Title: An Analog of the Rothe Method for Some Ill-Posed Problems for Parabolic Equations

Abstract: The classical method of Rothe proves existence and uniqueness theorems for initial boundary value problems for parabolic equations using the explicit finite difference scheme with respect to time. In this method, an elliptic boundary value problem is investigated on each time step. On the other hand, time dependent experimental data are always collected on discrete time grids, and the grid step size cannot be arranged to be infinitely small. The same is true for numerical studies. Therefore, it makes an applied sense to consider both unique continuation problems and coefficient inverse problems for parabolic equations, which are written in the form of finite differences with respect to time and without allowing the grid step size to tend to zero. This leads to a boundary value problem for a coupled system of elliptic equations with both Dirichlet and Neumann boundary data, which is somewhat similar to the Rothe's method. Dissimilarities are named as well. Two long standing open questions are addressed within this framework. A specific applied example of monitoring epidemics is discussed. In particular, a numerical method for this problem is constructed and its global convergence analysis is provided.