A singular limit problem for the porous medium equation (Q1263734)

This paper deals with limit problems of the following form: The function \(u_ m=u_ m(x,t)\in C([0,\infty);L^ 1({\mathbb{R}}^ N))\) denotes the unique semigroup solution of \[ u_ t=\Delta (| u|^{m- 1}u),\quad x\in {\mathbb{R}}^ N,\quad t>0;\quad u(x,0)=f(x),\quad x\in {\mathbb{R}}^ N,\quad f\in L^ 1({\mathbb{R}}^ N). \] Does the limit \(\lim_{m\to \infty}u_ m(x,t)\) exist in some sense and what is the limit? \textit{P. Benilan} and \textit{M. G. Crandall} [Indiana Univ. Math. J. 30, 161-177 (1981; Zbl 0482.35012)] and \textit{L. A. Caffarelli} and \textit{A. Friedman} [ibid. 36, 711-728 (1987; Zbl 0651.35039)] already obtained some convergence results. The author gives some generalizations of the above problem and the corresponding Dirichlet and Neumann problem.