Inverse problem with nonlocal observation of finding the coefficient multiplying ut in the parabolic equation

Resumo

We study the inverse problem of the reconstruction of the coefficient ϱ(x, t) = ϱ⁰(x, t) + r(x) multiplying u_t in a nonstationary parabolic equation. Here ϱ⁰(x, t) ≥ ϱ⁰ > 0 is a given function, and r(x) ≥ 0 is an unknown function of the class L_∞(Ω). In addition to the initial and boundary conditions (the data of the direct problem), we pose the problem of nonlocal observation in the form ∫₀^Tu(x, t) dμ(t) = χ(x) with a known measure dμ(t) and a function χ(x). We separately consider the case dμ(t) = ω(t)dt of integral observation with a smooth function ω(t). We obtain sufficient conditions for the existence and uniqueness of the solution of the inverse problem, which have the form of ready-to-verify inequalities. We suggest an iterative procedure for finding the solution and prove its convergence. Examples of particular inverse problems for which the assumptions of our theorems hold are presented.