Seminários de Pesquisa

Inscrição de e-mails/ E-mail Subscription 

Próximos Seminários (Next Seminars)






Seminários Anteriores (Previous Seminars)

2024


Title: An introduction to modeling random and stochastic perturbations in differential systems with real applications.


Abstract: A chemostat is a laboratory device used to investigate the growth of populations of species that consume a certain nutrient. It has many applications in real life, such as wastewater treatment, antibiotic production, renewable energy development or fermentation processes. For many years, researchers have used the classical chemostat model, which is a system of nonlinear ODEs, to obtain detailed information about the biological process that occurs in the chemostat. Nevertheless, biologists claim that real devices are subject to randomness and then new mathematical models, more realistic, must be developed.


In this talk the chemostat device will be introduced and the classical chemostat model will be derived. In addition, two different approaches to introduce randomness/stochasticity in the chemostat model will be shown. On the one hand, the common way in the literature, based on the standard Wiener process, that produces important drawbacks from the biological point of view. Secondly, a new promising manner, based on a bounded noise, that has been proved to be very realistic from the mathematical and the biological points of view. In both cases, the resulting random/stochastic systems will be carefully analyzed, accompanied with biological interpretations and numerical simulations that illustrate the theoretical results.


Title: Generalized φ-pullback attractors for evolution processes.


Abstract: The framework of pullback attractors is perhaps the most usual form of generalizing the theory of global attractors for semigroups, to describe the asymptotic behavior of nonautonomous problems. The problem that we face is the following: assuming that a pullback attractor A exists, there is no qualitative information regarding the rate of attraction of A. To that end, many authors have worked with the notion of a pullback exponential attractor. As an extension of this theory, in this talk we will define the generalized φ-pullback attractors for evolution processes in complete metric spaces, with rate of pullback attraction determined by the behavior of a decreasing function φ that vanishes at infinity. We will find conditions under which a given evolution process has a generalized φ-pullback attractor, both in the discrete and in the continuous cases. We will present results for the special cases to obtain a generalized polynomial or exponential pullback attractors, and apply these results to obtain such objects for a class of nonautonomous wave equations.



Title: Shadowing on Hilbert spaces.


Abstract: The Shadowing property has proved to be a powerful tool on the theory of dynamical systems on finite and infinite dimension. This property has been intensively studied and it is well known for diffeomorphisms defined on a finite dimensional manifold. In this talk we will show an extension of some of these results for Hilbert spaces, that is, we will show that some Morse-Smale maps defined on a Hilbert space have the Lipschitz property on the global attractor. Moreover, we also have Holder-Shadowing on a neighborhood of the attractor.


Title: Long-time behavior for evolution processes associated with non-autonomous nonlinear Schrödinger equation.

Abstract: We consider the non-autonomous non-linear Schrödinger equation with homogeneous Dirichlet boundary conditions (NNLS):

u_t − iβ(t)∆u − iγ(t) g(|u|^2) u + iδ(t) u  +  η(t) u = ε(t) f(x, t),            x ∈ Ω,   t > τ,

u(x, t) = 0,        x ∈ ∂Ω,    t > τ,

               u(x, τ ) = u_τ (x),      x ∈ Ω,    

where Ω = (0, T) for T < ∞, when N = 1 and Ω is a bounded smooth domain if N = 2, τ ∈ R, i = √−1 with 

u(x, t) : Ω × R → C. We impose some growth conditions for g and regularity conditions for f and assume that β, γ, δ, η and ε are continuous functions of real values of a variable real that satisfy some suitable conditions. This equation is a non-autonomous version of the equation studied in[3]. In this work (see [1]), we study the local and global existence of solutions to the problem (NNLS) using time rescaling and the theory of evolutionary processes (see [2]), and the existence of a pullback attractor in a suitable phase space. Furthermore, we will show that the pullback attractor has finite fractal dimension.

This work is together with Professor Phd. Marcelo J. D. Nascimento (UFSCar, Brasil).

[1] R. N. Figueroa-López and M. J. D. Nascimento, Long-time behavior for evolution processes associated with non-autonomous nonlinear Schrödinger equation, J. Differential Equations 386 (2024), 80-112.

[2] A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-dimensional Non-autonomous Dynamical Systems, Applied Mathematical Sciences 182, Springer Verlag, New York, 2012.

[3] J. M. Ghidaglia, Finite dimensional behavior for weakly damped driven Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire, 5 (4) (1988), 365-405.


Title: Pullback attractors with finite fractal dimension for a semilinear transfer equation with delay in some non-cylindrical domain.

Abstrac: We will study a semilinear transfer equation with a delay term defined over a non-cylindrical domain. We prove the existence and regularity of weak solutions as well as the existence, regularity and finiteness of the fractal dimension of pull- back attractors on tempered universes that depend on a non-increasing function. We address the problem of estimating the fractal dimension of pullback attrac- tors, under current techniques, which consists of readjusting the proof presented in [1, Lemma 1.3] and extending it to families of normed spaces parameterized in time.



Title: Study of a nonlocal quasilinear variation of the Chafee-Infante problem via Semigroup Theory.

Abstrac:  In this lecture, we will present a study of a one-dimensional nonlocal quasilinear problem of the form u_t - a(l(u))u_{xx} = λf(u) + h(t) with Dirichlet boundary conditions on the interval (0,1). We use a change in the time scale for reformulate the problem into a semilinear one, thus study a general equation through semigroup theory. Our focus extends to establishing the existence of a classical solution for an abstract problem, particularly when the forcing term f has a weaker modulus of continuity than continuous Hölder. We will obtain comparison results that will be helpful in ensuring the existence of a pullback attractor. 



2023


Title:  Fractional power of linear operator and its connection with Partial Differential Equations

Abstract: In this talk we study fractional powers of linear operators and its connections with PDEs, focusing on using this theory to approximate hyperbolic problems by parabolic ones. To illustrate this idea, we consider cascade systems of PDEs, where the fractional approximations are explicitly calculated and we discuss local solvability of the fractional equation with subcritical nonlinearity. As an example, a cascade system of Schrödinger equation is analyzed and a connection between the fractional system and the original system is established.

