Abstract
This paper is concerned with a reaction-diffusion-ODE system in non-uniform media. The system consists of a diffusive species and a non-diffusive species and models biological pattern formation. We prove the existence of (i) stationary solutions for all values of the diffusion coefficient D by applying a generalized mountain pass lemma, (ii) stationary solutions with transition layer for D sufficiently small by applying the singular perturbation method, and (iii) monotone stationary solutions for D sufficiently large by the upper and lower solution method. Also, the asymptotic stability of stationary solutions is studied.
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