Estimates of initial conditions of parabolic equations and inequalities via lateral Cauchy data

A parabolic equation and, more generally, parabolic inequality is considered in the cylinder QT = Ω × (0, T), where Ω ⊂ Rn is a bounded domain. Cauchy data, i.e., both Dirichlet and Neumann data are given at the lateral surface ST = ∂Ω × (0, T). Logarithmic stability estimates are obtained for the unknown initial condition at {t = 0}. These estimates enable one to establish the convergence rate of a numerical method for the inverse problem of the determination of that initial condition.