# 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

### Cite this

Sign-changing blowing-up solutions for the critical nonlinear heat equation. / Musso, Monica; Del Pino, Manuel; Wei, Juncheng; Zheng, Youquan.

In: Annali della Scuola Normale Superiore di Pisa, 21.03.2019.

Research output: Contribution to journalArticle

@article{b192a455e9574cd494f62d920915c77e,
title = "Sign-changing blowing-up solutions for the critical nonlinear heat equation",
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 $u_q(x, t)$.",
author = "Monica Musso and {Del Pino}, Manuel and Juncheng Wei and Youquan Zheng",
year = "2019",
month = "3",
day = "21",
language = "English",
journal = "Annali della Scuola Normale Superiore di Pisa",
issn = "0391-173X",
publisher = "Scuola Normale Superiore",

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TY - JOUR

T1 - Sign-changing blowing-up solutions for the critical nonlinear heat equation

AU - Musso, Monica

AU - Del Pino, Manuel

AU - Wei, Juncheng

AU - Zheng, Youquan

PY - 2019/3/21

Y1 - 2019/3/21

N2 - 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 $u_q(x, t)$.

AB - 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 $u_q(x, t)$.

M3 - Article

JO - Annali della Scuola Normale Superiore di Pisa

JF - Annali della Scuola Normale Superiore di Pisa

SN - 0391-173X

ER -