Pattern formation in hyperbolic models with cross-diffusion: Theory and applications

A class of hyperbolic reaction–diffusion models with cross-diffusion is derived within the context of Extended Thermodynamics. Linear stability analysis on the uniform steady states is performed to derive the conditions for the occurrence of Hopf, Turing and wave instabilities. The weakly nonlinear analysis is then employed to describe the time evolution of the pattern amplitude close to the stability threshold. The effects of the inertial times on the pattern formation as well as on the transient regimes are highlighted. As an illustrative example, our analysis is applied to the prototype Schnakenberg model and the theoretical results are illustrated both analytically and numerically.