  Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^n$ and denote the
regular part of the Green's function on $\Omega$ with Dirichlet boundary
condition as $H(x,y)$. Assume that $q \in \Omega$ and $n\geq 5$. We prove that
there exists an integer $k_0$ such that for any integer $k\geq k_0$ there exist
initial data $u_0$ and smooth parameter functions $\xi(t)\to q$, $0<\mu(t)\to
0$ as $t\to +\infty$ such that the solution $u_q$ of the critical nonlinear
heat equation \begin{equation*} \begin{cases} u_t = \Delta u +
|u|^{\frac{4}{n-2}}u\text{ in } \Omega\times (0, \infty),\\ u = 0\text{ on }
\partial \Omega\times (0, \infty),\\ u(\cdot, 0) = u_0 \text{ in }\Omega,
\end{cases} \end{equation*} has the form \begin{equation*} u_q(x, t) \approx
\mu(t)^{-\frac{n-2}{2}}\left(Q_k\left(\frac{x-\xi(t)}{\mu(t)}\right) - H(x,
q)\right), \end{equation*} where the profile $Q_k$ is the non-radial
sign-changing solution of the Yamabe equation \begin{equation*} \Delta Q +
|Q|^{\frac{4}{n-2}}Q = 0\text{ in }\mathbb{R}^n, \end{equation*} constructed in
\cite{delpinomussofrankpistoiajde2011}. In dimension 5 and 6, we also prove the
stability of $u_q(x, t)$.
