Singular limit in a parabolic equation

Abstract

Our aim here is to study the singular limit, when e —• 0, in a parabolic problem ' ut = s2Au +f(t,u), x € Í2 C Rn, f > 0, (1) u(0,x) = t¿o(a;) for x £ Í2, Bu = 0 for x e dQ, with a bounded smooth domain Q, £ > 0 and the boundary operator B of either the Dirichlet type (B = Id) or the Neumann type (B = n is the normal vector to dQ). We want to estimate the difference between u and the solution y — y(t,x) of the limit problem (2) ÍVt = f(t,y), t> 0, \»(0,a:) = «o(a), x £ Í2, the central interest being estimates on bounded time interval [0,T]. The problem of long time behaviour (when t —> oo) of a pair u, y has been considered previously, e.g. in [4, 7] (Fragment tekstu).