23/01/2024

Titulo: Uniqueness and nondegeneracy for fractional Dirichlet problems.
Palestrante: Isabella Ianni (Sapienza Università di Roma)

Data: 23/01/2024
Horário: 11h
Sala: C-116

Resumo: We discuss some recent uniqueness and nondegeneracy results for non-negative solutions of some fractional semilinear problems in bounded domains with Dirichlet exterior condition.

In particular we can consider least energy solutions in balls or in more general symmetric domains, for problems with power nonlinearities. The symmetry properties of the solutions of the associated linearized equation are also investigated.

The talk is mainly based on the following joint works:

[1] A. Dieb, I. Ianni, A. Saldaña, Uniqueness and nondegeneracy for Dirichlet fractional problems in bounded domains via asymptotic methods, Nonlinear Analysis, 236, 2023,

https://doi.org/10.1016/j.na.2023.113354

[2] A. Dieb, I. Ianni, A. Saldaña, Uniqueness and nondegeneracy of least-energy solutions to fractional Dirichlet problems, preprint arXiv:2310.01214

23/01/2024

Titulo: Propriedade de Liouville para equações não lineares degeneradas.
Palestrante: Disson dos Prazeres (Universidade Federal de Sergipe)

Data: 23/01/2024
Horário: 12h
Sala: C-116

Resumo: Nesta palestra, apresentaremos uma nova estratégia para obtermos a propriedade de Liouville para equações não-lineares degeneradas. Como aplicação iremos obter resultados de existência para problemas de Dirichlet.

24/01/2024

Titulo: Time optimal controls for infinite dimensional systems.
Palestrante: Sorin Micu - (Universidade de Craiova)

Data: 24/01/2024
Horário: 12h
Sala: C-116

Resumo: We consider the time optimal control problem, with a point target, for an infinite dimensional system described by the Kirchhoff plate equation with distributed control. We prove that  time optimal controls  have a bang-bang property and, consequently, that they are unique. The main ingredients used to achieve this goal is a new approximate observability property  from measurable sets for the system described by the Kirchhoff equation and an abstract result for systems with skew-adjoint generator.

29/11/2023

Titulo: A scattering conjecture about the nonlinear Schrodinger equation
Palestrante: Carlos Guzman (UFF)

Data: 29/11/2023
Horário: 12h
Sala: C-116

Resumo: Clique aqui

06/12/2023

Titulo: Uma introdução matemática à localização de Anderson.
Palestrante: Rodrigo Matos (PUC-Rio)

Data: 06/12/2023
Horário: 12h
Sala: C-116

Resumo: Há cerca de 65 anos, em seu artigo seminal intitulado " Absence of Diffusion in Certain Random Lattices", o físico P.W. Anderson argumentou que a presença de desordem pode afetar drasticamente as propriedades de transporte em um sistema quântico. Esta descoberta levou ao desenvolvimento de uma área da física-matemática que estuda os operadores de Schrödinger com potencial aleatório e, em particular, ao conceito de localização de Anderson. Nessa palestra, iremos discutir noções de localização bem como teoremas, conjecturas e objetos matemáticos pertinentes a essa teoria. Se o tempo permitir, irei comentar sobre avanços recentes e também algumas técnicas utilizadas nessa área, as quais situam-se na interface entre análise, equações diferenciais e probabilidade.

22/11/2023

Titulo: Lax equivalence principle for problems in fluid dynamics.
Palestrante: Eduard Feireisl (Institute of Mathematics of the Czech Academy of Sciences)

Data: 22/11/2023
Horário: 12h
Sala: C-116

Resumo: We discuss possible generalizations of the celebrated Lax equivalence principle: ``stability+consistency yields convergence'' to a class of nonlinear problems arising in fluid mechanics. In particular, we establish convergence of approximate schemes for problems with uncertain data.

23/08/2023

Titulo: Ingham type theorems and observability problems
Palestrante:  Vilmos Komornik (University of Strasbourg)

Data: 13/09/2023
Horário: 12h
Sala: C-116

Resumo: We report on some joint works with Claudio Baiocchi and Paola Loreti on the application of non-harmonic Fourier series to control theory.

