Summary: We reduce the Cauchy problem for a heat equation with nonlinear right-hand side which depends on some functionals acting on both, the unknown function and its gradient, to an equivalent system of integral equations. Considering mainly Banach spaces of continuous, bounded and/or exponentially bounded functions, we give some natural sufficient conditions for the existence and uniqueness of solutions to these equations. We prove that the solution of this system of integral equations consists of a weak solution to the differential-functional equation and its spatial derivatives.