Applied Analysis Seminar Pattern selection in reaction-diffusion-ODE systems: Turing patterns, far-from-equilibrium patterns and nonlinear stability

Reaction-diffusion-ODE systems couple diffusing species with non-diffusive components, such as cell-bound receptors or intracellular signals. This extreme form of spatial scale separation gives rise to a rich variety of stationary patterns, including classical Turing patterns bifurcating from a destabilized homogeneous state and far-from-equilibrium patterns with jump discontinuities. Which of them is observed is a question of stability.

In this talk, I present results from the first part of my thesis. After a brief introduction to reaction-diffusion-ODE systems and spatial scale separation, I discuss conditions for diffusion-driven instability in systems with non-diffusive, slow- and fast-diffusive components. The mechanism matters: if the instability originates in the non-diffusive subsystem, all nearby Turing patterns are unstable, whereas instability of the joint non-diffusive and slow-diffusive subsystem still permits stable Turing patterns. Jump-type patterns, in contrast, exist largely independently of the diffusion coefficients and form continua. 

I then present a general framework for nonlinear stability and instability of bounded, possibly discontinuous steady states of reaction-diffusion-ODE systems with arbitrarily many components. Since the linearization has non-trivial essential spectrum and the generated semigroup is not strongly continuous on L^infinity, classical arguments break down. Nevertheless, the sign of the spectral bound determines nonlinear stability and instability in L^infinity, without requiring a spectral gap.