Homogenization over the spatial variable in nonlinear parabolic systems

S. A. Kashchenko

Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 80 (2019) no. 1, pp. 63-86

Résumé

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.