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

Monica Musso, Manuel Del Pino, Juncheng Wei, Youquan Zheng

Research output: Contribution to journalArticle

Abstract

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 $H(x,y)$. Assuming the integer $k_0$ is sufficiently large, $q \in \Omega$ and $n\geq 5$. For $k\geq k_0$, we prove that there exist initial data $u_0$ and smooth parameter functions $\xi(t)\to q$, $0<\mu(t)\to 0$ ($t\to +\infty$) 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 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 investigate the stability of $u_q(x, t)$.
Original language English Annali della Scuola Normale Superiore di Pisa Accepted/In press - 21 Mar 2019