Programação

Local: Auditório Ricardo de Carvalho Ferreira, CCEN/UFPE

Name: Armando Augusto de Castro (UFBA)

Title: Técnicas para gap espectral para operadores em espaços de Banach

Abstract: O gap espectral de operadores de transferência é o ponto de partida para provar muitas propriedades estatísticas de medidas de estado de equilíbrio, desde decaimento de correlações até linear response formula. Nessa palestra, veremos duas das técnicas para provar gap espectral, Lasota -Yorke e Cones projetivos.

Name: Carlos Bocker (UFPB)

Title: Regularity of the density of SRB measures for solenoidal attractors

Abstract: We show that a class of higher-dimensional hyperbolic endomorphisms admit absolutely continuous invariant probabilities whose density are regular. The maps we consider are given by T(x; y) = (E(x);C(y) + f(x)), where E is a linear expanding map of T^u, C is a linear contracting map of R^d, f is in C^r(Tu;Rd) and r > 2. We prove that if |(detC)(detE)| ||C^(-1)||^(-2s) > 1 for some s < r-1- (u+d)/2 and T satisfi es a certain transversality condition, then the density of the SRB measure of T is contained in the Sobolev space H^s(T^u x R^d), in particular, if s > (u+d)/2 then the density is C^k for every k < s - (u+d)/2. We also exhibit a condition involving E and C under which this tranversality condition is valid for almost every f.

Name: Yuri Gomes (UFC)

Title: Dinâmica simbólica para bilhares

Abstract: Rufus Bowen (1947-1978) deixou um caderno com 157 problemas em dinâmica, que cobrem uma grande variedade de tópicos. No problema 17, ele escreveu "symbolic dynamics for billiards". Essa pergunta permaneceu aberta até o começo dos anos 90, quando Bunimovich, Chernov e Sinai construíram partições de Markov para a medida de Liouville em bilhares dispersivos. Recentemente, Carlos Matheus e eu construímos partições de Markov bem mais gerais, que funcionam para uma classe maior de bilhares (por exemplo, o estádio de Bunimovich) e de medidas (por exemplo, as medidas construídas recentemente por Baladi e Demers). Nessa palestra, tentarei explicar o contexto geral da prova, bem como as principais dificuldades encontradas.

Name: Davi Lima (UFAL)

Title: Sobre a estrutura topológica de M\L

Abstract: Nesta palestra apresentaremos os espectros de Markov e Lagrange, subconjuntos fechados da reta que aparecem naturalmente em aproximações diofantinas. Interpretações dinâmicas de tais espectros permitiram diversos avanços recentes na teoria, incluindo a prova da conjectura de Cusick de mais de 40 anos, por C. Matheus e C. G. Moreira. Em colaboração com C. Matheus, C. G. Moreira e S. Vieira encontramos o menor elemento conhecido para M\L e mais recentemente mostramos que M\L não é um subconjunto fechado da reta, respondendo negativamente a uma pergunta de T. Bousch.

Name: Krerley Oliveira (UFAL)

Title: Random dynamical systems and its thermodynamical formalism

Abstract: In this talk we discuss some results about maximal entropy measures and equilibrium states for some random dynamical systems.

Name: João Marcos do Ó (UFPB)

Title: Trudinger-Moser inequalities on the upper half-space

Abstract: We establish some sharp embedding results including trace embeddings from a certain Sobolev space, defined on the upper half-space, into the Lebesgue and Orlicz spaces. Moreover, some Trudinger- Moser type inequalities for functions defined on the upper half- space are obtained. As an application, we prove the existence of solutions for two classes of nonlinear elliptic problems involving non- linear boundary conditions.

Name: Uberlândio Severo (UFPB)

Title: Ground state solutions for a nonlocal equation in the plane involving vanishing potentials and exponential critical growth

Abstract: In this talk, we analyze the following class of nonlinear equations: -\Delta u+V(x) u = \left[|x|^{-\mu}*(Q(x)F(u))\right]Q(x)f(u),\quad x\in\mathbb{R}^2, where $V$ and $Q$ are continuous potentials, which can be unbounded or vanishing at infintiy, $f(s)$ is a continuous function, $F(s)$ is the primitive of $f(s)$, $*$ is the convolution operator and $0<\mu<2$. Assuming that the nonlinearity $f(s)$ has exponential critical growth, we establish the existence of ground state solutions by using variational methods. For this, we prove a new version of the Trudinger-Moser inequality for our setting, which was necessary to obtain our main results.

