We deal with a weakly-coupled system of semilinear parabolic equations, namely a competition-diffusion system, and prove the existence of a stable spatially-inhomogeneous equilibrium solution on the assumption that the spatial domain is far from being convex and that the corresponding system of ODEs in the absence of diffusion posesses at least two distinct asymptotically stable equilibria. We also consider a non-weakly coupled system of competition type involving cross-diffusion terms, which leave the system quasilinear but no longer semilinear. The point of interest is to see how the shape of the spatial domain or the presence of cross-diffusion terms contributes to the occurrence of pattern formation.
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Hiroshi Matano, Masayasu Mimura, Pattern Formation in Competition-Diffusion Systems in Nonconvex Domains. Publ. Res. Inst. Math. Sci. 19 (1983), no. 3, pp. 1049–1079DOI 10.2977/PRIMS/1195182020