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Conclusion

The main conclusion is that slow diffusion can act as a stabilization mechanism in nonlinear reaction-diffusion equations. Patterns that are unstable solutions of a scalar equation (1) can be stabilized by the a priori negligible effects of coupling a slow diffusion equation to this scalar problem (as in (3)). The general theory developed in [4,3] can be used to determine whether this `control mechanism' is effective; it can be applied to large classes of singularly perturbed reaction-diffusion equations, including the well-studied Gray-Scott and Gierer-Meinhardt models. Moreover, the theory of [4,3] is formulated in the setting of the Evans function approach. As a consequence, it is possible to extend the method both to more general classes of `patterns' (including multi-pulse and periodic patterns) and to more general systems than two-component reaction-diffusion equations in one spatial dimension [5].