The inverse problem of recovering the source in a parabolic equation under a condition of nonlocal observation

14 сентября 2018
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Контактные данные автора публикации Kostin, A.B.
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We study the inverse problem for a parabolic equation of recovering the source, that is, the right-hand side F(x, t) = h(x, t)f(x), where the function f(x) is unknown. To find f(x), along with the initial and boundary conditions, we also introduce an additional condition of nonlocal observation of the form Z T 0 u(x,t) d(t) = (x). We prove the Fredholm property for the problem stated in this way, and obtain sufficient conditions for the existence and uniqueness of a solution. These conditions are of the form of readily verifiable inequalities and put no restrictions on the value of T > 0 or the diameter of the domain O under consideration. The proof uses a priori estimates and the qualitative properties of solutions of initialboundary value problems for parabolic equations.
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