Regularization method for parabolic equation with variable operator

Consider the initial boundary value problem for theequation ut=−L(t)u, u(1)=w on an interval [0,1] for t>0, where w(x) is a given function in L2(Ω) and Ω is a bounded domain in ℝn with a smooth boundary ∂Ω. L is the unbounded, nonnegative operator inL2(Ω) corresponding to a selfadjoint, elliptic boundaryvalue problem in Ω with zero Dirichlet data on∂Ω. The coefficients of L are assumed to be smoothand dependent of time. It is well known that this problem isill-posed in the sense that the solution does not dependcontinuously on the data. We impose a bound on the solution att=0 and at the same time allow for some imprecision in the data.Thus we are led to the constrained problem. There is built anapproximation solution, found error estimate for the appliedmethod, given preliminary error estimates for the approximatemethod