On parabolic final value problems and well-posedness

TL;DR

This paper establishes well-posedness for a broad class of parabolic final value problems using explicit Hilbert space characterizations, introducing a new compatibility condition for data and extending results to the heat equation with non-zero boundary data.

Contribution

It introduces a novel framework for final value problems of parabolic equations, defining explicit data spaces and compatibility conditions for existence and stability of solutions.

Findings

Well-posedness established for a large class of parabolic final value problems.

Explicit Hilbert space characterizations of data spaces are provided.

Extension of compatibility conditions to non-zero Dirichlet data for the heat equation.

Abstract

We prove that a large class of parabolic final value problems is well posed.This results via explicit Hilbert spaces that characterise the data yielding existence, uniqueness and stability of solutions. This data space is the graph normed domain of an unbounded operator, which represents a new compatibility condition pertinent for final value problems. The framework is evolution equations for Lax--Milgram operators in vector distribution spaces. The final value heat equation on a smooth open set is also covered, and for non-zero Dirichlet data a non-trivial extension of the compatibility condition is obtained by addition of an improper Bochner integral.