# A class of well-posed parabolic final value problems

## TL;DR

This paper establishes well-posedness for a broad class of parabolic final value problems using Hilbert space frameworks, revealing new compatibility conditions and extending results to heat conduction problems with non-zero boundary data.

## Contribution

It introduces a novel Hilbert space approach to characterize data ensuring existence, uniqueness, and stability in parabolic final value problems, including heat conduction with non-zero boundary conditions.

## Key findings

- Well-posedness proved for a large class of parabolic final value problems.
- New compatibility condition involving an improper Bochner integral for non-zero Dirichlet data.
- Extension of results to final value heat conduction problems on smooth domains.

## Abstract

This paper focuses on parabolic final value problems, and well-posedness is proved for a large class of these. The clarification is obtained from Hilbert spaces that characterise data that give existence, uniqueness and stability of the solutions. The data space is the graph normed domain of an unbounded operator that maps final states to the corresponding initial states. It induces a new compatibility condition, depending crucially on the fact that analytic semigroups always are invertible in the class of closed operators. Lax--Milgram operators in vector distribution spaces constitute the main framework. The final value heat conduction problem on a smooth open set is also proved to be well posed, and non-zero Dirichlet data are shown to require an extended compatibility condition obtained by adding an improper Bochner integral.