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

## TL;DR

This paper establishes well-posedness for parabolic final value problems involving coercive Lax--Milgram operators, extending classical results to unbounded semigroups and providing a Duhamel formula within this framework.

## Contribution

It introduces a novel approach to well-posedness for coercive Lax--Milgram operators using an isomorphism between data and solution spaces, extending invertibility to unbounded semigroups.

## Key findings

- Proves well-posedness for a large class of final value problems.
- Extends invertibility of analytic semigroups to unbounded semigroups.
- Derives a Duhamel formula for the set-up.

## Abstract

This paper treats parabolic final value problems generated by coercive Lax--Milgram operators, and well-posedness is proved for this large class. The result is obtained by means of an isomorphism between Hilbert spaces containing the data and solutions. Like for elliptic generators, the data space is the graph normed domain of an unbounded operator that maps final states to the corresponding initial states, and the resulting compatibility condition extends to the coercive context. Lax--Milgram operators in vector distribution spaces is the main framework, but the crucial tool that analytic semigroups always are invertible in the class of closed operators is extended to unbounded semigroups, and this is shown to yield a Duhamel formula for the Cauchy problems in the set-up. The final value heat conduction problem with the homogeneous Neumann boundary condition on a smooth open set is also proved to be well posed in the sense of Hadamard.