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\title{Attractors for a class of Degenerate Parabolic Equations}

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\author{Le Thi {\sc Thuy}}

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\begin{abstract}
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~\cite{CalMu}. 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 $\mathbb{R}^N$ (see, for instance, \cite{AnhBinhThuy, ABT2, ABT3, ABT4, CTT, CT}). 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.
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\noindent
Email: thuylt@epu.edu.vn, thuylephuong@gmail.com
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% institution (including address):
Electric Power University \\
235 Hoang Quoc Viet, Nghia Do \\
10000 Ha Noi \\
Viet Nam \\

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\begin{thebibliography}{9}
\bibitem{AnhBinhThuy}
    Cung The Anh, Nguyen Dinh Binh and Le Thi Thuy,
     On the global attractors for a class of semilinear degenerate parabolic equations,
   Annales Polonici Mathematici,
    2010,
    71-89,
    98:1.
\bibitem{ABT2} Cung The Anh, Nguyen Dinh Binh and Le Thi Thuy, Attractors for quasilinear parabolic equations involving weighted p-laplacian operators, Vietnam Journal of Mathematics, 2010, 261-280, 38:1.
\bibitem{ABT3} Cung The Anh and Le Thi Thuy, Global attractors for a class of semilinear degenerate parabolic equations on $R^N$, Bulletin of the Polish Academy of Sciences Mathematics, 2013, 47-65, 61:1.
\bibitem{ABT4} Cung The Anh, Tang Quoc Bao and Le Thi Thuy, Regularity and fractal dimension of pullback attractors for a non-autonomous semilinear degenerate parabolic equation, Glasgow Mathematical Journal, 2013, 431-448, 55:2.
\bibitem{CalMu} P. Caldiroli and R. Musina, On a variational degenerate elliptic problem, Nonlinear Differential Equations Appl., 2000, 187–199, Vol 7.
\bibitem{CTT} Tran Thi Quynh Chi, Le Thi Thuy and Nguyen Xuan Tu, Existence and asymptotic behavior of solutions to a class of semilinear degenerate parabolic equations with nonlinearities of arbitrary order, Dynamics of Continuous, Discrete and Impulsive Systems, 2022, 77-89, 29:1.
\bibitem{CT} Le Thi Thuy and Tran Thi Quynh Chi, Global attractors for a class of quasilinear degenerate parabolic equations, The Journal of Analysis, 2025, 3069–3089, Vol 33.
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