Name: Maxwell Lizete da Silva (UFG)

Title: Ground states for a class of critical quasilinear coupled superlinear elliptic systems

Abstract: In this work we consider a quasilinear coupled system whole space. We consider positive potentials and a coupling term. We deal with nonlinearities g and h being subcritical or critical. The coupling term is a subcritical function which is superlinear at infinity. Our main theorem is stated without the well known Ambrosetti-Rabinowitz condition at infinity. Using a change of variable, we turn the quasilinear coupled system into a nonlinear coupled system, where we establish a variational approach based on Nehari method.

Name: Edcarlos Domingos da Silva (UFG)

Title: Existence of ground states for a class of fractional coupled systems involving critical coupling terms

Abstract: In this talk we study the existence of radial ground state solutions for critical coupled systems involving the fractional Laplacian. We deal with the critical case in the coupling term. We also consider continuous nonlinearities which it not necessarily satisfy the Ambrosetti-Rabinowitz condition. Our approach is based on the method of Nehari manifold jointly together with Schwarz symmetrization arguments, fractional versions of Polya-Szego type inequality and Strauss Lemma.

Name: Diego Ferraz (UFRN)

Title: Ground state for Schrödinger-Poisson systems involving oscillatory nonlinearity

Abstract: In this talk, we study the existence of standing waves for a class of fractional Schrödinger-Poisson systems with an oscillatory source term. The proofs of our main results are established via minimax arguments using a new kind of approach based on a refined version of the concentration-compactness method introduced by M. Struwe for Palais-Smale sequences of some semilinear elliptic equations. We perform a concentration-compactness analysis for bounded Palais-Smale sequences of the related energy functional, and we obtain ground state solutions for a broad class of sign-changing potentials.

Name: Rodrigo Clemente (UFRPE)

Title: On p-harmonic functions on the upper half-space

Abstract: In this paper, we investigate the existence, nonexistence and qualitative properties for p-harmonic functions on the half-space with nonlinear boundary conditions. Moreover, we obtain symmetry of positive solutions by using the classical moving plane method.

Name: Felipe Linares (IMPA)

Title: Asymptotic behavior of solutions of the dispersive generalized Benjamin-Ono equation

Abstract: In this talk we discuss the asymptotic behavior of uniformly bounded in time $ H^1\cap L^1$ solutions of the dispersive generalized Benjamin-Ono equation. We will show that the limit infimum as time t goes to infinity, converges to zero locally in an increasing-in-time region of space of order $ t/\log t $. This result agrees with the one recently obtained by Muñoz and Ponce for the Benjamin-Ono equation. In particular, this result implies that the existence of breathers or any other solution for the generalized Benjamin-Ono equation moving with a speed slower than a soliton is discarded.

Name: Jaime Angulo (USP)

Title: Sine-Gordon equation on star graphs

Abstract: The aim of the talk is to establish a linear instability criterium of stationary solutions for the sine-Gordon model on a star graph of Y- junction type. By considering boundary conditions of delta type- interaction at the graph-vertex, we show that the continuous tail and bump profiles are linearly unstable. The use of the analytic perturbation theory of operators and the extension theory of symmetric operators is a piece fundamental in our stability analysis. The arguments to be presented here, it have already been applied for other nonlinear evolution equations on star graphs, such as the KdV and the BBM models.

