Removable singularities of semilinear parabolic equations

Abstract

We prove that the parabolic equation $$f_t=\Delta f+F(x,f,\nabla f,t), $$ in $(\mathbb R^m\setminus\{0\})\times(0,T)$, $m\ge 3$, has removable singularities at $\{0\}\times (0,T)$ if $\|f\|_{L^{\infty}(\mathbb{R}^m\setminus\{0\}\times (0,T))} <\infty$ and $\|\nabla f\|_{L^{\infty}(\mathbb{R}^m\setminus\{0\}\times (0,T))} <\infty$. We also prove that the solution $u$ of the heat equation in $(\Omega\setminus\{0\})\times (0,T)$ has removable singularities at $\{0\}\times (0,T)$, $\Omega\subset\mathbb{R}^m$, $m\ge 3$, if and only if for any $0 < t_1 < t_2 < T$ and $\delta\in (0,1)$ there exists $\overline{B_{R_0}(0)}\subset\Omega$ depending on $t_1$, $t_2$ and $\delta$, such that $|u(x,t)|\le\delta |x|^{2-m}$ for any $0 <|x|\le R_0$ and $t_1\le t\le t_2$.