## 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 are 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.

Original language | English |
---|---|

Title of host publication | Advances in Microlocal and Time-Frequency Analysis |

Editors | Paolo Boggiatto, Marco Cappiello, Elena Cordero, Sandro Coriasco, Gianluca Garello, Alessandro Oliaro, Jörg Seiler |

Number of pages | 22 |

Publisher | Birkhäuser Verlag |

Publication date | 2020 |

Pages | 259-280 |

ISBN (Print) | 978-3-030-36140-2, 978-3-030-36137-2 |

ISBN (Electronic) | 978-3-030-36138-9 |

DOIs | |

Publication status | Published - 2020 |

Series | Applied and Numerical Harmonic Analysis |
---|---|

ISSN | 2296-5009 |

### Bibliographical note

Publisher Copyright:© Springer Nature Switzerland AG 2020.

## Keywords

- Compatibility condition
- Final value data
- Parabolic
- Well-posed