Uniqueness and stability in inverse parabolic equations with memory

First we establish a Carleman estimate for parabolic equations with second order spatial memory. Then we prove the stability results for the coefficient q from a measurement of the solution with respect to the normal derivative on an arbitrary part of the boundary and certain spatial derivatives at t = θ . Further we deduce the uniqueness result under some equivalence conditions on the solutions about the potential q . The proof of the results rely on Carleman estimates and certain energy estimates for parabolic equations with memory.