Pattern formation in two-component reaction-diffusion systems with equal diffusion coefficients

In general, whether Turing instability can occur when the diffusion coefficients are equal is an important problem in pattern formation.Classical Turing instability requires different diffusion rates, and when the diffusion coefficients are equal, linear stability analysis predicts that the homogeneous equilibrium is stable.Nevertheless, instability may arise beyond the linear framework. In \cite{N}, an example was constructed in which the kinetic system has an asymptotically stable equilibrium, while the corresponding reaction-diffusion system admits unstable stationary solutions arbitrarily close to the homogeneous equilibrium.In this paper, we generalize this example and derive conditions on the homogeneous nonlinear terms under which such phenomena occur.