Backward Parabolicity, Cross-Diffusion and Turing Instability

Abstract

We show that the ill-posedness observed in backward parabolic equation, or cross-diffusion systems, can be interpreted as a limiting Turing instability for a corresponding semi-linear parabolic system. Our analysis is based on the, now well established, derivation of nonlinear parabolic and cross-diffusion systems from semi-linear reaction–diffusion systems with fast reaction rates. We illustrate our observation with two generic examples for $$2\times 2$$ and $$4\times 4$$ reaction–diffusion systems. For these examples, we prove that backward parabolicity in cross-diffusion systems is equivalent to Turing instability for fast reaction rates. In one dimension, the Turing patterns are periodic solutions which have frequencies which increase with the reaction rate. Furthermore, in some specific cases, the structure of the equations at hand involves classical entropy/Lyapunov functions which lead to a priori estimates allowing to pass rigorously to the fast reaction limit in the absence of Turing instabilities.