23/08/2023

Titulo: Asymptotic description for the localized solution of the Cauchy problem for the  wave equation with fast-oscillating coefficient.
Palestrante:  Sergey Sergeev (PUC-Rio)

Data: 23/08/2023
Horário: 12h
Sala: C-116

Resumo: We consider the Cauchy problem with localized initial conditions for the multidimensional wave equation. The coefficient of this wave equation is assumed to be fast-oscillating. We are interested in the asymptotic (while localization parameter of initial condition is small) description of the given Cauchy problem. Such formulation leads to the appearance of two small parameters: the localization parameter and the parameter of oscillating in the wave equation coefficient. The ratio between the given parameters is crucial and affects the form of the main part of the asymptotic solution. We use  the homogenization procedure which takes into account this ratio and as result we obtain the equation with smooth coefficients. This equation is of the form of the wave equation with dispersion correction, which appears due to the homogenization procedure. The main part of the asymptotic solution for the initial Cauchy problem thus can be described with the help of the asymptotic solution of the homogenized equation with the smooth coefficients  and can be presented in the analytical form with the help of the Airy functions and related to them.

09/08/2023

Titulo: Propriedades matem´aticas do fluxo de espuma em meios porosos
Palestrante:  Luis Fernando Lozano G. (LAMAP-UFJF)

Data: 09/08/2023
Horário: 12h
Sala: C-116

Resumo: Clique aqui

26/04/2023

Titulo: The limit shape of the critical front profile for vanishing diffusion in Born-Infeld models.
Palestrante:  Maurizio Garrione (Politecnico di Milano)

Data: 26/04/2023
Horário: 12h
Sala: C-116

Resumo: We deal with traveling fronts for reaction-diffusion models where the diffusive term is of relativistic (Born-Infeld) type. We show that, in case the reaction term is monostable, the critical front connecting 0 and 1 sharpens on one side only, near the equilibrium 0. The presence of a convective term may alter this picture, leading to fully sharp or fully regular limit profiles, as will be briefly shown. The technique relies on a careful analysis of the associated first-order reduction.

23/03/2023

Titulo: Large Harmonic Functions for Fully Nonlinear Fractional Operators.
Palestrante:  Alexander Quaas (UTFSM)

Data: 23/03/2023
Horário: 12h
Sala: C-116

Resumo: Clique aqui

22/03/2023

Titulo: Uncovering the mechanisms of pattern formation and emergent collective behaviors in myxobacteria.
Palestrante:  Oleg Igoshin (Rice University)

Data: 22/03/2023
Horário: 12h
Sala: C-116

Resumo: Collective cell movement is critical to the emergent properties of many multicellular systems including microbial self-organization in biofilms, wound healing, and cancer metastasis. However, even the best-studied systems lack a complete picture of how diverse physical and chemical cues act upon individual cells to ensure coordinated multicellular behavior. Myxococcus xanthus is a model bacteria famous for its coordinated multicellular behavior resulting in dynamic patterns formation. For example, when starving millions of cells coordinate their movement to organize into fruiting bodies – aggregates containing tens of thousands of bacteria. Relating these complex self-organization patterns to the behavior of individual cells is a complex-reverse engineering problem that cannot be solved solely by experimental research. In collaboration with experimental colleagues, we use a combination of quantitative microscopy, image processing, agent-based modeling, and kinetic theory PDEs to uncover the mechanisms of emergent collective behaviors.

08/02/2023

Titulo: Stabilization of a time delayed for a generalized dispersive system.
Palestrante:  Fernando Gallego (Universidad Nacional de Colombia sede Manizales)

Data: 08/02/2023
Horário: 11h
Sala: C-119

Resumo: Clique aqui

08/02/2023

Titulo: A coupling approach to quantify the transportation Wasserstein path-distance between heat equation and the Goldstein--Kac telegraph equation
Palestrante:  Gerardo Barrera (University of Helsinki)

Data: 08/02/2023
Horário: 12h
Sala: C-119

Resumo: In this talk, I will present a non-asymptotic process level control between the so-called telegraph process (a.k.a. Goldstein-Kac equation) and a diffusion process with suitable diffusivity constant (explicit) via a transportation Wasserstein path-distance with quadratic average cost.

We stress that the telegraph process solves a partial linear differential equation of the hyperbolic type for which explicit computations can be carried by in terms of Bessel functions. In the present talk, I will discuss the coupling approach, which is a robust technique that can be used for more general PDEs. The proof is done via the interplay of the following couplings: coin-flip coupling, synchronous coupling and the celebrated Komlós-Major-Tusnády coupling. In addition, non-asymptotic estimates for the corresponding L^p time average are given explicitly.

