Speaker
Description
Degenerate parabolic equations arise in a variety of models in which diffusion vanishes or weakens in certain regions, thereby giving rise to substantial analytical difficulties. In particular, degeneracy strongly affects compactness and dissipative properties, which play a crucial role in the study of long-time dynamics and asymptotic behavior.
In this talk, we focus on a class of degenerate parabolic equations of Caldiroli--Musina type, characterized by the diffusion operator
$$ -\operatorname{div}\bigl(\sigma(x)\nabla u\bigr), $$
where $\sigma$ is a non-negative measurable function that may vanish at finitely many points. More precisely, $\sigma$ satisfies the structural conditions introduced by Caldiroli and Musina (2000). In recent years, the existence and asymptotic behavior of parabolic equations of the form
\begin{equation}\label{e1}
u_t - \operatorname{div}\bigl(\sigma(x)\nabla u\bigr)
+ f(u) + g(x) = 0,
\tag{1}
\end{equation}
have attracted considerable attention. Equation \eqref{e1} can be viewed as a simplified model for neutron diffusion arising in feedback control of nuclear reactors, where $u$ and $\sigma$ denote the neutron flux and the neutron diffusion coefficient, respectively.
We investigate equation \eqref{e1} under various assumptions on the nonlinear terms, considering both autonomous and non-autonomous settings, as well as bounded and unbounded domains in $\mathbb{R}^N$. The main aim of the talk is to illustrate how appropriate energy methods combined with suitable functional frameworks ensure the existence of global attractors and allow for a qualitative description of their structure. Finally, we provide an overview of related open problems and outline several directions for future research. This presentation is intended for a broad audience interested in partial differential equations and infinite-dimensional dynamical systems.
Affiliation
https://scholar.google.com/citations?user=jScEXFoAAAAJ&hl=en