We prove existence of transition fronts for a large class of reaction-diffusion equations in one dimension, with inhomogeneous monostable reactions. We construct these as perturbations of corresponding front-like solutions to the linearization of the PDE at u = 0. While a close relationship of the solutions to the two PDEs has been well known and exploited for KPP reactions (and our method is an extension of such ideas from [15]), to the best of our knowledge this is the first time such an approach has been used in the construction and study of fronts for non-KPP monostable reactions.

We show bifurcation of localized spike solutions from spatially constant states in systems of nonlocally coupled equations in the whole space. The main assumptions are a generic bifurcation of saddle-node or transcritical type for spatially constant profiles, and a symmetry and second moment condition on the convolution kernel. The results extend well known results for spots, spikes, and fronts, in locally coupled systems on the real line, and for radially symmetric profiles in higher space dimensions. Rather than relying on center manifolds, we pursue a more direct approach, deriving leading order asymptotics and Newton corrections for error terms. The key ingredient is smoothness of Fourier multipliers arising from discrepancies between nonlocal operators and their local long-wavelength approximations.

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