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 $H(x,y)$. Assume that $q\in \Omega$ 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 $t\to +\infty$ 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 $u_q(x,t)$.

60 pages; comments welcome

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Cited by in corpus (1)

• New type II Finite time blow-up for the energy supercritical heat equation