Kernel determination problem for a integro–differential heat equation with a variable thermal conductivity.

The inverse problem of finding a one-dimensional memory kernel of a time convolution integral depending on a time variable t and one-dimensional spatial variable x in the two-dimensional heat equation with a time-dependent coefficient of thermal conductivity is studied. The direct problem is the Cauchy problem for this equation. The integral term has the time convolution form of kernel and an elliptic operator of direct problem solution. As additional information, the solution of the direct problem on the x=0 is given. The problem reduces to an auxiliary problem which is more convenient for further consideration. Then the auxiliary problem is replaced by an equivalent system of Volterra-type integral equations with respect to unknown functions. Applying the method of contraction mappings to this system in the Hölder class of functions, it is proved the main result of the paper representing a local existence and uniqueness theorem. [ABSTRACT FROM AUTHOR]