SEMINAR #1: Nonlocal Parabolic Systems (Siham Boukarabila)
We study a nonlocal version of the Kardar–Parisi–Zhang systems where both equations involve fractional diffusion and nonlinear terms depending on nonlocal gradients. The problem is considered in a bounded domain with initial conditions, and with nonnegative sources.
Our goal is to identify the conditions on the data and on the nonlinear exponents that guarantee the existence of nonnegative weak solutions.
These results are contained in a joint paper with B. Abdellaoui (Abou bekr Belkaid University of Tlemcen, Algeria) and E. Laamri (Institut Élie Cartan de Lorraine, Nancy, France). Parabolic Systems
SEMINAR #2: The Goedel Completeness Theorem (Giovanni Molica Bisci)
We briefly describe the celebrated Soundness Theorem in the Tarski scheme.
SEMINAR #3: The Joy of Cylinders (Danilo Gregorin Afonso)
Radial solutions to differential equations of the type -?u = f(u) in radially symmetric domains have been much considered since the seminal works of Serrin and Gidas, Ni and Nirenberg. In this talk, we do not talk about them. Instead, we analyse the symmetry/break of symmetry and rigidity questions, but in another geometric setting: cylinders. We begin with some heuristic motivations for our study, with the hope of convincing those who are asking themselves “Ok, but why cylinders?”. Then, we illustrate many instances where cylindrical domains and one-dimensional solutions (i.e., solutions that depend solely on the axial coordinate) behave differently than their radial counterparts. Although we do not give any detailed proofs, we comment on the main ideas, which range from the hard analysis of domain derivatives to the more topological theory of bifurcation.