Sign-changing blowing-up solutions for the critical nonlinear heat equation
Let $Ω$ be a smooth bounded domain in $\mathbb{R}^n$ and denote the regular part of the Green's function on $Ω$ with Dirichlet boundary condition as $H(x,y)$. Assume that $q \in Ω$ 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 $ξ(t)\to q$, $0<μ(t)\to 0$ as $t\to +\infty$ such that the solution $u_q$ of the critical nonlinear heat equation \begin{equation*} \begin{cases} u_t = Δu + |u|^{\frac{4}{n-2}}u\text{ in } Ω\times (0, \infty),\\ u = 0\text{ on } \partial Ω\times (0, \infty),\\ u(\cdot, 0) = u_0 \text{ in }Ω, \end{cases} \end{equation*} has the form \begin{equation*} u_q(x, t) \approx μ(t)^{-\frac{n-2}{2}}\left(Q_k\left(\frac{x-ξ(t)}{μ(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*} ΔQ + |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 $u_q(x, t)$.