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

TL;DR

This paper constructs sign-changing blow-up solutions for the critical nonlinear heat equation in bounded domains, demonstrating their asymptotic form and stability in certain dimensions, extending understanding of complex solution behaviors.

Contribution

It introduces a method to produce sign-changing blow-up solutions with precise asymptotics for the critical heat equation, including stability results in dimensions 5 and 6.

Findings

Existence of sign-changing blow-up solutions with prescribed asymptotics

Construction of solutions using non-radial solutions of Yamabe equation

Stability of solutions in dimensions 5 and 6

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 $H(x,y)$. 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 $u_q(x, t)$.