Homogenization of semilinear parabolic equations with asymptotically degenerating coefficients

Анотація

An initial boundary value problem for semilinear parabolic equation :

FORMULA, x Î Ω, t Î(0,T);

with the coefficients a_ij^e(x) depending on a small parameter e is considered. We suppose that a_ij^e(x) are of the order of e^3+g( 0 ≤ g < 1) ) on a set of spherical annuluses G_e^a thickness d_e=de^2+g. The annuluses are periodically with a period e distributed in Ω. On the set Ω\UaG_e^a these coefficients are constants.We study the asymptotical behaviour of the solutions ue(x,t) of the problem as e ® 0. It is shown that the asymptotic behaviour of the solutions is described by a system of a parabolic p.d.e. coupled with an o.d.e.