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?