Is there a variational characterization of the heat equation?

Question

Is there a variational characterization of the heat eqauation? Suppose $\Omega$ be a bounded domain in $\mathbb{R}^n$, we konw that we can treat the solution of the Laplacian equation \begin{equation} \left\{\begin{split} \Delta u=0\,\,\,in\,\,\Omega,\\ u=\varphi\,\,\,\,\,\, on \,\,\partial\Omega. \end{split} \right. \end{equation} as the critical point of the functional $I(w)=\int_{\Omega}|\nabla w|^2dx$ in $A=\{w\in C^2(\bar{\Omega})|w=\varphi\,\,on\,\,\partial\Omega \}$. Does anyone konw if there is a similar characterization of the solution of the heat equation ? \begin{equation} \left\{\begin{split} u_t=\Delta u\quad(x,t)\in\Omega\times(0,T)\\ u=\phi\quad\,\,\,(x,t)\in\partial\Omega\times(0,T]\\ u(x,0)=\varphi(x)\,\quad\,\,\qquad x\in\Omega\\ \end{split} \right. \end{equation} If so, what is the functional $I(w)$ for it? We may take $\Omega=\mathbb{R}^n$ and omit the bounday condition $u=\phi$ on $\Omega\times(0,T]$ or take $T=+\infty$ for simplicity.