[1] Belluzi, M. B., Bezerra, F., and Nascimento, M. J. D. On spectral and fractional powers of damped wave equations. Commun. Pure Appl. Anal. 21, 8 (2022).

[2] Belluzi, M., Nascimento, M. J. D., and Schiabel, K. On a cascade system of Schrödinger equations. Fractional powers approach. J. Math. Anal. Appl. 506, 1 (2022).


Title:  Existência e estabilidade de uma família de atratores exponenciais pullback para uma equacão de evolucão semilinear não autônoma de segunda ordem 

Abstract: Neste trabalho, consideramos um problema de evolução semilinear não autônomo que surge em modelos de propagação em hastes elásticas não lineares e ondas íon-acústicas não lineares. Investigamos a existência e estabilidade de uma família de atratores exponenciais pullback para o nosso problema sob condições adequadas de crescimento e dissipatividade. Além disso, também provamos a semicontinuidade superior e inferior desta família de atratores exponenciais pullback no tempo zero. Como caso particular, obtemos a existência do atrator pullback em um espaço apropriado, provamos sua semicontinuidade superior e, por último, obtemos um resultado de regularidade desse atrator pullback. 



Title: Dynamical analysis for a semilinear parabolic equation on a domain with moving boundary.


Abstract: We present a semilinear mathematical model for thermal conduction problems defined on one-dimensional moving boundary domains, known in the literature as moving boundary problems or problems on non-cylindrical domains. We prove the existence and finite fractal dimension of the pullback attractors on tempered universes. Regarding the finiteness of the fractal dimension of the pullback attractor, we apply the method of Lyapunov exponents, associated with uniformly differentiable processes defined on families of Hilbert spaces that are parametrized in time.




       Title:  Structural stability for scalar reaction-diffusion equations.


Abstract:  In talk, we prove the structural stability of a family of scalar reaction-diffusion equations. Our arguments consist of using invariant manifolds to reduce the problem to a finite dimension and, we use the structural stability of Morse-Smale flows in a finite dimension to obtain the corresponding result in an infinite dimension. As a consequence, we obtain the optimal rate of convergence of the attractors and we estimate the Gromov-Hausdorff distance of the attractors using continuous $\varepsilon$-isometries.


Title:Trajectory Attractors and Some Approaches for Multivalued Problems.


Abstract: The trajectory attractor theory is one of the approaches to study the asymptotic behavior of autonomous and non-autonomous problems without uniqueness of solutions. We will talk about the other approaches to study the dynamics of non-autonomous problems without uniqueness of solution and their relation with the theory of trajectory attractors. As an application, we guarantee the existence of a trajectory attractor for a non-autonomous problem with p-Laplacian operator and with dynamic boundary conditions.



[1] R. A. SAMPROGNA, C. B. GENTILE MOUSSA, T. CARABALLO, K. SCHIABEL, Trajectory and global attractors for generalized processes, Discrete Continuous Dynamical Systems Series B, 24(8), 2019.



       Title: Dinâmica assintótica de uma classe de problemas parabólicos em domínios com um pequeno buraco.

Abstract: Neste trabalho estudamos a dinâmica assintótica de uma classe de problemas parabólicos semilineares com condição de contorno de Dirichlet em domínios com um pequeno buraco, cujo tamanho é proporcional a um parâmetro ε positivo pequeno. Em outras palavras, veremos que a família de atratores se comportam continuamente quando o parâmetro ε → 0, bem como, obteremos as taxas de convergência em termos do parâmetro. 



Title:  Abstract differential equations and L^{q, alpha} $-Hölder functions.


Abstract: We introduce the class of L^{q,\alpha} - Hölder functions and study the local and global existence and uniqueness of solution for abstract evolution differential equations assuming that the non-linear term is an L^{q, alpha}-Hölder function.



Title:  Hausdorff dimension and injective orthogonal projections in Hilbert spaces.

Abstract: We present new findings regarding how we can use the Hausdorff dimension of a subset A of an Euclidean space to find projections onto lower-dimensional spaces that are injective in A. We first introduce a version of Mañe's theorem for finite dimensional spaces and orthogonal projections, showing that the proof in this case is more intuitive, and the hypothesis over the Hausdorff dimension is improved and becomes optimal. We also show that information on Hausdorff dimension is not enough to achieve Hölder continuity on the inverse of the injective projections, but the box-counting dimension can be used in this sense.


Title: Impulsive dynamical systems and their global attractors.


Abstract: The theory of impulsive dynamical systems describes the evolution of systems whose continuous dynamics is interrupted by abrupt changes of state that are considered instantaneous, called impulses. In this talk we present some fundamental properties of such theory (autonomous case) in order to approach the existence of global attractors for these systems.




Title:  Kirchhoff-Boussinesq tipe problems with positive and zero mass. 

Abstract:  In this presentation we will treat the question existence of solution for the following class of elliptic Kirchhoff-Boussinesq type problems given by

                                                     ∆^2  u − ∆_p  u + u = h(u)    in R^N

and

                                                           ∆^2  u  − ∆_p  u = f(u)              in R^N,         

 where 2 < p ≤ 2N/(N−2) for N ≥ 3 and 2∗∗ = ∞ for N = 3, N = 4,  2∗∗ = 2N/(N−4) for N ≥ 5 and f and h are            continuous functions that satisfy hypotheses considered by Berestycki and Lion [ 3] . More precisely, the problem with nonlinearity f is realted to Zero mass case. The main argument is to find a Palais-Smale sequence satisfying a property related to Pohozaev identity, as in [ 7] , which was used for the first time by [ 8]. This work was a collaboration with Giovanny Figueiredo and Ricardo Ruviario, for more details you can see [1].     

[1] Carlos, Romulo D and Figueiredo, Giovany M and Ruviaro, Ricardo, Kirchhoff–Boussinesq-type problems

with positive and zero mass. , Applicable Analysis. (2023) 1–13. 1

[2] V. Ambrosio, Zero mass case for a fractional Beresticky-Lions type results, Advances in Nonlinear Analysis

https://doi.org/10.1515/anona-2016-0153

[3] H. Berestycki and P.L. Lions, Nonlinear Scalar Field, Arch. Rational Mech. Anal. 82 (1983), 313-345 1

[4] H. Berestycki, T. Gallouet and O. Kavian, Equations de Champs scalaires euclidiens non lin ́eaires dans le

plan. , C. R. Acad. Sci. Paris Ser. I Math. 297, 307–310 (1984)

[5] E. Gagliardo, Ulteriori propriet`a di alcune classi di funzioni in pi`u variabili, Ricerche Mat., 8:24-51, 1959.

