Simultaneous Reconstructions of (x,t) -Dependent Coefficients and Initial Conditions of Parabolic Equations

Abstract

This is the first publication, in which an inverse problem of the simultaneous reconstruction of an unknown (x,t) - dependent coefficient and an unknown initial condition in a general linear parabolic equation of the second order is considered. The input data depend on (n+1) variables and are, therefore, formally determined ones. Stability estimates and uniqueness theorem are obtained for this inverse problem. As a by-product, logarithmic stability estimates for initial conditions of parabolic equations and inequalities with time reversed data are obtained for the first time. All known publications about inverse problems for parabolic equations with formally determined input data and unknown coefficients assume that those coefficients depend either only on x or only on t. In addition, the initial condition is assumed to be known, except of two publications cited in the text. The inverse problem of this paper has two potential applications. The first one is in forecasting of public opinions in the framework of the Mean Field Games theory. The second one is in tracking spatiotemporal outbreaks of epidemics.