Final value problems for parabolic differential equations and their well-posedness

TL;DR

This paper investigates the well-posedness of parabolic final value problems, establishing explicit conditions and spaces for existence, uniqueness, and stability, with a detailed example involving the heat equation.

Contribution

It introduces a new framework using Hilbert spaces and compatibility conditions to analyze the well-posedness of parabolic final value problems.

Findings

Large class of problems proved well-posed

Explicit data space characterized by graph norm

Heat equation example with extended compatibility condition

Abstract

This article concerns the basic understanding of parabolic final value problems, and a large class of such problems is proved to be well posed. The clarification is obtained via explicit Hilbert spaces that characterise the possible data, giving existence, uniqueness and stability of the corresponding solutions. The data space is given as the graph normed domain of an unbounded operator occurring naturally in the theory. It induces a new compatibility condition, which relies on the fact, shown here, that analytic semigroups always are invertible in the class of closed operators. The general set-up is evolution equations for Lax--Milgram operators in spaces of vector distributions. As a main example, the final value problem of the heat equation on a smooth open set is treated, and non-zero Dirichlet data are shown to require a non-trivial extension of the compatibility condition by addition of an improper Bochner integral.