On the short-time behaviour of the free boundary of a porous medium equation
Let \(\Omega\) be a bounded open set in \(\mathbb{R}^N\), \(N\geq 2\), with a smooth boundary \(\Sigma\) and \(m>1\). Consider the problem \[ u_t= \Delta u^m \text{ in } \Omega \times (0,+\infty), \quad u(x,t)=1 \text{ in } \Sigma\times (0,+\infty), \quad u(x,0)=0 \text{ in } \Omega. \] For every \(x_0\) in \(\Omega\) there exists \(T(x_0)\) such that \(u(x_0,t)=0\) for \(0<t< T(x_0)\) and \(u(x_0,t)>0\) for \(t>T(x_0)\). The authors give a precise estimate of \(T(x_0)\) in terms of the geometric properties of \(\Omega\).
scientific article; zbMATH DE number 1011426