The talk is based on joint work with Jani Lukkarinen, University of Helsinki, Finland.

27/01/2023

Titulo: Boundary homogenization problems with high contrasts: the elasticity system & the local problems
Palestrante:  María Eugenia Pérez Martínez (Universidad de Cantabria)

Data: 27/01/2023
Horário: 12h
Sala: C-116

Resumo: We consider the homogenization problem for the elasticity operator posed in a bounded domain of the upper half-space, a part of its boundary being in contact with the plane. We assume that this surface is free outside small regions in which we impose Robin-Winkler boundary conditions linking stresses and displacements by means of a symmetric and positive definite matrix and a reaction parameter. These small regions are periodically placed along the plane while its size is much smaller than the period. We look at the asymptotic behaviour of spectrum and provide all the possible spectral homogenized problems depending on certain asymptotic relations between the period, the size of the regions and the reaction-parameter. We state the convergence of the eigenelements, as the period tends to zero, which deeply involves the corresponding microscopic stationary problems obtained by means of asymptotic expansions.
We compare results and techniques with those for the Laplace operator and outline some possible extensions (under consideration) of the problem.

Some references:

[1] D. Gómez, S.A. Nazarov, ; M.-E. Pérez-Martínez. Asymptotics for spectral problems with rapidly alternating boundary conditions on a strainer Winkler foundation. Journal of Elasticity, 2020, V. 142, p. 89-120.

[2] D. Gómez, S.A. Nazarov ; M.-E. Pérez-Martínez. Spectral homogenization problems in linear elasticity with large reaction terms concentrated in small regions of the boundary. In: Computational and Analytic Methods in Science and Engineering. Birkäuser, Springer, N.Y., 2020, pp. 121-143

[3] D. Gómez; M.-E. Pérez-Martínez. Boundary homogenization with large reaction terms on a strainer-type wall. Z. Angew. Math. Phys. Vol. 73, 28p 2022.

[4] M.-E. Pérez-Martínez. Homogenization for alternating boundary conditions with large reaction terms concentrated in small regions. In: Emerging problems in the homogenization of Partial Differential Equations. ICIAM2019 SEMA SIMAI Springer Series 10, 2021, pp. 37-57.

18/01/2023

Titulo: STABILITY OF MKDV BREATHERS ON THE HALF-LINE
Palestrante:  Márcio Cavalcante (UFAL)

Data: 18/01/2023
Horário: 12h
Sala: C-116

Resumo: In this talk I will discuss the stability problem for mKdV breathers on the left half-line. We are able to show that leftwards moving breathers, initially located far away from the origin, are strongly stable for the problem posed on the left half-line, when assuming homogeneous boundary conditions. The proof involves a Lyapunov functional which is almost conserved by the mKdV flow once we control some boundary terms which naturally arise. Also, recent results about orbital and asymptotic stability of solitons on the positive half-line will be discussed. This is a joint work with Miguel Alejo and Adán Corcho.

07/12/2022

Titulo: Desigualdades tipo Caffarelli-Kohn-Nirenberg com expoentes fixos e expoentes variáveis.
Palestrante:  Aldo Bazán (UFF)

Data: 07/12/2022
Horário: 12h
Sala: C-116

Resumo: Uma desigualdade tipo Caffarelli-Kohn-Nirenberg (CKN) é uma desigualdade funcional de interpolação. São conhecidas diferentes versões desta desigualdade na literatura, considerando derivadas de ordem superior, integrais definidas em domínios limitados, e outros. Nesta palestra consideramos o domínio das funções o Rn, e apresentamos uma análise da relação entre os expoentes, e os seus efeitos na CKN quando estes são constantes, e quando eles são variáveis, apresentando algumas semelhanças e diferenças entre estes casos.

