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Global uniqueness for inverse parabolic problems with final observation. (English) Zbl 0840.35120

A rather general uniqueness theorem is given for the inverse source problem with final overdetermination according to the evolution equation \[ {\partial u\over \partial t}+ Au= F(x, t)\quad\text{on} \quad Q\tag{1} \] provided that some growth assumptions on the right-hand side are fulfilled. The proof and the discussion of assumptions to be stated give interesting insights into the problem structure. Really, in the second part of this note it can be shown that this global theorem on the uniqueness of \(F\) in (1) implies a series of uniqueness theorems for the parabolic equations. So, the parameter functions \(c(x)\) in \[ u_t- \Delta u+ cu= 0\quad\text{on}\quad Q \] and \(d(x)\) in \[ du_t- \Delta u= 0\quad\text{on}\quad Q \] can be determined in a unique manner for appropriate initial and boundary data.

MSC:

35R30 Inverse problems for PDEs
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
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