Summary: We consider nonlinear diffusion equations of the form \(\partial_t u=\Delta\phi(u)\) in \(\mathbb{R}^N\) with \(N\geq 2\). When \(\phi(s)\equiv s\), this is just the heat equation. Let \(\Omega\) be a domain in \(\mathbb{R}^N\), where \(\partial\Omega\) is bounded and \(\partial\Omega= \partial(\mathbb{R}^N\setminus\overline\Omega)\). We consider the initial-boundary value problem, where the initial value equals zero and the boundary value equals \(1\), and the Cauchy problem where the initial data is the characteristic function of the set \(\Omega^c= \mathbb{R}^N\setminus\Omega\). We settle the boundary regularity issue for the characterization of the sphere as a stationary level surface of the solution \(u\), no regularity assumption is needed for \(\partial\Omega\).