30/11/2022

Titulo: Models of mosquito population control strategies for fighting against arboviruses.
Palestrante:  Michel Duprez (Inria, Université de Strasbourg, ICUBE,équipe MIMESIS)

Data: 30/11/2022 (quarta)
Horário: 12h
Sala: C-116

Resumo: In the fight against vector-borne arboviruses, an important strategy of control of epidemic consists in controlling the population of the vector, Aedes mosquitoes in this case. Among possible actions, a technique consist in releasing sterile mosquitoes to reduce the size of the population (Sterile Insect Technique). This talk is devoted to studying the issue of optimizing the dissemination protocol for each of these strategies, in order to get as close as possible to these objectives. Starting from a mathematical model describing the dynamic of a mosquitoes population, we will study the control problem and introduce the cost function standing for sterile insect technique. In a second step, we will consider a model with several patchs modeling the spatial repartition of the population. Then, we will establish some properties of these two optimal control problems. Finally, we will illustrate our results with numerical simulations.  

29/11/2022

Titulo: Rapid stabilization of linearized water waves and Fredholm backstepping for critical operators
Palestrante:  Ludovick Gagnon (Université de Lorraine, CNRS, Inria équipe SPHINX)

Data: 29/11/2022 (terça)
Horário: 14h
Sala: C-119

Resumo: The backstepping method has become a popular way to design feedback laws for the rapid stabilization of a large class of PDEs. This method essentially reduces the proof of exponential stability to the  existence and invertibility of a transformation. Initially applied with a Volterra transformation, the Fredholm alternative, introduced by Coron and Lü, allows to overcome some existence issues for the Volterra transformation. This new approach also has the advantage of having a systematic methodology, but the methods known until now were only applicable to differential operators D_x^a with a>3/2. In this talk, we present the duality/compactness method to surmount this threshold and show that the Fredholm-type backstepping method applies for anti-adjoint operators i |D_x|^a, with a >1. We will demonstrate the application of this result for the rapid stabilization of the linearized water waves equation.

23/11/2022

Titulo: Modelos de predador-presa e epidemiologia usando equações de transporte.
Palestrante:  Paulo Amorim (UFRJ)

Data: 23/11/2022
Horário: 12h
Sala: C-116

Resumo: Vou apresentar e analisar um modelo de predador-presa em que o predador tem uma estrutura de fome. O modelo tem a forma de uma equação de transporte com termos não locais, acoplada a uma EDO para a presa. Mostramos resultados de boa colocação e comportamento assintótico. Usando uma filosofia semelhante, apresento em seguida um modelo de epidemiologia em que a população saudável é estruturada por sua suscetibilidade e pelo conhecimento, ou consciência, da doença (disease awareness). Neste modelo, analisamos as propriedades de um sistema de EDOs para quantidades integrais e concluímos um resultado de comportamento assintótico para a solução da equação de transporte. Em todos os casos serão apresentados exemplos de aplicação e simulações numéricas. 

19/10/2022

Titulo: Magic Functions
Palestrante:  Felipe Gonçalves (IMPA)

Data: 19/10/2022
Horário: 12h
Sala: C-116

Resumo: We will talk about some of the challenging problems in different areas of mathematics that were solved by constructing certain "magic" functions with constraints on physical and/or frequency space (we shall focus slightly on the bandlimited case). The talk will be based entirely on examples, one of which is the sphere packing problem (its solution awarded the Fields medal to the Ukrainian female mathematician M. Viazovska).

05/10/2022

Titulo: Solução do Problema de Riemann na Variedade de Onda
Palestrante:  Marlon M. Lopes-Flores (IMPA)

Data: 05/10/2022
Horário: 12h
Sala: C-116

Resumo:  Clique aqui

28/09/2022

Titulo: TBA
Palestrante: Julio C. Correa Hoyos (UERJ)

Data: 28/09/2022
Horário: 12h
Sala: C-116

Resumo: TBA

21/09/2022

Titulo: Mergulhos de álgebras de von Neumann em álgebras de Roe uniformes.
Palestrante:  Bruno de Mendonça Braga (PUC-Rio)

Data: 21/09/2022
Horário: 12h
Sala: C-116

Resumo: Dado um espaço métrico localmente finito $X$, sua álgebra de Roe uniforme, denotada por $C^*_u(X)$, é uma álgebra-$C^*$ de operadores no espaço de Hilbert $\ell_2(X)$ e tem como objetivo capturar características da geometria de larga escala de $X$. Essa álgebra foi introduzida por John Roe em 1988 e vem atraindo a atenção de matemáticos de diversas áreas como álgebras de operadores, teoria geométrica de grupos e física matemática. Nessa palestra, introduzirei essa álgebra e falarei sobre avanços recentes no entendimento da estrutura de $C^*_u(X)$. Mais precisamente, discutirei sobre quais álgebras de von Neumann podem ser encontradas dentro de $C^*_u(X)$.

