Asymptotics of solutions of a perturbed heat equation

We consider the Neumann problem for the heat equation perturbed by a dissipation term auau, where aa is a function of space and time variables, small in some integral sense, and uu is the temperature. We derive a two term asymptotic representation for the solution for large time which can be used, in particular, to study boundedness and stability properties of the solution in the case when the leading term of the asymptotic expansion does not allow to do this analysis.