Existence, asymptotics and Lyapunov's stability of solutions of periodic parabolic boundary -value problems for Tikhonov-type systems

We consider a periodic parabolic singularly perturbed boundary value problem for the Tikhonov system: a singularly perturbed system with fast and slow equations. An asymptotic approximation of the solution to the problem is constructed, conditions for the existence of a solution and its asymptotic stability according to Lyapunov are obtained for these solutions as solutions to the corresponding initial-boundary value problems for this system, both in the case of various types of quasi-monotonicity and in the case of its violation. The results are generalized to initial-boundary value parabolic problems, including problems with quadratic nonlinearities (the so-called KPZ diffusion-advection reaction systems).
The work is a further development of the asymptotic method of differential inequalities (see [1] and references in this work) to new classes of systems.
[1]. Nefedov N. N. Development of methods for asymptotic analysis of transition layers in the reaction-diffusion-advection equations: theory and application", Zhurn. Comput. Math. and Math. Phys., 61:22 (2021)