Anomalous large-time behaviour of the \(p\)-Laplacian flow in an exterior domain in low dimension (Q845293)

The authors study the asymptotic behaviour of the nonnegative solutions of the exterior Dirichlet problem \[ \begin{aligned} {\partial u\over\partial t}=\nabla\cdot (|\nabla n|^{p-2}\nabla u)\quad &\text{in }\Omega\times (0,\infty),\\ u= 0\quad &\text{on } \partial\Omega\times(0, \infty),\\ u(x,0)= u_0(x)\quad &\text{in }\Omega,\end{aligned} \] where \(\Omega= \mathbb{R}^N\setminus\overline G\) and the hole \(G\) is a bounded domain. It is assumed that \(p> N\), where \(N\) is the number of independent space variables. Detailed asymptotic profiles are described. Also the moving boundary is studied.