Continuous dependence and uniqueness for lateral Cauchy problems for linear integro-differential parabolic equations

TL;DR

This paper establishes uniqueness and continuous dependence for lateral Cauchy problems involving linear integro-differential parabolic equations using Carleman estimates, without requiring initial conditions, based on boundary derivative data.

Contribution

It introduces new Carleman estimate techniques to prove uniqueness and stability for these problems without initial data, expanding the theoretical understanding.

Findings

Proved uniqueness of solutions under given boundary conditions.

Established continuous dependence on boundary data.

Extended the theory to integro-differential parabolic equations.

Abstract

Via Carleman estimates we prove uniqueness and continuous dependence results for lateral Cauchy problems for linear integro-differential parabolic equations without initial conditions. The additional information supplied prescribes the conormal derivative of the temperature on a relatively open subset of the lateral boundary of the space-time domain.