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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 $H=L^2$).

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  • $\begingroup$ Typically you will need to know something about the smoothness of the boundary of your domain. $\endgroup$ Feb 1, 2014 at 1:33

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take a look here. their setting is more abstract, but your specific equation should be just a special case of it.

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