Let $\zeta, u\_0\in L^2(\Omega)$, with $\zeta \geq 0$ and $\Omega\subset \Bbb R^d$ open and bounded.  

\begin{equation}\label{Star-3.7}
\begin{cases}
\partial\_t u -\Delta u + \zeta u=0 &\mbox{ in }\; \Omega\times (0, T),\\
u = 0 &\mbox{ in }\; \partial\Omega\times (0, T), \\
u(\cdot,0) = u\_{0}, &\mbox{ in }\; \Omega,
\end{cases}
\end{equation}
We say that $u: \Omega\times (0, T)\to \Bbb R$ is a weak solution if: $u \in L^2(0,T; H\_0^1(\Omega) \cap L^2(\Omega, \zeta dx))$ and we have
$$\int\_\Omega\partial\_t uv +\int\_\Omega\nabla u\nabla v+ \int\_\Omega\zeta uv \qquad\text{ for all $\quad v \in L^2(0,T; H\_0^1(\Omega) \cap L^2(\Omega, \zeta dx))$}. $$


Question: Is it possible to show the existence and uniqueness of a weak solution?


If Yes which method could be suitable here or what are the right references for such equations?



Looking for a reference or the procedure on how to solve the parabolic equation with $L^2$-weight
Here is a functional analytic approach (Kato's book on perturbation theory is a good reference):


Let
$$
a\colon D(a)\times D(a)\to \mathbb{R},\,(u,v)=\int \nabla u\cdot \nabla v+\int \zeta uv.
$$
with $D(a)=H^1\_0(\Omega)\cap L^2(\zeta\,dx)$. It is not hard to see that $a$ is closed, that is, $D(a)$ endowed with the inner product $\langle\cdot,\cdot\rangle\_2+a$ is complete. Therefore there exists a positive self-adjoint operator $A$ on $L^2(\Omega)$ with $D(A)\subset D(a)$ such that $a(u,v)=\langle Au,v\rangle$ for $u\in D(A)$, $v\in D(a)$.


Let $u(t,\cdot)=e^{-tA}u\_0$. By functional calculus, $u\in C^\infty([0,\infty);H^1\_0(\Omega)\cap L^2(\zeta\,dx))$ and
$$
\partial\_t u=-Au.
$$
In particular, $u$ is a weak solution of your PDE by the definition of $A$.


To prove uniqueness, you can apply the usual energy method, i.e., show that $a(u,u)$ is decreasing along solutions.


