Speakers

A unified theory for inertial manifolds, exponential dichotomy and saddle point property

Inertial manifold theory, saddle point property and exponential dichotomy have been treated as different topics in the literature with different proofs. As a common feature, they all have the purpose of ‘splitting’ the space to understand the dynamics. We give a unified proof for the inertial manifold theorem that, as a consequence, yields the roughness of exponential dichotomy (global In nature) and the saddle-point property and the fine structure within the stable and unstable manifolds (local in nature). In particular, we use these tools in order to establish the hyperbolicity of certain global solutions for a non-autonomous one dimensional scalar parabolic partial differential equations.

Problemas de Cauchy abstractos de ordem superior e potências fracionárias

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.

Pullback attractors for a semilinear parabolic equation with Neumann boundary conditions and time varying domains

In this research we are interested in studying a semilinear parabolic partial differential equation with Neumann boundary conditions and time varying domains. We will verify the existence and uniqueness of solutions for this equation and we will prove the existence of pullback attractors.

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

In this paper, 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 λk(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 λk. 

This is a joint work with Marcus Marrocos (UFAM) and Cleiton L. Cunha (UFAM).

The Nehari manifold for a degenerate logistic parabolic equation

We analyse the behavior of solutions to a degenerate logistic equation with a nonlinear term of the form $b(x)f(u)$, where the weight function b is assumed to be nonpositive. We exploit variational techniques and comparison principle in order to study the evolutionary dynamics. A crucial role is then played by the Nehari manifold, as we note how it changes as the parameter $\lambda$ in the equation or the function $b$ vary, affecting the existence and non-existence of stationary solutions. We describe a detailed picture of the positive dynamics and also address the local behavior of solutions near a nodal equilibrium, which sheds some further light on the study of the evolution of sign-changing solutions. Co-Author(s):  L. Maia.

Breve histórico da trajetória matemática do Professor Antônio Luiz Pereira

Nesta palestra faremos um breve histórico da trajetória matemática do Professor Antônio Luiz Pereira, exaltando sua carreira e contribuições para o desenvolvimento da Matemática brasileira. Sua  principal área de atuação encontra-se em Equações Diferenciais Parciais e Funcionais com forte influência de matemáticos renomados como D. Henry e J. Hale. Seus temas de interesse passam por Sistemas Dinâmicos de Dimensão Infinita e Perturbação da Fronteira em Problemas com Valores de Contorno. Destacam-se também as colaborações com outros matemáticos e  suas contribuições para a formação de novas gerações de matemáticos atuantes nas mais diversas universidades brasileiras.

Elliptic problems in thin domains with doubly weak oscillatory boundary and term concentrating on the boundary beyond the periodic setting

In this talk we will analyze the limit behavior of a family of solutions of the Laplace operator with homogeneous Neumann boundary conditions, set in a two-dimensional thin domain which presents weak oscillations on both  boundaries and  with terms concentrated in a narrow oscillating neighborhood of the top boundary. 

The aim of this problem is to study the behavior of the solutions as the thin domain presents oscillatory behaviors beyond the classical periodic assumptions. We then prove that the family of solutions converges to the solution of a 1-dimensional limit equation capturing the geometry and oscillatory behavior of boundary of the domain and the narrow strip where the concentration terms take place. In addition we will report some numerical experiments for visualizing the results.

Some results on a non-local evolution equation

In this talk we present some results on the asymptotic behavior of an evolution equation related to neuronal activity. We study the existence of Lyapunov functionals and the existence and continuity of global attractors with respect to the parameters present in the equation.