14/09/2022

Titulo: Decaimento das equações de Navier-Stokes-Coriolis em espaços críticos.
Palestrante:  César J. Niche (UFRJ)

Data: 14/09/2022
Horário: 12h
Sala: C-116

Resumo: Nos Fluidos Geofísicos, a rotação da Terra tem um papel importante na modelagem e análise das equações que descrevem esses fluidos. Nosso objetivo nesta palestra é o de quantificar o decaimento das soluções das equações de Navier-Stokes-Coriolis (NSC), modelo matemático mais simples a levar em conta a rotação do domínio ao redor de um eixo.  Os dois pontos mais importantes a respeito do trabalho a ser apresentado são: 1) alguns dos resultados que descreveremos (em certos espaços críticos que aparecem de maneira natural)  são qualitativamente bem diferentes dos análogos obtidos para as equações de Navier-Stokes; 2) o método e técnicas usados para provar as estimativas podem ser usados em muitas equações não necessariamente vinculadas à Dinâmica dos Fluidos.

Trabalho em colaboração com Leonardo Kosloff (Case Western Reserve University) e Gabriela Planas (Unicamp).

27/07/2022

Titulo: On chemical flooding models: Riemann problem solutions and viscous fingering phenomenon.
Palestrante:  Yulia Petrova (IMPA)

Data: 27/07/2022
Horário: 12h
Sala: C-116

Resumo: We are interested in mathematical modelling of oil recovery. Injection of a less viscous fluid into a more viscous one (like water into oil) in porous media generates instabilities, which are often called “viscous fingers”. To stabilize the displacement front petroleum engineers sometimes use chemical flooding enhanced oil recovery methods (injection of water with dissolved chemicals).

In the first part of the talk we will discuss the stable displacement and how to construct a unique solution to a corresponding one-dimensional Riemann problem. To distinguish physically meaningful weak solutions we use vanishing viscosity admissibility criterion. We demonstrate that when the flow function depends non-monotonically on the chemical agent concentration (which corresponds to the surfactant flooding), non-classical undercompressive shocks appear. They correspond to the saddle-saddle connections for the traveling wave dynamical system and are sensitive to precise form of the dissipation terms.

In the second part of the talk we will focus on unstable displacement (viscous fingering). There is a lot of experimental and numerical evidence of the linear growth of the mixing zone, but the mathematically rigorous proof is an open problem. We will talk about possible directions of attacking the problem.

08/06/2022

Titulo: A nonlocal version of the inverse problem of Donsker and Varadhan.
Palestrante: Erwin Topp Paredes (Universidad de Santiago de Chile)

Data: 08/06/2022
Horário: 12h
Sala: C-116

18/05/2022

Titulo: Reaction-diffusion equations with hysteresis.
Palestrante: Sergey Tikhomirov (St Petersburg State University & IMPA)

Data: 18/05/2022
Horário: 12h
Sala: C-116

Resumo: Hysteresis naturally appears as a mechanism of self-organization and is often used in control theory. Important features of hysteresis are discontinuity and memory. We consider reaction-diffusion equations with hysteresis. Such equations describe processes in which diffusive and non-diffusive instances interacts according to a hysteresis law. Due to the discontinuity of hysteresis, these equations are not always well-posed.

We consider a spatial discretization of the problem and present a new mechanism of pattern formation, which we call rattling. The profile of the solution forms two hills propagating with non-constant velocity. The profile of hysteresis forms a highly oscillating quasiperiodic pattern, which explains mechanism of illposedness of the original problem and suggests a possible regularization. Rattling is very robust and persists in arbitrary dimension and in systems acting on different time scales.

The talk is based on joint works with P. Gurevich. 

11/05/2022

Titulo: ON THE CONVERSE OF HARTMAN'S THEOREM .
Palestrante: Nilson C. Bernardes Jr. (UFRJ) 

Data: 11/05/2022
Horário: 12h
Sala: C-116

Resumo: Let us recall the following classical result from the 1960's in the interface between the areas of Dynamical Systems and Operator Theory:

Theorem. If an invertible bounded linear operator T on a Banach space X is hyperbolic, then it is structurally stable.