[6] D. Hu and Q. Zhang, Existence ground state solutions for a quasilinear Schr ̈odinger equation with Hardy

potential and Berestycki-Lions type conditions. Appl. Math. Lett. 123 (2022), Paper No. 107615, 7 pp.

[7] J. Hirata, N. Ikoma, K. Tanaka, Nonlinear scalar field equations in RN: mountain pass and symmetric

mountain pass approaches, Topol. Methods Nonlinear Anal. 35 (2010) 253-276. 1

[8] L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type

problem set on RN , Proc. Roy. Soc. Edinburgh Sect. A 129 (1999) 787-809. 1

[9] L. Jeanjean and K. Tanaka, A Remark on least energy solutions in RN . Proc. Amer. Math. Soc. 131, 2399-2408 (2002)

      



Title: Sobre equações diferenciais com retardo dependendo do estado.


Abstract: O objetivo desta palestra é apresentar resultados de existência de solução para algumas classes de equações diferenciais abstratas com retardo dependendo do estado, utilizando para isso teorema de ponto fixo e teoria de operadores. 



Title: Atrator global para um modelo de um termostato rudimentar.


Abstrac: Neste seminário, apresentarei um estudo que fizemos há 20 anos, [2], sobre uma equação de reação-difusão unidimensional com condições de fronteira não linear e não local envolvendo um parâmetro positivo, que modela um termostato rudimentar. Provamos a existência de atrator e também a estabilidade global da solução trivial para um certo intervalo do parâmetro. Esse trabalho, respondeu algumas questões provenientes de um artigo de Guidotti e Merino, [1], que recentemente, em 2021, publicaram outros resultados, [3], sobre o mesmo problema. O intuito da apresentação é embasar e motivar os ouvintes para um possível estudo deste último artigo.


[1]  P. Guidotti and S. Merino, Hopf bifurcation in a scalar reaction-diffusion equation, Journal  of Differential Equations, 140, no. 1, (1988), 201-269.


[2] R. C. D. S. Broche, L. A. F. de Oliveira and A. L. Pereira, Global attractor for an equation modelling a thermostat, Electronic Journal of Differential Equations, vol. 2003, no. 100, 2003, 1-7.


[3] P. Guidotti and S. Merino, On the maximal parameter range of global stability for a nonlocal thermostat model, Journal of Evolution Equations, 2021, 3205-3241.

2022 


Title:  Asymptotic behavior for nonautonomous semilinear evolution equation of second order

Abstract: Click here


Title:  An energy formula for fully nonlinear degenerate parabolic equations in one spatial dimension

Abstract: Energy (or Lyapunov) functions are used in order to prove stability of equilibria, or to indicate a gradient-like structure of a dynamical system. Matano constructed a Lyapunov function for quasilinear non-degenerate parabolic equations with gradient dependency. We modify Matano’s method to construct an energy formula for fully nonlinear degenerate parabolic equations. In particular, we provide a new energy formula for the porous medium equation. This is the fruition of joint explorations with Bernold Fiedler (FU-Berlin) and Ester Beatriz (ICMC-USP).


Title:  Parabolic Kirchhoff equations with flux boundary conditions: well-posedness, regularity and asymptotic behavior

Abstract: In this talk we are concerned with parabolic Kirchhoff equations involving the Laplacian and non-homogeneous boundary conditions of Neumann or Robin type. We describe results regarding to the existence, uniqueness, continuous dependence on data, priori estimates and higher regularity for the parabolic PDE. Conditions ensuring the existence of isolated global or local minimizers of the energy are also provided. Such minimizers are established as asymptotically stable stationary solutions with respect to the evolutionary equation.


Title: Finite dimensional attractors for some variants of the Navier-Stokes equations

Abstract: Click here. 


Title:  Fractional powers of damped wave equations

Abstract: In this talk we explore the theory of fractional powers of  positive operators, more precisely, we use the Balakrishnan formula to obtain parabolic approximations of  (damped) wave equations in bounded smooth domains in RN.  We also explicitly calculate the fractional powers of wave operators in terms of the fractional Laplacian in bounded smooth domains  and characterize the spectrum of these operators. 


Title:  Shadowing Lemma on a neighborhood of the Attractor 

Abstract: Shadowing Lemma is a well known result from the theory of dynamical systems which has several applications, such as continuity of attractors. In this seminar we will show that Morse Smale semigroups in Rn admit the shadowing property in a neighborhood of the global attractor.


Title:  Continuity of attractors for a family of highly oscillatory perturbations of the square

Abstract: In this talk, our goal is to obtain results about : a) the well - posedness, both local and global, for a family of semilinear parabolic problems obtained by oscillatory perturbations of the unit square, which do not converge to the identity in C1 norm, b) the existence of global attractors for these problems and c) the continuity of this family of attractors.


Title: Aproximação de problemas singulares unidimensionais por problemas não singulares em domínios finos

Abstract: Estudaremos a aproximação de um problema singular definido no intervalo (0, 1) por uma família de problemas não singulares definidos em domínios finos limitados de Rn+1. Para tanto vamos introduzir espaços de Hilbert convenientes de forma a obter a convergência das soluções de problemas elípticos. Alguns comentários e discussões também serão realizados com relação ao problema parabólico associado na direção de se obter a convergência da família de semigrupos.


Title:  Probabilistic Takens time-delay embeddings

Abstract: Consider a dynamical system (X,T) consisting of a compact set X in the Euclidean space and a transformation T on X. Takens-type time-delay embedding theorems state that for a generic real-valued observable h on X, one can reconstruct uniquely the initial state x of the system from a sequence of values of h(x), h(Tx), ..., h(Tk-1x), provided that k is large enough. In the deterministic setting, the number of measurements k has to be at least twice the dimension of the state space X. This was proved in several categories and can be seen as dynamical versions of the classical (non-dynamical) embedding theorems. We provide a probabilistic counterpart of this theory, in which one is interested in reconstructing almost every state x, subject to a given probability measure. We prove that in this setting it suffices to take k greater than the Hausdorff dimension of the considered measure, hence reducing the number of measurements at least twice. Using this, we prove a related conjecture of Shroer, Sauer, Ott and Yorke in the ergodic case (and construct an example showing that the conjecture does not hold in its original formulation). Time permitting, I will also describe a work in progress on quantitative aspects of probabilistic time-delay embeddings. This is based on joint works with Krzysztof Barański and Yonatan Gutman.


