to be critical, locally minimizing or locally maximizing for λ_k. This is a joint work with José N. V. Gomes (UFSCar) and Cleiton L. Cunha (UFAM).

• June 29 at 14h00, Alessandra Verri (UFSCar) 

Title: Dirichlet Laplacian in broken sheared waveguides

Abstract: Let −∆DΩ be the Dirichlet Laplacian in a three dimensional waveguide Ω. If Ω is bounded, it is known that the spectrum of −∆DΩ is purely discrete. The situation changes when Ω is unbounded. For example, if Ω is a straight infinite waveguide, the operator −∆DΩ does not have discrete eigenvalues. However, local deformations in Ω can create them. In this talk some constructions for Ω will be presented to show how its geometry can affect the spectral behavior of −∆DΩ . In particular, when Ω is a broken sheared waveguide, we will show that the discrete spectrum is non empty.

Video: Link

• June 15 at 14h00, Daniel Marroquin (UFRJ) 

Title: Stochastic two-scale Young measures and homogenization of stochastic conservation laws.

Abstract: We consider the generalized almost periodic homogenization problem for two different types of stochastic conservation laws with oscillatory coefficients and multiplicative noise. In both cases the stochastic perturbations are such that the equation admits special stochastic solutions which play the role of the steady-state solutions in the deterministic case. Our homogenization method is based on the notion of stochastic two-scale Young measures, whose existence we establish. This is a joint work with Hermano Frid and Kenneth Karlsen.

Video: Link

• June 01 at 14h00, Rafael Obaya (Universidad de Valladolid) 

Title: Uniform stability and chaotic dynamics in nonhomogeneous linear dissipative scalr ordinary differential equations

Abstract: The paper analyzes the structure and the inner long-term dynamics of the invariant compact sets for the skewproduct flow induced by a family of time-dependent ordinary differential equations of nonhomogeneous linear dissipative type. The main assumptions are made on the dissipative term and on the homogeneous linear term of the equations. The rich casuistic includes the uniform stability of the invariant compact sets, as well as the presence of Li-Yorke chaos and Auslander-Yorke chaos inside the attractor. This is a join paper with Juan Campos and Carmen Nuñez.

Video: Link

• Exceptionally on (Friday) May 20 at 14h00, Xiaoying Han (Auburn University) 

Title: The lottery competition model in random environments

Abstract: We will introduce a mathematical model that describes the competition for resources among different ecological species in random environments.  Starting from a discrete time model over observation periods, continuous models are developed using diffusion approximation.  The resulting system is a system of nonlinear stochastic differential equations, with complex nonlinear structures.  Dynamics of the SDE system will be presented to show how environmental fluctuations affect coexistence among species.  In particular, we will discuss a two species system under non-stationary environments, and an N species system (N > 2) under stationary environments.

Video: Link

• May 04 at 14h00, Tomasz Dłotko (University of Silesia) 

Title: Dirichlet problem for 2D Quasi-geostrophic equation with large data

Abstract: The 2D Quasi-geostrophic equation attracts attention of mathematicians through recent years. While the sub-critical problems are rather standard (but not classical), the critical equation contains nonlinearity of the same order as the main dissipative half negative Laplace operator. Therefore we face a balance of the two terms in that case, which makes the problem interesting. We discuss existence of a weak solution of the critical problem, and associate it with a multivalued semiflow. A compact global attractor is shown to exist for that  multivalued semiflow.

Video: Link 

• April 20 at 14h00, Tiago Pereira (ICMC-USP) 

Title: Coherence resonance in networks

Abstract: Complex networks are abundant in nature and many share an important structural property: they contain a few nodes that are abnormally highly connected (hubs). Some of these hubs are called influencers because they couple strongly to the network and play fundamental dynamical and structural roles. Strikingly, despite the abundance of networks with influencers, little is known about their response to stochastic forcing. Here, for oscillatory dynamics on influencer networks, we show that subjecting influencers to an optimal intensity of noise can result in enhanced network synchronization. This new network dynamical effect, which we call coherence resonance in influencer networks, emerges from a synergy between network structure and stochasticity and is highly nonlinear, vanishing when the noise is too weak or too strong. Our results reveal that the influencer backbone can sharply increase the dynamical response in complex systems of coupled oscillators.

Video: Link 

• April 6 at 14h00, Antoine Laurain (IME-USP) 

Title: Optimal control of volume-preserving mean curvature flow

Abstract: We develop a framework and numerical method for controlling the full space-time tube of a geometrically driven flow. We consider an optimal control problem for the mean curvature flow of a curve or surface with a volume constraint, where the control parameter acts as a forcing term in the motion law. The control of the trajectory of the flow is achieved by minimizing an appropriate tracking-type cost functional. The gradient of the cost functional is obtained via a formal sensitivity analysis of the space-time tube generated by the mean curvature flow. We show that the perturbation of the tube may be described by a transverse field satisfying a parabolic equation on the tube. We propose a numerical algorithm to approximate the optimal control and show several results in two and three dimensions demonstrating the efficiency of the approach.

