Inverse Problems of Heat and Mass Transfer and Filtration Theory

Abstract

— This is a survey article. We describe results on well-posedness in the Sobolev spaces of inverse problems with pointwise overdetermination for mathematical models of convection-diffusion and filtration described by parabolic equations and systems. The unknowns are time-dependent functions occurring in the source function and the operator itself as coefficients. The overdetermination conditions are the values of a solution at some collection of interior points of the domain. Conditions are imposed that guarantee the local-in-time well-posedness of the problem. It is shown that in linear problems the solvability is global in time. Stability estimates for solutions hold in all cases. Some attention is paid to the case of point sources. We describe the known results and, in particular, those obtained by the authors. The results presented are in many cases sharp. They can serve as a base for creating software packages of recovering pollution sources which can be included into regional intelligent decision support systems.