Title: Rapid boundary stabilization for longitudinal vibrations of a bar

Abstract: Consider the longitudinal vibrations of the cross-sections of a bar fixed at one end, with a mass attached to the other end. This lecture investigates the exponential decay of the solution to the physical problem, with an arbitrary decay rate. To obtain the decay, we use Komornik's work.

Title: Carleman estimates for parabolic equations with super strong degeneracy in a set of positive measure

Abstract: This work is concerned with the obtainment of new Carleman estimates for linear parabolic equations, where the second-order differential operator brings a super-strong degeneracy in a positive measure subset of the spatial domain. In order to prove our main result, the control domain must contain the set of degeneracies.  As a well-known consequence, we achieve a null controllability result in the current context.

Title: On the role of the viscosity parameters in the large time asymptotics of 2D micropolar flows

Abstract: We investigate the role of the four viscosity parameters, in fluids where the particles possess a microstructure (micropolar flows) and are allowed to rotate in a two-dimensional setting. We first establish the existence of global finite energy solutions, satisfying the classical energy equality, for arbitrary initial data in L^2, in the case of a spin viscosity \gamma\ge 0, and we construct the asymptotic profiles of the solution as t \to +\infty. We deduce the remarkable fact that the large time behavior only depends on the kinematic viscosity \mu, and not on the other parameters \chi (vortex-viscosity), \gamma (spin viscosity) and \kappa (gyroviscosity) of the model. Our primary tool is a new enstrophy-like identity of independent interest, involving the difference between the fluid vorticity and the micro-angular velocity. Another consequence of our analysis is the identification of scenarios where the presence of micro-rotational effects significantly enhances dissipation, thereby slowing down the fluid motion at large times. This is a joint work with L. Brandolese (Institut Camille Jordan, Université Lyon 1), A. V. Busuioc (Université Jean Monnet) and D. Iftimie (Institut Camille Jordan, Université Lyon 1).

Title: Time decay for second-grade fluids

Abstract: We study the long-time behavior of incompressible second-grade fluid flows in the whole space R^3. These models incorporate elastic effects through a regularization of the velocity field, leading to dynamics that differ from the classical Navier-Stokes equations. For initial data with finite energy and a mild low-frequency integrability assumption, we establish explicit algebraic decay rates for the velocity in the natural energy space of the model. We show that, despite the presence of the regularizing term, the decay rate remains the same as in the Navier-Stokes case. This phenomenon is governed by the low-frequency part of the solution and highlights the lack of full parabolic smoothing in the second-grade fluid equations. Our approach relies on a Fourier splitting method adapted to the structure of the modified linear operator. This is a joint work with César Niche (UFRJ), Cilon Perusato (UFPE) and Marko Rojas Medar (Universidad de Tarapacá, Chile).

Title: Pullback attractors for a semilinear parabolic equation with homogeneous Neumann  boundary conditions and time-varying domains

Abstract: In this work, we consider a non-autonomous semilinear parabolic partial differential equation with homogeneous Neumann boundary conditions and time-varying domains. Using a coordinate transformations technique, we will show that the non-autonomous problem in a time-varying domain is equivalent to a non-autonomous problem in a fixed domain, and we will show the existence of pullback attractors.

Title: Higher-order Cauchy problems and fractional powers of operators

Abstract: In this talk, we present several results concerning the existence and regularity of solutions to certain higher-order Cauchy problems, using an approach based on fractional powers of linear operators.

Title: Internal stability of a network of nonuniform timoshenko beam system

Abstract: In this talk, we will study the internal stability of a non-uniform Timoshenko beam system on a network. For star-shaped networks, we apply internal feedback on the rotation angle of all edges except one. Under certain assumptions regarding the time-varying physical coefficients, we prove the exponential stability of the system if the wave speeds are equal, and polynomial stability otherwise. The proof is based on the frequency domain method combined with the multiplier approach. We conclude the presentation by generalizing the results to general networks. I will present these results in collaboration with Julie Valein from Université de Lorraine and Mohammad Akil from Université Polytechnique Hauts-de-France.

Title: Title-Controllability problem of an evolution equation with singular memory

Abstract: In this talk, we discuss control problems governed by a semilinear evolution equation with memory kernel locally integrable.  A linear-quadratic regulator problem to determine the optimal control that yields approximate controllability for the linear control system. Furthermore, sufficient conditions for the existence of a mild solution and the approximate controllability of a semilinear system in a reflexive Banach space having uniformly convex dual has been discussed. Finally, we apply our theoretical findings to investigate the approximate controllability of the heat equation with singular memory.