Regularization method for parabolic equation with variable operator

Abstract

Consider the initial boundary value problem for the equation $u_t=-L\left(t\right)u$, $u\left(1\right)=w$ on an interval $\left[0,1\right]$ for $t>0$, where $w\left(x\right)$ is a given function in $L^2\left(\Omega \right)$ and $\Omega$ is a bounded domain in ${ℝ}^n$ with a smooth boundary $\partial \Omega$. $L$ is the unbounded, nonnegative operator in $L^2\left(\Omega \right)$ corresponding to a selfadjoint, elliptic boundary value problem in $\Omega$ with zero Dirichlet data on $\partial \Omega$. The coefficients of $L$ are assumed to be smooth and dependent of time. It is well known that this problem is ill-posed in the sense that the solution does not depend continuously on the data. We impose a bound on the solution at $t=0$ and at the same time allow for some imprecision in the data. Thus we are led to the constrained problem. There is built an approximation solution, found error estimate for the applied method, given preliminary error estimates for the approximate method.