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In this paper we study a parabolic inverse problem of finding
$a(t)$ such that $u_t = a(t) u_{xx}$ subject to initial-boundary
value conditions and an over-specified condition. We use the
over-specified information to solve for the unknown function and
then transform these inverse problems into some nonclassical
equations in which a trace type of functional is involved. By
applying the maximum principle, we deduce some a priori estimates
for the solution of our nonclassical problem. Then the continuity
methods can be applied to establish the global existences of
solutions to these problems.
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