Stability Analysis of Nonlocal Reaction–Diffusion Systems: The Role of Memory

Abstract: Classical stability theory provides a powerful framework for understanding the long-time behavior of reaction–diffusion systems. A natural question is whether similar principles remain valid when memory effects are incorporated into the dynamics.

In this talk, I will present an abstract stability and instability theory for a class of nonlocal-in-time evolution equations. The main result is a linearization principle showing that, under suitable assumptions, the stability or instability of the linearized problem determines the behavior of the corresponding nonlinear system.

As an application, I will consider reaction–diffusion systems with memory and show how the abstract theory can be used to recover analogues of several classical results. In particular, I will discuss instability of nonconstant stationary solutions obtained through spectral properties of the linearized operator, extending classical results due to Chafee, Casten–Holland, and Matano to the nonlocal-in-time setting. I will also briefly discuss the implications of memory effects for diffusion-driven (Turing) instability and pattern formation.