Inverse Problems of Finding the Absorption Parameter in the Diffusion Equation

The paper is devoted to the study of inverse problems of finding, together with a solution $u(x,t)$ of the diffusion equation
$$ u_t-\Delta u +[c(x,t)+aq_0(x,t)]u=f(x,t), $$
the parameter $a$ characterizing absorption ($c(x,t)$ and $q_0(x,t)$ are given functions). It is assumed that, on the function $u(x,t)$, nonpercolation conditions and some special overdetermination conditions of integral form are imposed. We prove existence theorems for solutions $(u(x,t),a)$ such that the function $u(x,t)$ has all Sobolev generalized derivatives appearing in the equation and $a$ is a nonnegative number.