On one time-optimal control problem for a parabolic equation with involution in a bounded domain

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.