Title:  Spectral Analysis of a Third Order Equation

Abstract: In this presetation we will study the third order differential equation

uttt + Aθutt + Aρut + Au= 0, 

where A is a closed, densely defined, self-adjoint and positively defined operator and 0 ≤ θ ≤ ρ ≤1. In particuar, we want to know under which conditions of θ and ρ our problem can be well defined (in the sense of semigroup theory) and when the semigroup can be analytic.


Title:  Robustness with respect to variable exponents for a system of inclusions.  

Abstract: In this talk I will show results on stability of the solutions and the robustness of the global attractors with respect to the variation of spatially variable exponents for a coupled system with three inclusions. I will present a sketch of the proof for the continuity of the flows and upper semicontinuity of the global attractors. Trabalho apoiado pela FAPEMIG - Processo APQ-01601-21. 


Title:  - Extensible Beams with Balakrishnan-Taylor Damping: Long-time Dynamics

Abstract: - In this talk, we present some recent results on long-time dynamics of a class of hyperbolic evolution equations related to extensible beams with nonlocal nonlinear damping term. To the proofs, we provide several technical results by means of refined estimates that open up perspectives for a branch of nonlinearly damped problems.


Title:  Teoria espectral para semigrupos de operadores lineares e limitados

Abstract: Discutiremos sobre a teoria espectral para semigrupos de operadores lineares e limitados; a saber, faremos uma breve apresentação sobre a teoria espectral de operadores fechados e densamente definidos em espaços de Banach, semigrupos de operadores lineares e limitados, operadores setoriais no sentido de Henry, e a teoria de potências fracionárias para operadores lineares do tipo K-positivo no sentido do Amann. Com isso, apresentaremos alguns resultados relevantes para a área introduzidos pelos matemáticos ShuPing Chen e Roberto Triggiani.


Title:  -  Some recent results on the stability of Lamé systems with delay

Abstract: - This talk is dedicated to showing various results on the study of long-time dynamics for weakly damped Lam´e elasticity systems exposed to constant and variable delay effects, which allow modeling the behavior of seismic waves on isotropic elastic bodies.


Title:  Nonautonomous perturbations of Morse-Smale semigroups: stability of the phase diagram 

Abstract: In this lecture we study Morse-Smale semigroups under nonautonomous perturbations, which leads us to introduce the concept of Morse-Smale evolution processes of hyperbolic type, associated with nonautonomous evolutionary equations. They are amongst the dynamically gradient evolution processes with a finite number of hyperbolic global solutions, for which the stable and unstable manifolds intersect transversally.  We prove the stability of the phase diagram of the attractors for a small continuously differentiable nonautonomous perturbation of a Morse-Smale semigroup with a finite number of hyperbolic equilibria. This is a joint work with Alexandre N. Carvalho (ICMC-USP), José A. Langa (Universidad de Sevilla) and G. Raugel (in memoriam).



Title:  Stochastic and random fluctuations in biological systems 

Abstract: Bioreactors are very used in practice to model competition between species due to its large number of applications in real processes such as antibiotic production, waste water treatment processes and fermentation of wine and beer, to name some of the most representative. Even though most of models in the existing literature are deterministic, it is very well known that real biological processes are subject to suffer random disturbances and this fact can be easily observed in labs. Motivated by this, in this talk we propose two different ways of modeling random fluctuations in the input flow in bioreactors: the first one uses the typical Wiener process, which leads in important drawbacks from the biological point of view (see [1]), and the second one makes use of the Ornstein-Uhlenbeck process to model bounded perturbations, as in real experiments (see [1, 2, 3, 4, 5]). We provide theoretical results concerning persistence of species, which is the most interesting issue for practitioners, and illustrate our results with several numerical simulations.

[1] T. Caraballo, M. J. Garrido-Atienza, J. López-de-la-Cruz and A. Rapaport, Modeling and analysis of random and stochastic input flows in the chemostat model, Discrete & Continuous Dynamical Systems - Series B, vol. 24 (2018), pp. 3591–3614.

[2] T. Caraballo, R. Colucci, J. López-de-la-Cruz and A. Rapaport, A way to model stochastic perturbations in population dynamics models with bounded realizations, Communications in Nonlinear Science and Numerical Simulation, vol. 77 (2019), pp. 239–257.

[3] T. Caraballo and J. López-de-la-Cruz, Survey on chemostat models with bounded random input flow, Mathematical Modelling and Control, vol. 1, no. 1, (2021) pp. 52–78.

[4] T. Caraballo and J. López-de-la-Cruz, Bounded random fluctuations on the input flow in chemostat models with wall growth and non-monotonic kinetics, AIMS Mathematics, vol. 6, no. 4(2021) pp. 4025-4052.

[5] T. Caraballo and J. López-de-la-Cruz and A. Rapaport, Study of the dynamics of two chemostats connected by Fickian diffusion with bounded random fluctuations, Stochastics and Dynamics, vol. 22, no. 3 (2022) 2240002.



2021


Title:  Well-posedness and asymptotic behavior of global solutions of abstract differential equations with state-dependent delay

Abstract: In this talk, I will present some results related to the existence of global solutions and well-posedness of a class of abstract differential equations with state-dependent delay of the form

u'(t) = Au(t) + F(t, u(t), u(t − σ(t, u_t))), t ≥ 0, (1)

u_0 = φ ∈ C([−p, 0]; X), (2)

in which A : D(A) ⊂ X → X is the generator of an analytic C0-semigroup of bounded linear operators  {T(t)}_{t≥0} defined on a Banach space (X, ||· ||),  u_t : [−p, 0] → X denotes the history function defined by ut(θ) = u(t+θ), for θ ∈ [−p, 0], and the functions F(·) and σ(·) satisfy a general Lipschitz-type condition. I will also present results related to the existence of global attractor for the autonomous version of problem (1)-(2).

