Many nonlinear evolution equations couple processes acting on widely separated time scales, suggesting a reduced description in terms of the slow variables. For partial differential equations, however, unbounded spatial operators complicate the separation of fast and slow dynamics and create obstacles to applying classical geometric singular perturbation theory.

 

In this talk, I will present a framework for constructing slow manifolds for evolution equations in Banach spaces, extending key aspects of Fenichel theory to an infinite-dimensional setting. Under suitable stability and spectral separation assumptions, this approach yields locally invariant, attracting slow manifolds and quantitative comparisons with the limiting dynamics. I will explain the main ideas behind the construction, with particular attention to the interaction between temporal and spatial scales.

 

The main emphasis will be on applications to reaction–diffusion equations on thin domains and to continuum limits of adaptive networks. In the first setting, slow manifolds provide a geometric framework for dimensional reduction and the computation of corrections to the limiting equations. In the second, fast adaptation generates effective higher-order interactions whose persistence in the continuum limit can be studied through the compatibility of reduction and passage to the continuum. These examples illustrate how slow-manifold theory connects singular limits, effective evolution equations, and quantitative comparisons of dynamics.