In this paper, the solvability of some inverse problems for a nonlocal analogue of a fourth-order parabolic equation was studied. For this purpose, a nonlocal analogue of the biharmonic operator was introduced. When defining this operator, transformations of the involution type were used. In a parallelepiped, the eigenfunctions and eigenvalues of the Dirichlet type problem for a nonlocal biharmonic operator were studied. The eigenfunctions and eigenvalues for this problem were constructed explicitly and the completeness of the system of eigenfunctions was proved. Two types of inverse problems on finding a solution to the equation and its righthand side were studied. In the two problems, both of the righthand terms depending on the spatial variable and the temporal variable were obtained by using the Fourier variable separation method or reducing it to an integral equation. The theorems for the existence and uniqueness of the solution were proved.

In this paper, we study the problem of time-optimal control for the parabolic equation with involution in a multidimensional parallelepiped domain. A generalized solution to the initial boundary value problem is found, and the control problem is reduced to a first-order Volterra integral equation. To prove the existence and uniqueness of the solution to this integral equation, necessary estimates are obtained for its kernel. The existence of a solution to the integral equation, i.e., the admissibility of the control function, is proven, and an optimal estimate of the minimum time required to heat the domain to a certain average temperature is found.