Sign-changing blowing-up solutions for the critical nonlinear heat equation

Let $\mathrm{\Omega }$ be a smooth bounded domain in ${\mathbb{R}}^n$ and denote the regular part of the Green’s function on $\mathrm{\Omega }$ with Dirichlet boundary condition as $H(x,y)$. Assume that $q\in \mathrm{\Omega }$ and $n\ge 5$. We prove that there exists an integer $k_0$ such that for any integer $k\ge k_0$ there exist initial data $u_0$ and smooth parameter functions $\xi (t)\to q$, $0<\mu(t)\to 0$as$t\to +\infty$suchthatthesolution$u_q$ 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 $u_q(x,t)$.