Homogenization over the spatial variable in nonlinear parabolic systems

Kashchenko, S.

doi:10.1090/mosc/288

Homogenization over the spatial variable in nonlinear parabolic systems
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by S. A. Kashchenko
Translated by: V. E. Nazaikinskii

Trans. Moscow Math. Soc. 2019, 53-71

DOI: https://doi.org/10.1090/mosc/288

Published electronically: March 31, 2020

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Abstract:

We consider boundary value problems for nonlinear parabolic systems whose coefficients are periodic rapidly oscillating functions of the spatial variable. Results on the closeness of time-periodic solutions of an original boundary value problem and the problem homogenized over the spatial variable are presented. The dynamic properties of these equations are studied in near-critical cases of the equilibrium stability problem. Algorithms for constructing the asymptotics of periodic solutions and for calculating the coefficients of the so-called normal forms are developed. In particular, we show that an infinite process of bifurcation and disappearance of a stable cycle can occur with increasing oscillation degree of the coefficients. In addition, we study some classes of problems with a deviation in the spatial variable as well as with a large diffusion coefficient. Logistic delay equations with diffusion and logistic equations with a deviation in the spatial variable, which are important in applications, are studied as examples. The coefficients of these equations are assumed to be rapidly oscillating in the spatial variable.

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