On unstable contrast structures in one-dimensional reaction–diffusion–advection problems with discontinuous sources

A new method is developed for studying unstable contrast structures (solutions with an internal transition layer), based on the construction of sufficiently accurate unordered upper and lower solutions and the application of a corollary of the Krein–Rutman theorem. Conditions are formulated for the existence of Lyapunov-unstable one-dimensional step-type contrast structures as stationary solutions of singularly perturbed parabolic reaction–diffusion equations with a discontinuous right-hand side. It is shown that the results obtained can be extended to other singularly perturbed one-dimensional reaction–diffusion–advection problems with discontinuous nonlinearities.