Logarithmic stability of parabolic Cauchy problems

Abstract

The uniqueness of parabolic Cauchy problems is nowadays a classical problem and since Hadamard \cite{Ha} these kind of problems are known to be ill-posed and even severely ill-posed. Until now there are only few partial results concerning the quantification of the stability for parabolic Cauchy problems. In the present article, we bring the complete answer to this issue, provided that the space domain has finite diameter with respect to the geodesic distance and assuming that solutions are sufficiently smooth.

References in corpus (1)

• Quantitative estimates of unique continuation for parabolic equations, determination of unknown time-varying boundaries and optimal stability estimates

Cited by in corpus (5)

• Identification of a boundary influx condition in a one-phase Stefan problem
• Global logarithmic stability results on the Cauchy problem for anisotropic wave equations
• Carleman estimates for the parabolic transmission problem and Hölder propagation of smallness across an interface
• The property of unique continuation for second order evolution PDEs
• Two-sided Gaussian bounds for fundamental solutions of non-divergence form parabolic operators with Hölder continuous coefficients