Inverse Source Problems for Kelvin–Voigt System With Final Overdetermination Condition

ABSTRACT This paper is devoted to studying inverse source problems for the linear Kelvin–Voigt equations with memory, which govern the flow of incompressible viscoelastic fluids. The memory term is represented by an integral with convolution and reflects the elastic properties of the fluid. The inverse problems consist of recovering the spatial distribution of external forces by measurements for the velocity and gradient of pressure at the final moment . These inverse problems are investigated in the following cases: with and without the memory term in the momentum equation, and under both sticking and sliding boundary conditions. In all cases, under suitable conditions on the data of the problem, the existence, uniqueness as well as the stability of strong solutions were established.