In the recent paper [Colombo, Physica D 236: 81–89, 2007] the author investigates an inverse problem arising in the theory of combustion. The problem studied is: determine the temperature u and the convolution memory kernel k in the evolution equation
given suitable initial-boundary conditions and the following additional restriction on u:
φ(x)u(t, x) dx = g(t),
where φ and g are given functions. The main results are a local in time existence theorem and a global in time uniqueness result. In this paper we complete the study considering the linearized version of the model. We prove that, if F(u(t, x), ∇u(t, x)) is sublinear, then the inverse problem has a unique global in time solution.