Parabolic equation with unknown absorption coefficient

A problem of finding of solution and absorption coefficient in parabolic equation is studied in the case when the coefficient has a form
$$q(x,t)=\sum\limits_{k=1}^mq_k(x)h_k(x,t)+h_0(x,t)$$
with known functions $h_k(x,t)$ and with unknown $q_k(x)$. Theorems of existence, uniqueness and stability of solutions are proved if natural boundary conditions, some overdetermination conditions, assumptions of belonging of input data to certain functional spaces are valid and input data satisfy some conditions of inequality type.