Título: "The limit shape of the critical front profile for vanishing diffusion in Born-Infeld models"

Data: 26/04/2023
Horário: 12:00h
Local: C-116
Palestrante: Maurizio Garrione (Politecnico di Milano)

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.

Título: Boundary homogenization problems with high contrasts: the elasticity system & the local problems

Data: 27/01/2023
Horário: 11:00h
Local: C-116
Palestrante: María Eugenia Pérez Martínez (Universidad de Cantabria)

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.

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Palestra 1
Titulo: Stabilization of a time delayed for a generalized dispersive system.
Palestrante: Fernando A. Gallego (Universidad Nacional de Colombia)
Data: 08/02/2023
Horário: 11:00
Local: sala C-119
Resumo: In this talk we study the asymptotic behavior of the solution of the time–delayed higher order dispersive systems posed in the real line. Under suitable assumptions on the time delay coefficients we prove that the system under consideration is exponentially stable in two different ways. First, if the coefficient of the delay term is bounded from below by a positive constant, we use the Lyapunov approach to prove that the energy associated to the solution of the higher order dispersive system decays exponentially. After that, we extend this result to the case in which the coefficient of the undelayed feedback is also indefinite. Both problems are investigated when the exponent p in the nonlinear term ranges over the interval [1, 2j) where 2j + 1 is the order of the dispersive system.

Palestra 2
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: 12:00
Local: 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."

Título: Stability of Mkdv Breathers on The Half-Line

Palestrante: Márcio Cavalcante (UFAL)
Data: 18/01/2023
Horário: 12:00h
Local: 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.