This paper concerns the final value problem for the heat equation under 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 problem was proven well-posed in the original sense of Hadamard under an additional assumption of Holder continuity of the source term. The point of departure is an abstract analysis in spaces of vector distributions of final value problems generated by coercive Lax--Milgram operators, 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 its form domain norm). This result allows a direct application to the considered inverse Neumann heat problem.

中文翻译：

---

齐次Neumann条件热方程终值问题的同构适定性

本文研究欧氏空间光滑开集边界上齐次诺依曼条件下热方程的终值问题。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. 这改进了作者最近的工作，其中在源项的 Holder 连续性的附加假设下，该问题在 Hadamard 的原始意义上被证明是适定的。出发点是在由强制 Lax--Milgram 算子生成的终值问题的向量分布空间中进行抽象分析，从而为此类问题产生同构适定性。因此，数据空间是无界算子的图赋范域，它将最终状态映射到相应的初始状态，从而导致数据的非局部兼容性条件。作为一种新颖性，引入了更强版本的兼容性条件，目的是表征产生具有在生成器图范数（而不是其形式域范数）中平方可积的正则性的解的数据。该结果允许直接应用于所考虑的逆 Neumann 热问题。引入了更强版本的兼容性条件，目的是表征产生具有在生成器图范数（而不是其形式域范数）中平方可积的正则性的解的数据。该结果允许直接应用于所考虑的逆 Neumann 热问题。引入了更强版本的兼容性条件，目的是表征产生具有在生成器图范数（而不是其形式域范数）中平方可积的正则性的解的数据。该结果允许直接应用于所考虑的逆 Neumann 热问题。