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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
##### Keywords:
existence; uniqueness; nonlocal observation; monotone operators
Full Text:
##### References:
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