[1] Hernandez, E., Fernandes, D., Wu, J. Existence and uniqueness of solutions, well-posedness and  global attractor  for abstract differential equations with state-dependent delay, {\sl Journal of Differential Equations  } (2021), Volume 302,  753-806.


Title: Upper semicontinuity of pullback attractors of semilinear damped wave equation with time-dependent coefficients

Abstract:  In this talk we consider the semilinear damped wave problem of the form

(𝝰 (t)uₜ)ₜ -β(t)Δu+ ɣ(t)u ₜ + 𝛿 (t)u = β(t)f(u),  x ∈ Ω,  t> 𝜏,

u(x,t)=0,  x ∈ 𝜕Ω, t≥  𝜏,

u(x,𝜏)=u𝜏 (x),  uₜ(x,𝜏)=v𝜏 (x),  x∈ Ω,

where Ω is a bounded smooth domain in ℝᴺ, N ≥ 3, 𝜏 ∈ ℝ, f is a real valued function of a real variable with some suitable conditions of growth, regularity and dissipativity, and 𝝰, β, ɣ and 𝛿 are continuous real valued functions of a real variable with some suitable conditions of growth, regularity and signs. Using rescaling of time we prove existence, regularity, gradient-like structure, upper semicontinuity of the pullback attractors for the  evolution processes associated with this boundary initial value problem in a suitable phase space. 

This work is together with the professors PhD. Gleiciane S. Aragão (UNIFESP, Brasil), PhD. Marcelo J. D. Nascimento (UFSCar, Brasil) and PhD. Flank D. M. Bezerra (UFPB, Brasil). This work is financed by CAPES/Brazil (Finance Code 001/2019). 

 [1] G. S. Aragão, F. D. M. Bezerra, R. N. Figueroa-L\'opez and M. J. D. Nascimento, Continuity of pullback attractors for evolution processes associated with semilinear damped wave equations with time-dependent coefficients, Journal of Differential Equations 298 (2021), 30-67.

[2] H. Uesaka, A pointwise oscillation property of semilinear wave equations with time-dependent coefficients II, Nonlinear Anal., 47 (2001), 2563-2571.



Title: Parallelizable impulsive semidynamical systems

Abstract: In this talk, we introduce the concept of parallelizability for impulsive semi-dynamical systems. Two relationships are investigated in the study of impulsive semidynamical systems: the relationship between parallelizable systems and the existence of sections and the relationship between parallelizable systems and dispersive systems.

In addition, we present a study of parallelizability in the region of weak attraction of a compact, strongly invariant and asymptotically stable set.


Title: Pullback attractors for a non-autonomous Klein-Gordon-Zakharov system

Abstract: The Klein-Gordon-Zakharov system is a model that arises in the study of interaction of waves and it appears frequently in thermoelasticity, mechanics, plasma physics, and other areas alike. The aim of this talk is to address recent results concerning the long-time dynamics of solutions, in the sense of pullback attractors, for a non-autonomous formulation of the Klein-Gordon-Zakharov system. This is a joint work with professors Everaldo M. Bonotto (ICMC-USP) and Marcelo J. D. Nascimento (DM-UFSCar).


Title: Fractional powers of linear operators associated with cascade systems of PDEs 

Abstract: The goal is to study fractional powers of a cascade system of partial differential equations. We explicitly calculate the fractional powers of linear operators associated with this type of system and we discuss local solvability of the fractional equation with subcritical nonlinearity. As an example, a cascade system of Schrödinger equation is analyzed. A connection between the fractional system and the original system is established and we prove the convergence of the linear semigroups obtained by the fractional power operator to the original linear semigroup, as the power 𝝰  approaches 1. 


Title: Smoothing and finite-dimensionality of uniform attractors in Banach

Abstract: In this talk we find an upper bound for the fractal dimension of uniform attractors in Banach spaces. The main technique we employ is essentially based on a compact embedding of some auxiliary Banach space into the phase space and a corresponding smoothing effect between these spaces. Our bounds on the fractal dimension of uniform attractors are given in terms of the dimension of the symbol space and the Kolmogorov entropy number of the embedding. In addition, a dynamical analysis on the symbol space is also given, showing that the finite-dimensionality of the hull of a time-dependent function is fully determined by the tails of the function, which allows us to consider more general non-autonomous terms than quasi-periodic functions. As application we show that the uniform attractor of the reaction-diffusion equation is finite dimensional in L^2 and in L^p, with p > 2.

Title: Stability of an abstract viscoelastic model under small delay perturbations

Abstract: In this talk, we present a stability result for a class of viscoelastic delayed models with memory. Actually, following the lines of [1, 2], we characterize the exponential stability of such models. To do this, we consider kernels with weaker restrictions than the ones satisfying some differential inequalities and require some uniform boundness for the delay coefficient in terms of the memory kernel.

[1] V. V. Chepyzhov and V. Pata, Some remarks on stability of semigroups arising from linear viscoelasticity. Asymptot. Anal. 46 (2006), no. 3-4, 251-273.

[2] V. Pata, Exponential stability in linear viscoelasticity. Quarterly of applied mathematics 64.3 (2006), 499-513.

Link:  https://meet.google.com/hzd-pdto-fkd


Title: About the structure of attractors for a nonlocal Chafee-Infante problem

Abstract: We study the structure of the global attractor for the multivalued semiflow generated by this nonlocal reaction-diffusion equation in which we cannot guarantee uniqueness of the Cauchy problem. We analyse the existence and properties of stationary points, showing that the problem undergoes the same cascade of bifurcations as in the Chafee-Infante equation. Also, we prove that the attractor consists of the stationary points and their heteroclinic connections analysing some of the possible connections. 


