One Day CADEDIF Workshop
One Day CADEDIF-Workshop on Differential Equations and Dynamics
One day of seminars where our visitors and some local researchers from UCM and other universities in the area of Madrid will share their research, ideas and projects in the broad area of Differential Equations and Dynamics.
The Workshop will take place Thursday February 22, 2024 at Room 222 which is one of the Seminar Rooms of the Applied Mathematics and Math Analysis Department. This room is at the second floor of the School of Mathematics, Universidad Complutense de Madrid.
Program:
09:45 — 10:15. Nsoki Mavinga (Swarthmore College, Philadelphia)
"Nonlinear Elliptic Equations with growth involving critical Sobolev exponents"
10:15 — 10:45. Felipe Rivero (Univ. Politécnica de Madrid)
"Asymptotic pullback behavior and permanence of solutions for a non-autonomous prey-predator system"
10:45 –- 11:15. COFFE BREAK
11:15 –- 11:45. Maya Chhetri (Univ North Carolina Greensboro)
"Some uniqueness results for strongly singular problems"
11:45 — 12:15. Juan Carlos Felipe (Univ Complutense de Madrid)
"Reformulating fractional problems by using nonlocal regional-type operators"
12:15— 12:45. Sergio Junquera (Univ Complutense de Madrid)
"Quenching in a system with non-local diffusion"
13:00-14:00 Seminario del Departamento by Alexandre N. Carvalho (Univ. São Paulo)
"Inertial manifolds, exponential dichotomy and the saddle point property: a unified theory"
14:00— 15:30 LUNCH
15:30— 16:00. Manuel Villanueva-Pesqueira (Univ Pontificia Comillas, Madrid)
"Weak oscillatory boundaries in thin domains"
16:00— 16:30. Joaquín Domínguez (Univ Complutense de Madrid)
"Non-periodic homogenization in a weird brush"
16:30— 17:00. Jose Manuel Uzal (Univ Complutense de Madrid)
"Relations between different attractors related to impulsive dynamical systems"
Abstracts (in alphabetical order):
A.N. Carvalho, "Inertial manifolds, exponential dichotomy and the saddle point property: a unified theory"
Abstract: 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.
M. Chhetri, "Some uniqueness results for strongly singular problems"
Abstract: We consider a semilinear elliptic problems with strongly singular nonlinearity and discuss Brezis-Oswald type uniqueness results for positive solutions. Specifically, with further restriction on either the behavior of the nonlinearity near the origin or on the range of the singular exponent, the problem admits at most one positive solution.
J. Domínguez, "Non-periodic homogenization in a weird brush"
Abstract: In this brief talk, we will discuss some results that study the behavior of solutions to PDEs when a domain, or part of it, is thin, meaning it has a dimension negligibly small compared to the rest. We will address the case where this combines with periodic or non-periodic structures, giving rise to homogenization problems. The goal will be to introduce the audience to these types of problems without diving so much into the details, but also to motivate the study of a new project we are working on, the non-periodic weird brush problem, presenting the approaches we are using and the difficulties it entails.
J.C. Felipe, "Reformulating fractional problems by using nonlocal regional-type operators"
Abstract:This talk will be devoted to presenting how regional-type operators appear naturally when studying certain qualitative properties of solutions to equations driven by the fractional Laplacian. I will begin by introducing the fractional Laplacian and regional-type operators. Then, I will give two examples where a reformulation of the fractional problem turns it into regional-type so we can treat it in a simpler way. While the first example concerns the study of odd symmetry of solutions, the second one deals with a fractional Neumann condition.
S. Junquera, "Quenching in a system with non-local diffusion"
Abstract: The phenomenon of quenching in a dynamical system consists of the explosion of the velocity of the solution while the solution itself remains bounded. It was first assessed by Hideo Kawarada in 1974 for the equation u_t = u_xx + (1-u)^(-1), and the same phenomenon for other systems has been studied since then. The aim of this talk is to speak about the quenching that arises in a system of equations with intertwined absorption terms and a non-local diffusion operator of the type convolution with a smooth kernel. We will tackle the appearance of stationary solutions, the quenching rates of both components and the added difficulties this problem presents with respect to the single equation with non-local diffusion.
N. Mavinga, "Nonlinear Elliptic Equations with growth involving critical Sobolev exponents"
Abstract: We will present some recent results on the existence of weak minimal and maximal solutions between an ordered pair of sub- and super-solutions for nonlinear elliptic equations with nonlinearities in the differential equation and on the boundary. No monotonicity conditions (through one-sided Lipschitz condition, a linear shift or otherwise) are imposed on the nonlinearities. Unlike previous results, we allow the growth conditions in the nonlinearities to go all the way to the critical Sobolev exponents in the appropriate Lebesgue spaces in duality. The approach makes careful use of pseudomonotone coercive operators introduced by H. Brezis, the axiom of choice through Zorn’s lemma and a Kato inequality up to the boundary along with appropriate estimates.
Abstract: In this talk we are going to show the existence of the pullback attractor for a prey-predator ODE system with time depend coefficients using simple techniques coming from a previous study of a non-autonomous logistic equation. Furthermore, the permanence of the solutions will be shown under some conditions over the system's coefficients.
J.M. Uzal, "Relations between different attractors related to impulsive dynamical systems"
Abstract: In this talk we introduce impulsive dynamical systems and we study the existence of its global attractor. Moreover, we investigate the relationship between the global attractor of impulsive systems, the global attractor of the related continuous dynamical system which comes from the impulsive system. This is based on a joint work with Everaldo M. Bonotto (ICMC-USP).
M. Villanueva-Pesqueira, "Weak oscillatory boundaries in thin domains"
Abstract: In this talk, we will study the behavior of solutions to certain elliptic Partial Differential Equations (PDEs) that are defined on thin domains with boundaries characterized by weak roughness. Our exploration is based on the distinctive feature of such oscillations—where the oscillation order is less than the order of thickness of the domain. This allows us to extend our analysis beyond the conventional periodic conditions typically associated with homogenization theory. Specifically, we will analyze scenarios including double oscillatory thin domains, quasi-periodic oscillations, and even almost periodic oscillations.
For more information: arrieta@mat.ucm.es
This event is organized and financed by:
- UCM Research Group “Comportamiento Asintótico y Dinámica de Ecuaciones Diferenciales-CADEDIF"
- Projects PID2019-103860GB-I00 and PID2022-137074NB-I00 from Ministerio de Ciencia e Innovación, Spain.
- Applied Mathematics and Math Analysis Department, Universidad Complutense de Madrid.