Inverse Problem for a Multidimensional Integro-Differential Parabolic Equation with Nonlocal Initial and Boundary Conditions

We study the inverse problem of determination of the kernel of the convolution-type integral term in multidimensional second-order integro-differential parabolic with common uniformly elliptic operator depending on the spatial variables. The data of the problem include the nonlocal initial condition, the Dirichlet boundary condition, and the additional integral condition. First, the direct initial boundary-value problem is studied by using the spectrum of the corresponding elliptic operator, and the Fourier coefficients of the solution of the direct problem are found from the mixed Volterra–Fredholm type integral equation. Integral inequalities are used to verify the solvability of the direct problem. Next, using the additional condition on the unknown function of the inverse problem, we derive a nonlinear integral equation of the second kind equivalent to the inverse problem. This equation is studied via the Banach contraction principle. As a result, the inverse problem is shown to be locally solvable, which is our main result.