Based on the solution of the first initial-boundary value problem for an inhomogeneous two-dimensional heat equation, we state and study inverse problems, whose right-hand sides contain unknown factors depending on spatial and time variables. Preliminarily, we explicitly construct a solution to the direct initial-boundary-value problem. We prove the uniqueness of the solution to direct and inverse problems, making use of the completeness property of the system of eigenfunctions of the corresponding homogeneous Dirichlet problem for the Laplace operator. We also establish existence theorems for solutions of inverse problems and construct solutions explicitly.

中文翻译：

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右侧未知的二维热方程的反问题

基于非齐次二维热方程的第一个初边值问题的解，我们陈述和研究反问题，反问题的右手边包含取决于空间和时间变量的未知因素。初步，我们明确构造了直接初始边界值问题的解决方案。利用Laplace算子对应的齐次Dirichlet问题的本征函数系统的完备性，我们证明了正解和反问题解的唯一性。我们还建立了反问题解的存在性定理，并明确构造了解。