Inverse Problems of Finding the Absorption Parameter in the Diffusion Equation

Abstract

The paper is devoted to the study of inverse problems of finding, together with a solution u(x, t) of the diffusion equation

\({u_t} - \Delta u + [c(x,t) + a{q_0}(x,t)]u = f(x,t),\)

the parameter a characterizing absorption (c(x,t) and q₀(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.