Isomorphic Well-Posedness of the Final Value Problem for the Heat Equation with the Homogeneous Neumann Condition

Abstract

This paper concerns the final value problem for the heat equation subjected to the homogeneous Neumann condition on the boundary of a smooth open set in Euclidean space. The problem is here shown to be isomorphically well posed in the sense that there exists a linear homeomorphism between suitably chosen Hilbert spaces containing the solutions and the data, respectively. This improves a recent work of the author, in which the same problem was proven well-posed in the original sense of Hadamard under an additional assumption of Holder continuity of the source term. Like for its predecessor, the point of departure is an abstract analysis in spaces of vector distributions of final value problems generated by coercive Lax–Milgram operators, now yielding isomorphic well-posedness for such problems. Hereby the data space is the graph normed domain of an unbounded operator that maps final states to the corresponding initial states, resulting in a non-local compatibility condition on the data. As a novelty, a stronger version of the compatibility condition is introduced with the purpose of characterising the data that yield solutions having the regularity property of being square integrable in the generator’s graph norm (instead of in the form domain norm). This result allows a direct application to the class 2 boundary condition in the considered inverse Neumann heat problem.