On continuous dependence of solution of parabolic equations on coefficients

Abstract

It is well known that solutions of classical initial-boundary problems for second-order parabolic equations depend continuously on the coefficients if the coefficients converge to their limits in a strong enough topology.

In case of one spatial variable, we consider the question of the weakest possible topology providing convergence of the solutions. The problem is closely related to weak convergence of corresponding diffusion processes. Continuous Markov processes corresponding to the Feller operators D_vD_u arise, in general, as limiting processes. Solutions of the parabolic equations converge, in general, to the solutions of corresponding initial-boundary problems for the limiting operator D_vD_u.