Identification of two memory kernels and the time dependence of the heat source for a parabolic conserved phase-field model

We consider a system of two integro-differential evolution equations coming from a conservative phase-field model in which the principal part of the elliptic operators, involved in the two evolution equations, have different orders. The inverse problem consists in finding the evolution of the temperature, the phase-field,the two memory kernels and the time dependence of the heat source when we suppose to know additional measurements of the temperature on some part of the body. We prove that the inverse problem admits a local in time solution, but we are also able to prove a global in time uniqueness result.