This result is often called Hartman's Theorem and it is a major tool for the proof of the celebrated Grobman-Hartman Theorem. It was originally established by Philip Hartman in 1960 for operators on finite-dimensional euclidean spaces. The extension to arbitrary Banach spaces was independently obtained by Jacob Palis and Charles Pugh in the late 1960's, motivated by an argument due to Jürgen Moser. The basic question related to Hartman's Theorem is whether or not its converse holds. In other words, we have the following natural question:

Problem. If an invertible bounded linear operator T on a Banach space X is structurally stable, is it necessarily hyperbolic?

It was soon realized that the answer is "yes" in the finite-dimensional setting (Joel Robbin, 1972), but the full question remained open for more than 50 years! This problem was finally solved in a 2021 joint paper of the speaker with Ali Messaoudi. In our talk we will present an overview of this solution and its relationship with the notion of generalized hyperbolicity and the Generalized Grobman-Hartman Theorem.

04/05/2022

Titulo: The Virial/Morawetz and the role of the inhomogeneity for non-radial scattering in Schrödinger-type equations.
Palestrante: Luccas Campos

Data: 04/05/2022
Horário: 12h
Link de acesso: https://meet.google.com/rsq-subn-yeh
Sala: A palestra será remota, mas será transmitida na sala C-116

Resumo: The concentration-compactness-rigidity method, pioneered by Kenig and Merle, has become standard in the study of global well-posedness and scattering in the context of dispersive and wave equations. Albeit powerful, it requires building some heavy machinery in order to obtain the desired space-time bounds.

In this talk, we present a simpler method, based on Tao's scattering criterion and on Dodson-Murphy's Virial/Morawetz inequalities, first proved for the 3d cubic nonlinear Schrödinger (NLS) equation.

Tao's criterion is, in some sense, universal, and it is expected to work in similar ways for dispersive problems. On the other hand, the Virial/Morawetz inequalities need to be established individually for each problem, as they rely on monotonicity formulae.

However, to treat the non-radial case, the original Dodson-Murphy's approach for the NLS equation relies heavily on the conservation of momentum and on the Galilean invariance. This poses an obstacle for equations which do not present these features. We show how to circumvent this problem in the case of the inhomogeneous NLS, by exploiting the decay of the nonlinearity, making it possible to drop the radiality assumption.

The approach is versatile, as it can be shown to work in the energy-subcritical setting for different spatially decaying nonlinearities, as well as for higher-order equations.

27/04/2022

Titulo: On a study and applications of the Concentration-compactness type principle for Elliptic Systems.
Palestrante: Lauren Maria Mezzomo Bonaldo (UFRJ)

Data: 27/04/2022
Horário: 12h
Sala: C-116

Resumo:  In this talk, we obtain a concentration-compactness  type principle for fractional Sobolev spaces with variable exponents. As an application of the result, we obtain the existence of  solutions for a class of critical nonlocal systems with variable exponents.

12/04/2022

Titulo: Modelling chemotaxis with a nonlinear Schrödinger equation: solitary waves.
Palestrante: Miguel A. Alejo (Universidad de Córdoba, España)

Data: 12/04/2022
Horário: 12h
Sala: B106-B

Resumo: In this talk I will show how chemotaxis can be modelled by using a nonlinear Schrödinger equation with  well-known quantum dissipative mechanisms. This relation will allow us to find explicit new solitary wave solutions.

06/04/2022

No dia 6 de arbil teremos uma palestra remota pelo prof. Erwin Topp, seguida de uma homenagem ao prof. Luiz Adauto Meideros.

Titulo: On large solutions for fractional Hamilton-Jacobi equations
Palestrante: Erwin Topp (Universidad de Santiago de Chile)

Data: 06/04/2022
Horário: 12h
Link de acesso: https://meet.google.com/txg-gueo-rgq

Resumo: In this talk I will report some multiplicity results for large solutions of fractional Dirichlet problems associated with Hamilton-Jacobi equations on bounded domains. We construct large solutions using the method of sub and supersolutions, following the classical approach of J.M. Lasry and P.L. Lions for second-order, Hamilton-Jacobi equations with subquadratic gradient growth. We identify two classes of solutions: the one coming from the natural scaling of the problem, and a one-parameter family of solutions, consequence of the nonlocal nature of the problem. Joint work with Alexander Quaas and Gonzalo Dávila (UTFSM-Chile).