Uniqueness of one inverse problem for heat equation with a variable thermal conductivity

We study two problems of determining the kernel of the integral terms in a parabolic integro-differential equation. In the first problem the kernel depends on time t and x=(x1, …, xn) spatial variables in the multidimensional integro-differential equation of heat conduction. In the second problem the kernel is determined from one dimensional integro-differential heat equation with a time-variable coefficient of thermal conductivity. In both cases it is supposed that the initial condition for this equation depends on a parameter y=(y1, …, yn) and the additional condition is given with respect to a solution of direct problem on the hyperplanes x=y. It is shown that if the unknown kernel has the form k(x,t)=Σi=0Nai(x)bi(t)⁠, then it can be uniquely determined.

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