A Strongly Ill-Posed Integro-Differential Parabolic Problem with Integral Boundary Conditions

Abstract

Via Carleman estimates we prove uniqueness and continuous dependence results for an identification and strongly ill-posed linear integro-differential parabolic problem with the Dirichlet boundary condition, but with no initial condition. The additional information consists in a boundary linear integral condition involving the normal derivative of the temperature on the whole of the lateral boundary of the space-time domain.

Keywords:

• ill-posed problems
• linear parabolic integro-differential equations
• no initial condition
• integral boundary conditions
• uniqueness
• continuous dependence results

AMS Subject Classification:

• primary 35R30
• secondary 35K20
• 45K05
• 45Q05