Time-periodic solutions with internal and boundary layers to singularly perturbed parabolic problems: Existence, approximation and domain of attraction

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The project concerns boundary value problems for singularly perturbed time-periodic reaction-diffusion-advection equations in 1D.We will develop algorithms of analytic construction of sequences of lower and upper solutions with internal and boundary layers. The positions of the internal layers will periodically move in time, in general. The lower and upper solutions will be used for proving existence of exact solutions with internal and boundary layers, for uniform approximation of them, for verifying their stability and for estimation of their domain of attraction. Moreover, the lower and upper solutions are useful also for creating numerical algorithms for layered time-periodic solutions. The main technical tools are formal asymptotic expansions and using of stretched variables close to the layers. The terms in the asymptotic expansions as well as the additional, modifying terms, which create lower and upper solutions, can be calculated by solving linear inhomogeneous time-periodic parabolic problems on the half axis. operators on the half axis. The inverted linear parabolic partial differential operators have to be order preserving, and they should map exponentially decaying functions into exponentially decaying functions. For proving stability we use the Krein-Rutman Theorem.