I'm reading an article in which the authors present a "well-known" mixed variational formulation of the heat equation. If $\Omega$ is a bounded domain in $\mathbb{R}^n$, $n=1,2,3$, and we consider $T>0$, the heat equation reads as follows:
Find a function $u$ such that: $$u_t - \triangle u = f \quad\text{in }\Omega\times(0,T),$$ $$u=0 \quad\text{in }\partial\Omega\times(0,T),$$ $$u(\cdot,0) = g \quad\text{in } \Omega.$$
By introducing the gradient of $u$ as a new variable (i.e. $\sigma=\nabla u$) and by setting $V=L^2(\Omega)$ and $H=H(\text{div},\Omega):=\{\boldsymbol{w}\in(L^2(\Omega))^n : \text{div}\boldsymbol{w}\in L^2(\Omega)\}$, we can rewrite the heat equation in the following mixed variational formulation:
Find $(u,\sigma):(0,T)\to V\times H$ such that $$(u_t,v) - (\text{div}\sigma,v) = (f,v) \qquad \forall v\in V,\; \forall t\in(0,T),$$ $$(\sigma,\chi) + (u,\text{div}\chi) = 0 \qquad \forall \chi\in H,\; \forall t\in(0,T),$$ $$u(\cdot,0) = g,$$
where $(\cdot,\cdot)$ denotes the usual $L^2(\Omega)$ inner product.
I understand how to deduce the mixed variational formulation, it's actually just applying a Green's formula and the fact that $u=0$ in $\partial\Omega\times(0,T)$, but now I'm having a hard time finding a reference in which it is showed the well-posedness of this formulation (that is, existence and uniqueness of solution).
I assume such reference exists since the article I'm reading states it is a "well-known" mixed formulation, but the article doesn't mention a reference for this subject, and I've been looking for a few days with no success and that's why I'm asking for this here. If anyone know a reference for this and can share it with me I'd really appreciate it.
Thanks in advance.