Abstract: Advanced Topics in PDE Analysis focuses on the study of classical partial differential equations, with emphasis on elliptic and parabolic types. The workshop aims to strengthen understanding of key analytical tools—such as maximum principles, energy methods, and Sobolev space techniques—while connecting theory to modern research directions. Participants will explore both fundamental principles and advanced methods for analyzing and solving PDEs, gaining deeper insight into their structure and applications across science and engineering. This workshop is ideal for students and researchers seeking to build a strong foundation in PDE theory and analysis.

Speakers:

Emeritus Professor Laurent Véron — University of Tours

Emeritus Professor Marie-Françoise Bidaut-Véron — University of Tours

Professor Huyan Chen — Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS)

Professor Jingang Xiong — Beijing Normal University

Dr. Yuzhe Zhu — University of Chicago

Dr. Nguyen Ngoc Khanh — Academy of Mathematics and Systems Science，CAS Workshop Plan:

Speaker: M.F. Bidaut-Véron, University of Tours

Title: On local properties of the solutions of the equations $-\Delta u+m|\nabla u|^q=f(u)$ according to $f(u)=u^p$ or $e^u$

Abstract: We study the local properties of positive solutions of equations $-\Delta u+m|\nabla u|^q=f(u)$ with $m>0,q>1$ in a punctured ball or in an exterior domain, according to $f(u)=u^p$ with $p>1$ and $f(u)=e^u$. We compare techniques used to obtain upper estimates near singularities or infinity,including the Bernstein method, Keller-Osserman estimates, and the Doubling Lemma.

Speaker: Laurent Véron, University of Tours

Title: Initial trace of solutions of semilinear heat equation

Abstract: An overview of results on the initial trace of positive solutions of $\partial_tu-\Delta u+f(u)=0$ in $\mathbb{R}^N\times(0,T)$, focusing on $f(u)=u^p$. We discuss the critical exponent $p_c=(N+2)/N$ which affects the topology required to describe the initial trace when $p\ge p_c$.

Speaker: Jingang Xiong, Beijing Normal University

Title: Isolated singularity for the Yamabe equation

Abstract: We review the classification of solutions near isolated singularities of the Yamabe equation.

Building upon classical results by Caffarelli–Gidas–Spruck and later refinements by Korevaar–Mazzeo–Pacard–Schoen, we present new results in dimension 6 and in asymptotically flat metrics.

Speaker: Huyan Chen, SIMIS, Shanghai

Title: Solutions of Semilinear Equations on the Integer Lattice Graphs

Abstract: We present results on positive solutions to semilinear elliptic equations on lattice graphs,including Kazdan–Warner-type and Lane–Emden equations. Existence and nonexistence results are discussed in $\mathbb{Z}^d$ and its subdomains under Dirichlet boundary conditions.

Speaker: Yuzhe Zhu, University of Chicago

Title: Kinetic equations through the lens of Fisher information: Landau, multi-species, and Fermi-Dirac variants

Abstract: This talk explores Fisher information monotonicity in kinetic equations,particularly for the Landau equation and its quantum variants. Connections to the Bakry–Émery criterion and extensions to multi-species systems are discussed.

Speaker: Nguyen Ngoc Khanh, AMSS, CAS

Title: Lusin approximation for functions of bounded variation

Abstract: We introduce Lusin-type approximation results for functions of bounded variation on metric measure spaces, extending classical results for measurable and Sobolev-type functions. Joint work with Panu Lahti (AMSS, CAS).