The heat transfer equation with an unknown heat capacity coefficient

Under study are the inverse problems of finding, together with a solution $u(x,t)$ of the differential equation $cu_t -\Delta u + a(x,t)u = f(x,t)$ describing the process of heat distribution, some real $c$ characterizing the heat capacity of the medium (under the assumption that the medium is homogeneous). Not only the initial condition is imposed on $u(x,t)$, but also the usual conditions of the first or second initial-boundary value problems as well as some special overdetermination conditions. We prove the theorems of existence of a solution $(u(x,t),c)$ such that $u(x,t)$ has all Sobolev generalized derivatives entered into the equation, while $c$ is a positive number.