Title: Rate-induced tipping and saddle-node bifurcation in a class of scalar quadratic nonautonomous ODEs

Abstract: We carry out an in-depth analysis of the possible dynamical scenarios for scalar differential equations of the type x' = −x²+ q(t)x + p(t), where q : R → R and p : R → R are bounded and uniformly continuous. Particularly, the study of nonautonomous bifurcations of saddle-node type is fundamental to explain the absence or occurrence of rate-induced tipping for the differential equation y'= (y − 2/π arctan(ct))² + p(t) as the rate c varies in [0, ∞).  A classical attractor-repeller pair, whose existence for c = 0 is assumed, may persist for any c > 0, or disappear at a certain critical rate c = c_0, giving rise to rate-induced tipping. A suitable example demonstrates that it is possible to have an alternation between intervals of values of c in [0, ∞) where the attractor-repeller pair exists and intervals where it does not.

This is a joint work with

Carmen Nunez, University of Valladolid, Spain, carnun@wmatem.eis.uva.es

Rafael Obaya, University of Valladolid, Spain, obaya@wmatem.eis.uva.es

Martin Rasmussen, Imperial College London, UK, m.rasmussen@imperial.ac.uk

Link:  https://meet.google.com/zcg-rifv-hbt


Title: Dynamical systems in scalar field cosmology

Abstract: In this talk I will give an overview of dynamical systems in cosmology with scalar fields. This includes models of early inflation, quintessence and quintessential inflation.


Title: Approximating solutions of the 3D Navier-Stokes equations on R^3 using large periodic domains

Abstract: Title self explanatory.


Title: O método do operator unfolding em domínios finos tubulares e rugosos 

Abstract: Nesta palestra, discutiremos o comportamento assintótico de uma equação de convecção-reação-difusão com condições de contorno mistas e definido em um domínio tubular fino com fronteira rugosa. Na parte da fronteira onde há condição de fronteira de Robin, o mecanismo de reação depende de um parâmetro α ∈ R que estabelece diferentes regimes que também dependem do perfil e da geometria do tubo definido por uma função periódica g : R²→ R. Vemos que, se é ∂²g não nulo (isto é, quando g realmente depende de ambas as variáveis), então três regimes em relação a α são estabelecidos: como α < 2, α = 2 (o valor crítico) e α > 2. Por outro lado, se ∂²g ≡ 0, regimes semelhantes são obtidos, mas agora com um valor crítico diferente. De fato, se tivermos ∂²g ≡ 0, então o valor crítico do problema deve ser α = 1. Para cada um desses seis regimes obtemos o comportamento assintótico das soluções à medida que o domínio fino cilíndrico se degenera num intervalo estendendo resultados anteriores de [1, 2] e referências ali mencionadas.

[1] J. C. Nakasato, I. Pažanin and M. C. Pereira, Roughness-induced effects on the convection-diffusion-reaction problem in a thin domain, Applicable Analysis 100 (2021) 1107-1120.

[2] J.M. Arrieta and M. Villanueva-Pesqueira, Thin domains with non-smooth oscillatory boundaries, J. Math. Anal. Appl. 446 (2017) 130–164.


Title: On the connections of parabolic equations with discontinuous nonlinearity

Abstract: We consider a parabolic equation of reaction-diffusion type with a discontinuous nonlinearity, which can be expressed by means of a Heaviside functions as a differential inclusion. We show first that under mild assumptions this equation generates a multivalued semiflow that possesses a global attractor and study its structure, which is an interesting and challenging problem. It is noticeable that our problem is the limit of a sequence of Chafee-Infante problems that undergo an infinite sequence of bifurcations, so it is reasonable to expect that it inherits the structure of the attractor of the Chafee-Infante equation. In fact, we prove that there is an infinite (but countable) number of equilibria and that the sequence of equilibria of the approximative problems converges to the equilibria of the limit problem. Since a Lyapunov function exists, the attractor is characterized by the fixed points and their heteroclinic connections, so a full description of the dynamics is got if we determine which connections exist. We give a partial answer to this question. Finally, if we restrict the semiflow to the positive cone, then nice regularity properties of solutions are obtained. In particular, the structure of the global attractor, in this case, is fully understood.


Title: A dynamical system approach to radial fully nonlinear problems

Abstract: In this talk we discuss existence, nonexistence and classification of radial positive solutions of a class of weighted nonlinear equations of Lane-Emden type. Our approach is entirely based on the analysis of the dynamics induced by an autonomous quadratic system, which is obtained after a suitable transformation.This method allows to treat both regular and singular solutions in a unified way, without using energy arguments. We also compare known results of the standard Hénon-Lane-Emden equation and differences produced when one replaces the Laplacian operator by a Pucci fully nonlinear operator. Joint work with Liliane Maia (Universidade de Brasília) and Filomena Pacella (La Sapienza Università di Roma)


Title: Multiple radial solutions for some Neumann problems

Abstract: In this seminar, I will talk about some semilinear or quasilinear problems under Neumann boundary conditions. In the first part of the seminar, I will focus on supercritical p-Laplacian problems, set in a ball of R^N, and describe some existence results obtained via variational methods. In the second part, I will present some multiplicity results obtained via the shooting method for ODEs. In particular, I will describe a recent result on a one-dimensional mean-curvature problem, for which we find several positive oscillating BV-solutions. These results are contained in some joint papers with Alberto Boscaggin, Colette De Coster, and Benedetta Noris.


Title: Miscelânea de mini-palestras em dinâmica não-linear

Abstract: Neste seminário, quatro alunos de PIBIC apresentarão as suas respectivas pesquisas feitas até o presente momento em palestras curtas (de 15 minutos cada). 

Ester Beatriz: A Lyapunov function for fully nonlinear degenerate equations.

Giuliano Pantaroto: Dynamics of the Big Bang: a center manifold of unfoldings of the Taub points.

Victor Hugo: Chaos at the Big Bang: maps of the circle and an iterated function system.

Hauke Sprink: Extremal Horava-Lifshitz cosmology and perturbation of periodic orbits in general relativity. 


