Why does nobody consider the equation $-\Delta u = f$ in the space $L^2(0,T;H^{-1}(\Omega))$?
Eg. given $f \in L^2(0,T;L^2(\Omega))$ find a solution $u \in L^2(0,T;H^1_0(\Omega))$ such that $$\int_0^T \int_\Omega \nabla u(t) \nabla v(t) = \int_0^T\int_\Omega f(t)v(t)$$ for all $v \in L^2(0,T;H^1_0(\Omega))$. This solution exists by Poincare's inequality which applies on a.e. time: $$\int_\Omega |\nabla v(t)|^2 \geq C\int_\Omega |v(t)|^2\quad\text{for a.e. $t$}$$ and then we apply Lax-Milligram.
Of course I know that Poison's equation is an elliptic equation, and Bochner space is for parabolic equations. But is there anything wrong with what I wrote?