Backward Parabolicity, Cross-Diffusion and Turing Instability
Résumé
We show that the ill-posedness observed for cross-diffusion systems, that is due to backward parabolicity, can be interpreted as a limiting Turing instability of a corresponding semi-linear parabolic system. Our analysis is based on the, now well established, derivation of cross-diffusions from reaction-diffusion systems for fast reaction rates. We illustrate our observation with two generic examples for 2 × 2 and 4 × 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 frequency increases 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 rigorously pass to the fast reaction limit in the absence of Turing instabilities.
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