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

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.

39 pages. Revised version, with minor improvements. Essentially identical to the accepted version, which appeared in Axioms on 9 May 2018

References in corpus (4)

• On the trace problem for Triebel--Lizorkin spaces with mixed norms
• Anisotropic Lizorkin--Triebel Spaces with Mixed Norms --- Traces on Smooth Boundaries
• On the spectral properties of Witten-Laplacians, their range projections and Brascamp-Lieb's inequality
• On parabolic final value problems and well-posedness

Cited by in corpus (5)

• Time analyticity for the heat equation and Navier-Stokes equations
• Well-Posed Final Value Problems and Duhamel's Formula for Coercive Lax--Milgram Operators
• On parabolic final value problems and well-posedness
• A class of well-posed parabolic final value problems
• A formula for backward and control problems of the heat equation