Fix $u_0\in H^1(\Omega)$ and $f=f(x,y,t)\in L^2(\Omega\times [0,T])$ where $\Omega$ is a sufficiently smooth bounded domain in $\mathbb{R}^2$. Consider the problem of finding $u:\Omega\times[0,T]\to\mathbb{R}$ satisfying the following variational equation $$ \begin{cases} \langle \nabla u, \nabla v\rangle_{L^2(\Omega)}+\langle u,v\rangle_{L^2(\Omega)}=\langle f,v\rangle_{L^2(\Omega)},\;\;\forall v\in H^1(\Omega),\;\forall t\in(0,T], \\\;u(\cdot,0)=u_0, \; \end{cases} $$ It is clear that this variational problem comes from the equation

$$\begin{cases} -\Delta u+u=f, & \text{ on }\Omega\times(0,T],\\ u(0)=u_0. \end{cases}$$

I did not put boundary conditions, but any boundary condition for me works as long as $\int_{\partial\Omega}(\vec{n}\cdot\nabla u)v=0$ for all $v$.

**Note:** *All of the differential operators seen above are spacial operators, they involve no derivatives in time.*

**Questions:**

- What are appropriate function spaces to look for solutions?
- Can we have that $u(\cdot,t)\in L^2([0,T])$ for all $t\in[0,T]$?
- Can we have that $\text{essup}_{t\in[0,T]}\|u(\cdot,t)\|_{L^2(\Omega)}\leq C$ for some constant $C\in\mathbb{R}$ independent of time and space?