Heat equation with pure Neumann boundary conditions

[Ruslan]

I’m not an expert, but I would say yes.
See for instance:

Steady, nonlinear heat equation with pure neumann BCs

I believe there are two mistakes in the formulation of the weak problem. First, there is a sign error in the boundary term. If we derive a weak problem via integration by parts, we get the following: \int_\Omega -\nabla\cdot(k\nabla u)v\,d\mathbf{x} = \int_\Omega fv\,d\mathbf{x} \Rightarrow \int_\Omega k\nabla u\cdot\nabla v\,d\mathbf{x} ~-~\int_{\partial\Omega}(k\nabla u\cdot\mathbf{n})v\,d\mathbf{s}~=~\int_\Omega f v\,d\mathbf{x} \Rightarrow \int_\Omega k\nabla u\cdot\nabla v\,d\mathb…