Weak solvability Neumann problem for a global in time parabolic equation

The weak solvability of the Neumann problem for a global in time parabolic equation is proven. The globality means that there is a coefficient in the equation that depends on the integral of the solution over the entire time interval where the problem is being solved. The Galerkin method is used to prove the solvability. Besides, it is shown that the problem with the homogeneous Neumann conditions can have several solutions independent of the spatial variables.