Title: Stability for wave equation with localized damping revisited

Abstract: In this talk we revisit some classical problems involving the wave equation to show another way to prove the stability of the problems. We start considering the n-dimensional linear wave equation in a bounded domain subject to a locally distributed linear damping term. In this case, we proved, via semigroup results (Gearhart Theorem), that the energy decays exponentially. For our surprise, this result has never been proved by this methodology so far. After this, using the linear case, we proved the stability to the wave equation now subject to a locally distributed nonlinear damping term. The second case considered is when the domain is whole space $\mathbb{R}^N$. In this situation, we proved two results. First, using semigroups results with full damping out of a compact set. The second case (when the domain is whole space) the goal here is that we removed damping. Precisely, given a positive real number $M>0$, we show that for any compact set $K$ of $\mathbb{R}^N$, it is possible to build a region $\Xi$ free of damping, with $meas(\Xi)=M$ (measure of $\Xi$), such that $\Xi$ is globally distributed. Finally, we also considered a case with an unbounded domain with finite measure. In both cases where the undamped region is unbounded it possesses a finite measure we considered microlocal analysis tools combined with Egorov's theorem.


Title: Hyperbolicity of equilibria for a non-local autonomous one-dimensional parabolic problem

Abstract: Hyperbolicity is an important property of stationary solutions of semilinear problems in infinite dimensional spaces. It can be extended (under suitable conditions) to quasilinear problems.

In this lecture, we will present a non-local quasilinear problem for which a notion of hyperbolicity of equilibria is available. For that, we will study an associated non-local semilinear problem and analyse the spectrum of its linearization.  


Title: Spontaneous periodic orbits in the Navier-Stokes flow

Abstract: In this talk, we introduce a general method to obtain constructive proofs of existence of periodic orbits in the forced autonomous Navier-Stokes equations on the three-torus. After introducing a zero finding problem posed on a Banach space of geometrically decaying Fourier coefficients, a Newton-Kantorovich theorem is applied to obtain the (computer-assisted) proofs of existence. As applications, we present proofs of existence of spontaneous periodic orbits in the Navier-Stokes equations with Taylor-Green forcing.


Title: Bubbling phenomena in the critical nonlinear heat equation

Abstract: In this talk I will discuss some recent constructions of blow-up solutions for a Fujita type problem for power related to the critical Sobolev exponent. Both finite type blow-up (of type II) and infinite time blow-up are considered. This research program is in collaboration with C. Cortazar, M. del Pino and J. Wei.

2020

Título: Dynamics of a non-local scalar one-dimensional parabolic problem 

Resumo: In this talk we will present some results on a non-local and non-autonomous version of the Chafee-Infante equation. We want to evidence difficulties related to those modifications in the original version...Ver mais


Título: O Big Bang: um bilhar caótico em um conjunto de Cantor

Resumo: Os modelos de Bianchi descrevem uma singularidade espacial da relatividade geral (como o Big Bang) através de EDOs. Tal dinâmica contínua induz a construção de uma dinâmica discreta no círculo, que pode ser interpretada como um jogo de sinuca, e sabe-se que tal dinâmica discreta gerada é caótica...Ver mais


Título: A reaction-diffusion model with nonlocal viscosity and Lyapunov structure 

Resumo: During the last decades many mathematicians have been studying non local problems motivated by its various applications in physics, biology or population dynamics...Ver mais


Título: A stability study of nonuniform nonautonomous hyperbolicity 

Resumo: We study nonuniform exponential dichotomies for linear evolution processes in Banach spaces.  We establish theorems that allow us to compare discrete and continuous dynamical systems with this type of hyperbolicity. We use these results to prove robustness of nonuniform exponential dichotomy for continuous evolution processes...Ver mais


Título: Boa colocação para um fluxo de dois fluidos não isotérmicos, viscosos e incompreensíveis

Resumo: O estudo da dinâmica da interface de uma mistura de dois fluidos diferentes desempenha um papel importante na teoria da hidrodinâmica, devido às crescentes aplicações na engenharia. Neste trabalho, estudamos um modelo de interface difusa não isotérmico que descreve a mistura de dois fluidos incompressíveis...Ver mais


Título: Blow-up and grow-up for parabolic PDEs 

Resumo:  We discuss recent results on finite and infinite time blow-up solutions of parabolic PDEs. We are mainly interested in the limiting dynamics as it approaches the maximal time of existence...Ver mais


Título: PDE's in Singularly Perturbed Domains: Some recent results

Resumo: We want to present some domain perturbation problems in PDE's and techniques to deal with this kind of problems. We also present some recent results obtained in the context of elliptic equations... Ver mais


Título: Nonautonomous fractional oscillon equation

Resumo: In this work we are concerned with the asymptotic behavior of nonautonomous fractional approximations of oscillon equation...Ver mais 


Título: Generic dynamics of the scalar parabolic equations 

Resumo: The hyperbolicity of equilibrium points and periodic orbits, as well as the transversality of the intersection of their stable and unstable manifolds, are important features from the dynamical point of view...Ver mais


Título: Nonlinear Schrödinger Equations on compact metric graphs 

Resumo: Within this talk we overview some recent results on stationary solutions to nonlinear Schrödinger equations on compact metric graphs. We exploit two different variational approaches...Ver mais


Título: Remarks on stability of wave equations with damping-delay interaction 

Resumo: In this webinar we introduce a class of wave equations featuring delay effects on the velocity, as studied by Nicaise and Pignotti (SICON 2006). In order to show the asymptotic stability of the system, we recall the method of perturbed energy...Ver mais 


Título: Unbounded attractors for dynamical systems 

Resumo: Dynamical systems with bounded absorbing sets have been studied for several decades and model many real world phenomena. However, a great amount of interesting systems have unbounded solutions, and for those a bounded absorbing set cannot exist...Ver mais


Título: Spatio-temporal feedback control of partial differential equations 

Resumo: Noninvasive time-delayed feedback control ("Pyragas control") has been investigated theoretically, numerically and experimentally during the last twenty years, mostly for ordinary differential equations...Ver mais 


Título: An overview of Takeuchi - Yamada problem

Resumo: We discuss the properties of the reaction-diffusion equation with degenerate p-Laplacian studied by Takeuchi and  Yamada in 2000, its similarities and non-similarities to Chafee-Infante problem as well as the continuity properties with respect to the parameters p and q...Ver mais


Título: Lipschitz perturbations of Morse-Smale semigroups

Resumo: In this work we deal with Lipschitz continuous perturbations of gradient Morse-Smale semigroups (all critical elements are equilibria). We study the the permanence of connections between equilibrium points (structural stability) when subjected to Lipschitz perturbations...Ver mais


Título: Estrutura de atratores não autônomos através da configuração skew-product 

