Abstract In the Cartesian product $$\mathbb {R}^n\times (0,+\infty ) $$ , we consider an integro-differential heat equation with an integral term of the convolution type on the right-hand side. The direct problem is the Cauchy problem about determining the temperature of a medium given a known initial heat distribution (for the zero value of the time variable $$t $$ ). The inverse problem consists in determining the kernel of the integral term based on the solution of the direct problem known at the point $$x=0\in \mathbb {R}^n$$ for $$t>0 $$ . Using the resolvent of the kernel, we reduce the inverse problem to another inverse problem more convenient for the analysis. The latter is replaced by an equivalent system of integral equations for the unknown functions,and the unique solvability of this system is proved with the use of the contraction mapping principle.

中文翻译：

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确定导电介质的热记忆的问题

摘要 在笛卡尔积 $$\mathbb {R}^n\times (0,+\infty ) $$ 中，我们考虑了一个积分微分热方程，其右侧有一个卷积类型的积分项。直接问题是关于确定已知初始热分布（对于时间变量 $$t $$ 的零值）的介质温度的柯西问题。逆问题包括基于点 $$x=0\in \mathbb {R}^n$$ for $$t>0 $$ 处已知的直接问题的解来确定积分项的核。利用核的解算器，我们将逆问题简化为另一个更便于分析的逆问题。后者由未知函数的积分方程的等效系统代替，