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This paper treats parabolic final value problems generated by coerciveLax--Milgram operators, and well-posedness is proved for this large class. Theresult is obtained by means of an isomorphism between Hilbert spaces containingthe data and solutions. Like for elliptic generators, the data space is thegraph normed domain of an unbounded operator that maps final states to thecorresponding initial states, and the resulting compatibility condition extendsto the coercive context. Lax--Milgram operators in vector distribution spacesis the main framework, but the crucial tool that analytic semigroups always areinvertible in the class of closed operators is extended to unboundedsemigroups, and this is shown to yield a Duhamel formula for the Cauchyproblems in the set-up. The final value heat conduction problem with thehomogeneous Neumann boundary condition on a smooth open set is also proved tobe well posed in the sense of Hadamard.

Well-Posed Final Value Problems and Duhamel's Formula for Coercive Lax--Milgram Operators