Resumo: Iremos apresentar os semigrupos skew-product como forma de estudar problemas não autônomos e descrever a estrutura do atrator de uma equação diferencial ordinária planar difusivamente acoplada...Ver mais


Título: A equação $d^3u/dt^3+Au=0$ 

Resumo: Nesta palestra apresentaremos resultados sobre equação $d^3u/dt^3+Au=0$ em espaços de Hilbert...Ver mais


Título: The spectrum of a nonlocal Dirichlet problem 

Resumo:  In this talk we will discuss the spectrum set of a nonlocal equation with non-singular kernels and Dirichlet conditions in bounded open sets. We study continuity of the eigenvalues with respect to the domain of definition of the solutions getting a Hadamard formula for perturbations of simple eigenvalues given by diffeomorphisms...Ver mais


Título: Estrutura gradiente de sistemas do tipo cascata

Resumo: Neste seminário mostraremos condições suficientes para que um sistema do tipo cascata...Ver mais


Título:  Quantitative properties in the shadowing theory 

Resumo:  Shadowing property is related to the question under which conditions for any approximate trajectory there exists a close nearby trajectory. In this talk we will give an overview of quantitative results in this direction, and discuss its relation to hyperbolicity and non-uniform hyperbolicity...Ver mais


Título: A characterization of parallelizable dynamical systems in terms of fiber bundles

Resumo:  Given a dynamical system...Ver mais


Título: Compact convergence of operators, reaction diffusion equations with nonlinear boundary conditions and perturbations of attractors 

Resumo: In this talk I will present some results of an ongoing project in collaboration with Gleiciane S. Aragão and Simone M. Bruschi,  in which we try to address the issue of the behavior of the attractors of a reaction diffusion equation with nonlinear boundary condition when the domain undergoes a Lipschitz perturbation...Ver mais


Título: Persistence of hyperbolic equilibria under nonautonomous random perturbations

Resumo: In this work we study permanence of hyperbolicity for autonomous differential equations under nonautonomous random perturbations. For the linear case, we study robustness and existence of exponential dichotomies for nonautonomous random dynamical systems. Next, we establish a result on the persistence of hyperbolic equilibria for nonlinear differential equations...Ver mais


Título: A dynamical system approach to human consciousness 

Resumo: The complex and changing landscape from brain dynamics has received a huge research over the last twenty years. Recently, a novel mathematical approach based on dynamical system theory has been introduced to describe human consciousness. It is associated to structures, of informational nature, related to the existence of flows of global attractors...Ver mais


Título: Dynamical systems under impulse conditions 

Resumo: Impulsive systems are used to describe the evolution of processes whose continuous dynamics are interrupted by abrupt changes of state. In this talk, we present an overview of the theory of impulsive semidynamical systems ...Ver mais


Título: Stabilization by boundary noise: a Chafee-Infante equation with dynamical boundary conditions

Resumo: The stabilization of parabolic PDEs by multiplicative noise is a well know phenomenon that has been studied extensively over the past decades. However, the stabilizing effect of a noise that acts only on the boundary of a domain had not been investigated so far...Ver mais


2019

Título: Pullback attractors for a class of non-autonomous thermoelastic plate systems 

Resumo: In this article we study the asymptotic behavior of solutions, in the sense of pullback attractors, of the evolution system...Ver mais 


Título: Some examples of Topological Structural Stability 

Resumo: A semilinear parabolic problem with Neumann conditions in a Dumbbell domain... Ver mais 


Título: Global attractors for critical wave equations with locally distributed damping

Resumo:  In this talk we present some classical results on the existence of global attractors for 3D wave equations with Sobolev-critical forcing and locally distributed damping...Ver mais 


Título: A delayed Kaldor's model

Resumo: We deal with an economic model of the form...Ver mais 


Título: Application of the control theory in the study of asymptotic behavior for a Riemannian wave equation

Resumo: This talk is devoted to show the different control conditions and their implications in the Riemannian wave equations with respect to the study of the exact control in finite time, as in the asymptotic dynamics...Ver mais


Título: The p-Laplacian equation in oscillating thin domains 

Resumo: In this work we apply the unfolding operator method to analyze the asymptotic behavior of the solutions of the p-Laplacian equation with Neumann boundary condition set in a bounded thin domain of the type...Ver mais


Título: Stochastic Differential Equations via Generalized ODEs 

Resumo: In this talk, we present the theory of stochastic differential equations via the theory of generalized ordinary differential equations. We show that under some conditions, the Itô integral can be considered as a Kurzweil integral...Ver mais 


Título: Existence, uniqueness and asymptotic behaviour in a parabolic PDE driven by the p-Laplacian with indefinite logistic source 

Resumo: We are concerned with existence, uniqueness and asymptotic behaviour of positive solutions of a parabolic PDE driven by the p-Laplacian with indefinite and unbounded potentials besides a logistic source having weight which is also indefinite and unbounded...Ver mais


Título:  Parallelizable Semidynamical Systems

Resumo: In this work, we present the theory of parallelizable systems in the context of semidynamical systems. We establish necessary and sufficient conditions for a semidynamical system to be parallelizable...Ver mais


Título:  Periodic solutions of neutral functional differential equations

Resumo: This work concerns the study of periodic solutions for a class of neutral functional differential equations  (NFDEs) of type...Ver mais


Título:  Well-posedness of a nonautonomous semilinear wave equation

Resumo: In this talk we present an alternative proof on Hadamard well-posedness of a damped semilinear wave equation with sub-critical growth...Ver mais


Título: Well posedness and asymptotic behavior of parabolic problem with non local term and flux type  boundary condition 

Resumo: The purpose of this lecture is dedicated to the study of existence, uniqueness, continuous dependence and asymptotic behavior of the solution to  parabolic problem with non local term  and flux type boundary condition...Ver mais


Título: The Schrödinger-Debye system in compact riemannian manifolds 

Resumo:  In this talk we present the local well-posedness of the initial value problem associated with the Scrödinger-Debye system posed on a  d-dimensional compact Riemannian manifold...Ver mais 


Título: Finite dimension of negatively invariant subsets of Banach spaces 

Resumo: We give a simple proof of a result due to Mañé...Ver mais