Workshops

De Cadedif

Jornada de Dinámica Infinito Dimensional: Martes 27 de Noviembre de 2007, Departamento de Matemática Aplicada, UCM. 9:30-13:25

Jornada de Dinámica Infinito Dimensional: Martes 27 de Noviembre de 2007

Departamento de Matemática Aplicada Universidad Complutense de Madrid

Jornada de Dinámica Infinito Dimensional Martes 27 de Noviembre de 2007

Lugar: Sala 209,

Seminario del Departamento de Matemática Aplicada

Facultad de Ciencias Matemáticas, UCM

9:30-10:20. "Attractors for Parabolic Problems in dumbbell domains",

German Lozada, Univ. Del Estado de Sao Paulo, Brasil

10:20-11:10. "Semilinear Damped Wave Equations with nonlinearities",

Jan Cholewa, U. Silesia (Katowice), Polonia

11:15-11:45. Café

11:45-12:35. "Dynamical approach to elliptic BVP in asymptotically symmetric unbounded domains",

Messoud Efendiev, Technische Universistät München, Alemania

12:35-13:25. "Non simultaneous quenching in a system of heat equations coupled at the boundary",

Raul Ferreira, U. Complutense

Organiza: Grupo de Investigación CADEDIF de la UCM. Parcialmente financiado por: Proyecto MTM 2006-08262, "Programa de financiación de Grupos de Investigación UCM-Comunidad de Madrid GR69/06-920894" y Departamento de Matemática Aplicada, UCM Más información: José M. Arrieta arrieta@mat.ucm.es Anibal Rodriguez Bernal arober@mat.ucm.es

ABSTRACTS

"Attractors for Parabolic Problems in dumbbell domains"

German Lozada, Univ. del Estado de Sao Paulo, Brasil

In this talk we analyze the dynamics of a parabolic equation with homogeneous Neumann boundary conditions in the dumbbell domain. We provide an appropriate functional setting to treat this problem and show that the attractors behave upper semicontinuous as the channel shrinks to a line segment.

"Semilinear Damped Wave Equations with fast growing Cholewa, U. Silesia (Katowice), Polonia nonlinearities"

Jan Cholewa, U. Silesia (Katowice), Polonia

A class of the second order in time semilinear partial differential equations is considered in the Banach space setting. The results concerning local existence, regularity, bootstrapping continuation, and asymptotic properties of solutions are discussed in case when the nonlinear term satisfies certain critical growth conditions.

"Dynamical approach to elliptic BVP in asymptotically symmetric unbounded domains",

Messoud Efendiev, Technische Universistät München, Alemania

We consider dynamical approach to the elliptic problem in asymptotically symmetric unbounded domain and study the large-time behaviour of solutions. Due to the lack of the uniqueness of the solutions the standard approach based both on the semigroup theory and on elliptic machinery fails. Our approach based on the trajectory dynamical systems. Symmetrization and stabilization of the solitions as well as open problem will also be discussed.

"Non simultaneous quenching in a system of heat equations coupled at the boundary",

Raul Ferreira, U. Complutense

We study the formation of singularities in finite time for solutions of the heat equations coupled at the boundary through a nonlinear flux at one border and zero flux at the other border. We characterize, in terms of the parameters involved when non-simultaneous quenching may appear. Moreover, if quenching is non-simultaneous we find the quenching rate and the quenching set. We also find a possible continuation after quenching of the solutions. Joint work with A. de Pablo, Mayte Pérez-Llanos, F. Quirós and J. D. Rossi.

iMdea matemáticas:seminario 29 de noviembre 2007

Dpto. de Matemáticas, sala 520 Facultad de Ciencias -

UAM Ciudad Universitaria de Cantoblanco 28049 Madrid

10:30 · 11:10 Hardy inequalities in twisted waveguides

David KREJ CIRÍK

Department of Theoretical Physics, Nuclear Physics Institute, Academy

of Sciences, Rez, Czech Republic e-mail: krejcirik@ujf.cas.cz

The Dirichlet Laplacian in tubular domains is a simple but remarkably successful model for the quantum Hamiltonian in mesoscopic waveguide systems. We make an overview of geometrically induced Hardy-type inequalities established recently for the Laplacian in twisted tubes, and mention consequences for the electronic transport. We begin by recalling the classical Hardy inequality and its relation to geometric, spectral, stochastic and other properties of the underlying Euclidean space. After discussing the complexity of the problem when reformulated for quasi-cylindrical subdomains, we give a proof of the Hardy inequality due to a twist of three-dimensional tubes of uniform cross-section and use it to prove certain stability of the spectrum. We also discuss similar effects induced by curvature of the ambient space or switch of boundary conditions.

11:10 · 11:50 Existence and continuity of global attractors for a class of non local evolution equations

Antônio L. PEREIRA

Instituto de Matemática e Estatística-USP

Rua do Matão, 1010, Cidade Universitária, São Paulo-SP,

Brasil e-mail: alpereir@ime.usp.br

In this work we prove the existence of a compact global attractor for the flow of the equation

in L²(S¹). We also show that the flow is gradient and the global attractor depends continuosly on the parameters h and . AMS subject classification: 34G20,47H15.

11:50 · 12:10 Coffee break

12:10 · 13:10 Creating materials with desired refraction coefficient

A. G. RAMM

Mathematics Department, Kansas State University,

Manhattan, KS 66506-2602, USA

ramm@math.ksu.edu

A method is given for calculation of a distribution of small impedance particles, which should be embedded in a bounded domain, filled with material with known refraction coefficient, in order that the resulting new material would have a desired refraction coefficient. The new material may be created so that it has some desired wave-focusing properies. For example, it can scatter plane wave mostly in a fixed solid angle. The inverse scattering problem with scattering data given at a fixed wave number and at a fixed incident direction is formulated and solved.

iMdea