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- W2898664501 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 $ngeq 5$. We prove that there exists an integer $k_0$ such that for any integer $kgeq k_0$ there exist initial data $u_0$ and smooth parameter functions $xi(t)to q$, $0<mu(t)to 0$ as $tto +infty$ 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}}utext{ in } Omegatimes (0, infty), u = 0text{ on } partial Omegatimes (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_kleft(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 = 0text{ in }mathbb{R}^n, end{equation*} constructed in cite{delpinomussofrankpistoiajde2011}. In dimension 5 and 6, we also prove the stability of $u_q(x, t)$." @default.
- W2898664501 created "2018-11-09" @default.
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- W2898664501 date "2018-10-31" @default.
- W2898664501 modified "2023-10-17" @default.
- W2898664501 title "Sign-changing blowing-up solutions for the critical nonlinear heat equation" @default.
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- W2898664501 doi "https://doi.org/10.48550/arxiv.1811.00039" @default.
- W2898664501 hasPublicationYear "2018" @default.
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