Michael V. Klibanov
Estimates of Initial Conditions of Parabolic Equations and Inequalities Via the Lateral Cauchy Data
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ABSTRACT. A parabolic equation and, more generally, parabolic inequality is considered in the cylinder $Q_{T}=\Omega \times \left( 0,T\right) ,$ where $\Omega \subset R^{n}$ is a bounded domain. Cauchy data, i.e., both Dirichlet and Neumann data are given at the lateral surface $S_{T}=\partial \Omega \times \left( 0,T\right) $. Logarithmic stability estimates are obtained for the unknown initial condition at $\left\{ t=0\right\} $. These estimates enable one to establish convergence rate of a numerical method for the inverse problem of the determination of that initial condition.