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 qq 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=θt=θ. Further we deduce the uniqueness result under some equivalence conditions on the solutions about the potential qq. The proof of the results rely on Carleman estimates and certain energy estimates for parabolic equations with memory.