Limiting behavior of solutions of \(u_ t=\Delta{} u^ m\;as\;m\rightarrow{} \infty\) (Q1176416)

The author deals with the following porous medium equation, \[ u_ t=\Delta(| u|^{m-1}u), (x,t)\in\mathbb{R}^ N\times[0,\infty), u(x,0)=f, x\in\mathbb{R}^ N, \] The author considers the behavior of the solution \(u(\cdot,t)\) when \(m\to\infty\) for any fixed \(t>0\). Under a certain condition on \(f\), he shows that \(u\) approaches some \(u_ \infty\) which satisfies \(u_ \infty-\Delta\varphi_ \infty(u_ \infty)\ni f\), where \(\varphi_ \infty(s)\) is the maximal monotone graph \(\varphi_ \infty(s)=0\) if \(| s|<1,=\pm[0,\infty)\) if \( s=\pm1,\) and \(=\emptyset\) if \( | s|>1\).