Problem of Determining the Thermal Memory of a ConductingMedium

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.