Video: Link 

• November 24 at 11h00, Björn Sandstede (Brown University) 

Title: Homoclinic and heteroclinic bifurcations: from theory to applications

Abstract: I will provide an overview of homoclinic and heteroclinic bifurcations in differential equations, geometric and analytic approaches to study them, and their applications to traveling waves in spatially extended systems. I will start with bifurcations of codimension one, which lead to periodic orbits or to chaotic dynamics, and then discuss bifurcations of codimension two, which often serve as organizing centers for more complex dynamics in such systems. Amongst the approaches I will discuss are homoclinic center manifolds and homoclinic Lyapunov-Schmidt reduction. The applications will focus on the FitzHugh-Nagumo system and provide an overview of homoclinic orbits that arise in this model.

Video: Link

• November 10 at 11h00, Piotr Kalita (Jagiellonian University) 

Title: On attractors of non-autonomous subquintic wave equation

Abstract: We study the non-autonomous weakly damped wave equation with subquintic growth condition on the nonlinearity. We show the existence, smoothness, upper-semicontinuous dependence on the perturbation of the nonlinearity, and relations between pullback, uniform, and cocycle attractors for the class of Shatah–Struwe weak solutions, which, as we prove, coincides with the class of the solutions obtained by the Galerkin method. Talk will be based on the joint article with J. Banaśkiewicz (https://link.springer.com/article/10.1007/s00245-021-09790-8)

Video: Link

• October 27 at 11h00, Danielle Hillhorst (Université Paris-Saclay & CRNS) 

Title: Singular limit of an Allen-Cahn equation with nonlinear diffusion and a mild noise

Abstract: The objective of this talk is to understand the generation and  the propagation of the interface of an Allen-Cahn equation with nonlinear diffusion perturbed by a mild noise. The Allen-Cahn equation was introduced to understand the motion of the interface separating two different phases of a metal alloy. Instead of the usual linear diffusivity of the classical Allen-Cahn equation, we consider a density dependent diffusion term and perturb the reaction term by a mild noise. In particular, we describe how the nonlinear diffusion term affects the motion of the interface. This is joint work with Perla El Kettani, Yong Jung Kim and Hyunjoon Park.

Video: Link

• October 13 exceptionally at 9h00, Eiji Yanagida (Tokyo Institute of Technology) 

Title: Solvability of  the heat equation with a dynamic Hardy-type potential

Abstract: Motivated by the celebrated paper of Baras and Goldstein (1984), we study the heat equation with a Hardy-type singular potential. In particular, we are interested in the case where the singular point moves in time. In the subcritical case, when the motion of the singularity is not so quick, it is shown that there exist two types of positive solutions.  On the other hand, when the singularity moves like a fractional Brownian motion, there exists a positive solution in a wider range of parameters.

Video:  Link

• September 29 at 11h00, Sergey Zelik (University of Surrey) 

Title: Smooth extensions for inertial manifolds of semilinear parabolic equations

Abstract: We discuss the problem of smoothness  of inertial manifolds for abstract semilinear parabolic problems. It is well known that in general we cannot expect more than $C^{1,\epsilon}$-regularity for such manifolds (for some positive, but small $\epsilon$). Nevertheless,, under some  natural assumptions, the obstacles to the existence of a $C^n$-smooth inertial manifold (where $n\in\Bbb N$ is any given number) can be removed by increasing the dimension and by modifying properly the nonlinearity outside of the global attractor (or even outside the $C^{1,\epsilon}$-smooth IM of a minimal dimension). The proof is strongly based on the Whitney extension theorem.

Video: Link

• September 15 at 11h00, Bianca M. R. Calsavara (UNICAMP) 

Title: Existência de solução para um modelo de solidificação com convecção nas regiões não sólidas e movimento rígido nas regiões sólidas

Abstract: Neste trabalho é tratado um sistema de equações diferenciais com condições iniciais e de contorno, que modela um processo de solidificação/liquefação em domínios limitados $3D$, acoplando uma equação de campo de fase e um sistema de Navier-Stokes-Boussinesq de fronteira livre. Neste modelo, o calor latente é considerado via modificação do modelo de Caginalp. Além disso, a convecção nas regiões não sólidas são é tratada via viscosidade dependente da fase do material e que degenera na fase sólida, permitindo somente movimentos rígidos nesta fase. Então, provamos existência de solução fraca global no tempo para um modelo regularizado, por meio de convergência de soluções de problemas não-degenerados obtidos pelo truncamento da viscosidade.

Video: Link 

• September 01 at 11h00, Pedro T. P. Lopes (IME-USP) 

Title: On the Cahn-Hilliard equation on manifolds with conical singularities.

Abstract: The Cahn-Hilliard equation has been widely studied. Results on global solutions, existence of attractors and convergence to the equilibrium can be found in classical papers and monographs. Those results have been generalized in many directions, for instance, for different potentials and boundary conditions. However, a topic that has not yet been fully explored is how the regularity of a surface or a domain affects the dynamical properties of the solutions. In this presentation, we restrict our study to the most classical version of the Cahn-Hilliard equation and to one of the simplest types of singularities on a surface or a domain: the conical singularity. We show how new results on complex interpolation and Lp-regularity, obtained using pseudodifferential operators, can be combined with classical approaches in order to understand the dynamics of the solutions in this context. Joint work with Nikolaos Roidos (Patras University).