Name: Márcio Cavalcante (UFAL)

Title: Asymptotic stability of KdV solitons on the half-line

Abstract: In this talk I will discuss the asymptotic stability problem for KdV solitons on right half-line. Unlike standard KdV, these are not exact solutions to the equations posed on the half-line, and, contrary to NLS, no exact soliton solution seems to exist. In a previous result, we showed that solitons of the KdV equation posed in the entire line, placed sufficiently far from the origin, are stable in the half-line energy space, and assuming homogeneous boundary conditions. Now, we confirm these half-line KdV solitons are indeed asymptotically stable in the energy space. For the proof we follow in spirit the ideas by Martel and Merle, with some important differences coming from the fact that mass and energy are not conserved by the dynamics of the half-line KdV, and high regularity boundary terms modify the dynamics in the long time regime. Additionally, some bad behavior of the KdV soliton for the entire line must be cut off in order to ensure the correct convergence of the dynamics to a unique final state. This is a joint work with Claudio Muñoz (Universidad de Chile).

Name: Fernando Gallego (UNAL)

Title: Well-posedness and Controllability Aspects of the cubic fourth order Schrödinger equation on a star graph

Abstract: In this talk, we present some recent results related to the wellposedness and controllability results related to the cubic fourth order Schrodinger equation on star graph structure $\mathcal{G}$: $i\partial_tu -\partial_x^4u-\lambda |u|^2u=0.$ Here, we consider $\mathcal{G}$ composed by $N$ edges parameterized by half-lines $(0,+\infty)$ and/or $(a_i,b_i)$ attached with a common vertex $\nu$. With this structure we study the well-posedness and some controllability aspects of a dispersive model on star graphs with three appropriated vertex conditions by using the boundary forcing operator approach, more precisely, we give positive answer for the Cauchy problem in low regularity Sobolev spaces. We have noted that this approach seems very efficient, since this allows to use the tools of Harmonic Analysis, for instance, the Fourier restriction method of Bourgain, while for the other known standard methods to solve partial differential partial equations on star graphs are more complicated to capture the dispersive smoothing effect in low regularity. The arguments presented in this work has prospects to be applied for others nonlinear dispersive equations on the context of star graphs with unbounded edges.

Name: Miguel Loayza (UFPE)

Title: Existência local de soluções para problemas parabólicos semilineares em espaços de Lebesgue

Abstract: Nesta palestra caracterizamos as não linearidades de alguns problemas parabólicos com dados iniciais no espaços de Lebesgue. Usando o método das super/sub soluções respondemos a questão de existência e não existência local de soluções.

Name: Ademir Pampu (UFPE)

Title: Limites polinomiais para o crescimento das normas da solução da equação de Klein-Gordon semilinear em espaços de Sobolev

Abstract: Nesta palestra consideramos a equação de Klein Gordon semilinear em uma variedade Riemanniana de dimensão três com ou sem bordo, e analisamos o comportamento das normas de sua solução em espaços de Sobolev de ordem superior. A partir de um argumento de indução, combinado com as estimativas de Strichartz, provamos que estas normas podem ser limitadas por funções com crescimento polinomial.

Name: Yavidat Iiyasov (UFG)

Title: On nonlinear generalized Rayleigh quotient and its applications to nonlinear PDE

Abstract: In 1869, Heinrich Weber introduced the quotient R(u):=\frac{\int( (\frac{\partial u}{\partial x})^2+(\frac{\partial u}{\partial y})^2)dxdy}{ \int u^2 dxdy}, to show that the equation of vibrations of membrane possesses an infinite sequence $(\lambda_n)$ of positive eigenvalues. The rigorous proof of Weber's result was obtained by Poincar\'e (1896). Later this variational principle has been developed in works of Fisher (1875-1954), H. Weyl (1885-1955) and Courant (1988-1972). In modern, the function $R(u)$ is called Rayleigh's quotient due to the fact that it became widely known after the works of Rayleigh , who effectively used it in practice. The main purpose of the talk is to present a new conception for the generalization of Rayleigh's quotient applicable to nonlinear equations and coincide with classical Rayleigh's quotient in the case of linear spectral problems. The new generalized Rayleigh's quotient is, by its nature, a new variational tool in the study of nonlinear models, complementing the well-known fundamental principals such as the Lagrangian, Hamiltonian, the abbreviated action, conservation laws and etc. Theoretical results will be illustrated by examples of the application of the generalized Rayleigh's quotient for studying nonlinear boundary value problems.