On a domain $\Omega$ with $f \in L^2(0,T;H^{-1})$ such that $f(0) = f(T)$, consider
$$u\_t - \Delta u = f\quad\text{on $\Omega$}$$
$$\frac{\partial u}{\partial \nu} = 0\quad\text{on $\partial\Omega$}$$
$$u(0) = u(T)$$


Is there any well-posedness result in Sobolev/Bochner spaces for this problem? 


I am wondering whether the associated bilinear form $b(u,v)$ (in this case $b(u,v) = \int \nabla u \nabla v$) has to be coercive in the stronger sense $b(v,v) \geq C|v|\_{V}$ rather than just $b(v,v) + A|v|\_H \geq C|v|\_V$ (here $V=H^1$ and $H=L^2$).



well-posedness of heat equation with Neumann BC and periodic data
take a look [here](http://netlizama.usach.cl/Keyantuo-Lizama%28JEE-Final%29%282006%29.pdf). their setting is more abstract, but your specific equation should be just a special case of it.