Video:  Link 

• July 07 at 16h00, Artur Alho (Instituto Superior Técnico de Lisboa)

Title: Dynamical systems in scalar field cosmology

Abstract: In this talk I will give an overview of dynamical systems in cosmology with scalar fields. This includes models of early inflation, quintessence and quintessential inflation.

Video: Link 

• June 23 at 16h00, James Robinson (University of Warwick)

Title: Approximating solutions of the 3D Navier-Stokes equations on R^3 using large periodic domains

Abstract:  Title self explanatory.

Video: Link 

• June 09 at 16h00, José Valero (Universidad Miguel Hernández de Elche)

Title: On the connections of parabolic equations with discontinuous nonlinearity

Abstract: We consider a parabolic equation of reaction-diffusion type with a discontinuous nonlinearity, which can be expressed by means of a Heaviside functions as a differential inclusion. We show first that under mild assumptions this equation generates a multivalued semiflow that possesses a global attractor and study its structure, which is an interesting and challenging problem. It is noticeable that our problem is the limit of a sequence of Chafee-Infante problems that undergo an infinite sequence of bifurcations, so it is reasonable to expect that it inherits the structure of the attractor of the Chafee-Infante equation. In fact, we prove that there is an infinite (but countable) number of equilibria and that the sequence of equilibria of the approximative problems converges to the equilibria of the limit problem. Since a Lyapunov function exists, the attractor is characterized by the fixed points and their heteroclinic connections, so a full description of the dynamics is got if we determine which connections exist. We give a partial answer to this question. Finally, if we restrict the semiflow to the positive cone, then nice regularity properties of solutions are obtained. In particular, the structure of the global attractor, in this case, is fully understood.

Video: Link

• May 26 at 16h00, Francesca Colasuonno (Università di Bologna)

Title: Multiple radial solutions for some Neumann problems

Abstract: In this seminar, I will talk about some semilinear or quasilinear problems under Neumann boundary conditions. In the first part of the seminar, I will focus on supercritical p-Laplacian problems, set in a ball of R^N, and describe some existence results obtained via variational methods. In the second part, I will present some multiplicity results obtained via the shooting method for ODEs. In particular, I will describe a recent result on a one-dimensional mean-curvature problem, for which we find several positive oscillating BV-solutions. These results are contained in some joint papers with Alberto Boscaggin, Colette De Coster, and Benedetta Noris.

Video: Link

• May 12 at 16h00, Marcelo Cavalcanti (Universidade Estadual de Maringá)

Title: Stability for wave equation with localized damping revisited

Abstract: In this talk we revisit some classical problems involving the wave equation to show another way to prove the stability of the problems. We start considering the n-dimensional linear wave equation in a bounded domain subject to a locally distributed linear damping term. In this case, we proved, via semigroup results (Gearhart Theorem), that the energy decays exponentially. For our surprise, this result has never been proved by this methodology so far. After this, using the linear case, we proved the stability to the wave equation now subject to a locally distributed  nonlinear damping term. The second case considered is when the domain is whole space $\mathbb{R}^N$. In this situation, we proved two results. First, using semigroups results with full damping out of a compact set. The second case (when the domain is whole space) the goal here is that we removed damping. Precisely, given a positive real number $M>0$, we show that for any compact set $K$ of $\mathbb{R}^N$, it is possible to build a region $\Xi$ free of damping, with $meas(\Xi)=M$ (measure of $\Xi$), such that $\Xi$ is globally distributed. Finally, we also considered a case with an unbounded domain with finite measure. In both cases where the undamped region is unbounded it possesses a finite measure we considered microlocal analysis tools combined with Egorov's theorem.

Video: Link

• April 28 at 16h00, Jean-Philippe Lessard  (McGill University)

Title: Spontaneous periodic orbits in the Navier-Stokes flow

Abstract: In this talk, we introduce a general method to obtain constructive proofs of existence of periodic orbits in the forced autonomous Navier-Stokes equations on the three-torus. After introducing a zero finding problem posed on a Banach space of geometrically decaying Fourier coefficients, a Newton-Kantorovich theorem is applied to obtain the (computer-assisted) proofs of existence. As applications, we present proofs of existence of spontaneous periodic orbits in the Navier-Stokes equations with Taylor-Green forcing.

Video: Link

• April 14 at 15h00, Monica Musso  (University of Bath)

Title: Bubbling phenomena in the critical nonlinear heat equation

Abstract: In this talk I will discuss some recent constructions of blow-up solutions for a Fujita type problem for power related to the critical Sobolev exponent. Both finite type blow-up (of type II) and infinite time blow-up are considered. This research program is in collaboration with C. Cortazar, M. del Pino and J. Wei.