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Inverse problem with nonlocal observation of finding the coefficient multiplying \(u_t\) in the parabolic equation. (English. Russian original) Zbl 1365.35226
Differ. Equ. 52, No. 2, 220-239 (2016); translation from Differ. Uravn. 52, No. 2, 220-238 (2016).
Summary: We study the inverse problem of the reconstruction of the coefficient \(\varrho (x,t)= \varrho^{0}(x,t)+r(x)\) multiplying \(u_t\) in a nonstationary parabolic equation. Here \(\varrho^{0}(x,t) \geq \varrho_{0}>0\) is a given function, and \(r(x) \geq 0\) is an unknown function of the class \(L_\infty(\Omega)\). In addition to the initial and boundary conditions (the data of the direct problem), we pose the problem of nonlocal observation in the form \(\int_{0}^{T}u(x,t)d \mu(t)= \chi (x)\) with a known measure \(d \mu(t)\) and a function \(\chi (x)\). We separately consider the case \(d \mu (t)= \omega (t)dt\) of integral observation with a smooth function \(\omega (t)\). 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.
MSC:
35R30 Inverse problems for PDEs
35K10 Second-order parabolic equations
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