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.

An inverse problem and a time-like Carleman estimate for parabolic integro-differential equations

“…In relation to the Carleman estimates for the parabolic integro-differential equations, we refer to works by A. Lorenzi et al [25,26,27,28,29,30]. In their article, they treat the case of that the integrands of the Volterra type integral in the equations are the first or zero-th order in space, that is, the case of a ij = 0 for i, j = 1, .…”