Evolution Systems

This thematic session focuses on recent developments in Evolution Systems and related topics, highlighting their deep connections with nonlinear PDEs and modern analysis. The goal is to gather researchers working on parabolic, hyperbolic, and dispersive evolution equations, including reaction–diffusion systems, Schrödinger-type models, wave phenomena, and abstract evolution frameworks in functional analysis.

Our goal is to highlight recent results, foster new collaborations, and provide an accessible overview of current trends and open problems in the broad field of Evolution Systems, emphasizing its deep connections with nonlinear analysis, PDE theory, and applications in physics and engineering

Thursday - January 22 - Morning

Speaker: Marko Rojas Medar (Universidad de Tarapacá - Chile)

Hour and room: 9:00-9:40 - Department of Mathematics - 2nd Floor - 347

Title: Variable density incompressible asymmetric fluids

Abstract: In this talk, we consider the equations governing the motion of a density-dependent viscous incompressible asymmetric fluid in a regular bounded domain. We will attempt to present the most important known theoretical results, particularly those obtained by the author and his collaborators. We will pay special attention to a recently studied numerical scheme, which, it should be emphasized, is the first result in the literature for this fluid model.

Speaker: Ricardo Freire da Silva (Universidade Estadual do Sudoeste da Bahia)

Hour and room: 9:45-10:25 - Department of Mathematics - 2nd Floor - 347

Title: Global existence and nonexistence of solutions to semilinear parabolic equations on sub-Riemannian manifolds

Abstract: We study the global existence and nonexistence of global solutions to the semilinear heat equation $u_t−\Delta_M u = h(t) f(u)^{p(x)}$ on $Mx[0, T)$, where M is a sub-Riemannian manifold, and $f : [0, \infty) →[0, \infty)$ is a continuous function. The exponent function $p : M→(0, \infty)$ is also continuous and satisfies $0 < p^{-}= \inf_{x\in M} p(x)\leq p(x) \leq \sup_{x\in M} p(x) = p^{+} < \infty$. This equation extends the nonlinear heat equation $u_t−\Delta u = u^p $ to the sub-Riemannian setting. The results obtained are consistent with classical results in the Euclidean framework. Moreover, they allow us to recover Fujita-type results for connected unimodular Lie groups with polynomial or exponential volume growth.

Speaker: Brandon Carhuas (Universidade Federal de Pernambuco)

Hour and room: 10:30-11:10 - Department of Mathematics - 2nd Floor - 347

Title: Existence and non-existence of solutions for the Hardy parabolic equation with singular initial data.

Abstract: We establish local existence, non-existence, and uniqueness of the solutions of the Hardy parabolic equation $u_t - \Delta u = h(t)|\cdot |^{-\gamma}g(u)$ on  $\Omega \times (0,T) $ with Dirichlet boundary conditions. We assume that $\Omega$ is any smooth domain (bounded or unbounded), $h \in C(0,\infty)$, $g \in C([0,\infty))$ is a non-decreasing function, $0<\gamma<\min\{2,N\}$, and the initial data have a singularity at the origin.

Speaker: Gerardo Jonatan Huaroto Cárdenas (Universidade Federal de Alagoas)

Hour and room: 11:15-12:00 - Department of Mathematics - 2nd Floor - 347

Title: Uniqueness of entropy solution for doubly nonlinear degenerate fractional parabolic

Abstract: In this work, we investigate the uniqueness of solutions for the double nonlinear isotropic degenerate fractional parabolic problem within bounded domains, subject to homogeneous Dirichlet boundary conditions. Our study employs the mathematical analysis techniques, doubling variable techniques by Kruzkov.

Thursday - January 22 - Afternoon

Speaker: Clessius Silva (Universidade Federal Rural de Pernambuco)

Hour and room: 14:00-14:40 - Department of Mathematics - 2nd Floor - 347

Title:  Existência de soluções brandas em espaços de pseudo-medida para o sistema de Boussinesq com difusão e memória fracionárias

Abstract: O sistema de equações de Boussinesq modela o transporte de calor via convecção natural em fluidos viscosos incompressíveis. A estrutura matemática deste modelo compreende o acoplamento entre as equações de Navier-Stokes e a equação de difusão do calor. O presente trabalho investiga a existência de soluções para uma generalização deste sistema, a qual incorpora operadores de difusão e memória fracionárias. Estabelecemos a existência de soluções brandas (mild solutions) locais e globais no contexto dos espaços de pseudo-medida. Além disso, demonstramos que essas soluções globais satisfazem a propriedade de autossimilaridade, considerando dados iniciais com regularidade adequada nesses espaços funcionais.

Speaker: Jandeilson Santos da Silva (Sec. Estadual de Educação - PB)

Hour and room: 14:45-15:25 - Department of Mathematics - 2nd Floor - 347

Title: Observability for the KdV equation on star graphs

Abstract: This talk addresses control properties of the KdV equation posed on a star graph, modeled by N coupled KdV equations on bounded intervals with suitable boundary conditions. Motivated by recent applications of dispersive models on graphs—particularly the use of the KdV equation to describe pulsatile arterial flow—we study the problem of steering solutions to a desired state via boundary controls. The analysis relies on observability inequalities derived from associated spectral problems for the KdV operator. Different combinations of Dirichlet and Neumann controls are considered, leading to the identification of critical length sets. Exact controllability is shown to hold outside these critical lengths.

Speaker: Fernando Gallego (Universidad Nacional de Colombia - Colombia)

Hour and room: 15:30-16:10 - Department of Mathematics - 2nd Floor - 347

Title: Boundary observability for an internal waves model

Abstract: In this talk, we study the exact boundary controllability of a nonlinear internal-waves model described by a coupled Korteweg–de Vries type system posed on a bounded interval. The equations capture the interaction of two weakly nonlinear gravity waves propagating in a stratified fluid. Since the model involves two components, the well-posedness framework naturally requires six boundary conditions; nevertheless, from a control viewpoint, we investigate several reduced boundary actuation strategies, restricting ourselves to configurations involving no more than four independent controls. We begin by analyzing the linearized dynamics, where controllability is obtained through a duality method combined with the identification of hidden boundary regularity effects. This strategy allows us to reformulate the problem in terms of a suitable spectral characterization, whose resolution relies on techniques inspired by the Paley–Wiener approach developed by Rosier (1997). A key step consists of determining whether a quotient of entire functions preserves analyticity, which can be interpreted as a factorization problem in the class of entire functions. According to the selected control setting, this analysis reduces to the study of transcendental relations governing critical spectral parameters. Finally, exploiting the linear results and applying a fixed point argument, we prove local exact controllability for the original nonlinear system.