Visiting Xian Liao at Dalian University, China for two weeks in December 2025. I will be giving a minicourse at the Dynamical System seminar and presentations at the Differential Equations seminar.

Minicourse

Title: Orbital stability of propagated waves in parabolic systems

Abstract: In these two lectures, we will discuss a class of partial differential equations that models a large range of biological and physical phenomena. Some of their solutions are given as a fixed profile traveling with constant speed. We will discuss the long time stability of these specific solutions. Due to translational invariance, the linearized dynamics spectrum touches the imaginary axis. As a consequence, it is necessary to distinguish between shape and phase dynamics. I will present the different tools and approaches that where developed in the past fifty years to separate and control these different dynamics. If time allows, we will discuss possible extensions of these tools to open problems.

Keywords: solitons, kinks, periodic waves, spectral projection, resolvent kernel, Evans function, Floquet-Bloch transform.

Seminars

Title: Stability of propagated fronts in scalar balance laws

Abstract: Scalar balance laws are advection-reaction equations, that appear either in biology or physic when one mesures the variation of a quantity over time. In this presentation, we will focus on specific solutions of these equations, namely propagated waves that connect two distinct constant states. A large variety of such waves can be constructed, and we will discuss the stability of some of them. It is a joint work with L. M. Rodrigues.

Title: Large time dynamics in Klein-Gordon equations

Abstract: The Klein-Gordon equation is a wave equation with an additional mass damping term. In this presentation, I will review some literature about the dynamic of such equation when settled on an unbounded one-dimensional spatial domain. I will further present some new results regarding the global existence and long time behaviour of solutions that are initialy close to constant or periodic equilibria. Most notably, I will talk about a viscous approximation of this equation, as well as describe how uniformly local orbital stability can be obtained from polar decomposition. This is joint work with Björn de Rijk and Emile Bukieda.