Under study are the inverse problems of finding, together with a solution u(x,t) of the differential equation cu_t − Δu + a(x, t)u = f(x, t) describing the process of heat distribution, some real c characterizing the heat capacity of the medium (under the assumption that the medium is homogeneous). Not only the initial condition is imposed on u(x, t), but also the usual conditions of the first or second initial-boundary value problems as well as some special overdetermination conditions. We prove the theorems of existence of a solution (u(x, t), c) such that u(x, t) has all Sobolev generalized derivatives entered into the equation, while c is a positive number.

中文翻译：

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热容量系数未知的传热方程

研究中的逆问题是，找到描述热分布过程的微分方程cu _t −Δu + a（x，t）u = f（x，t）的解u（x，t），一些真实的c表征了介质的热容量（假设介质是均匀的）。不仅对u（x，t）施加了初始条件，而且还对第一或第二个初始边界值问题施加了通常的条件以及某些特殊的超确定条件。我们证明了解的存在性定理（u（x，t），c）使得u（x，t）将所有Sobolev广义导数输入方程，而c为正数。