Non-negative solutions to fast diffusions (Q921255)

The authors consider the initial value problem of following nonlinear evolution equation (which is, so to speak, a generalized porous media equation). They deal with the problem of uniqueness, existence and regularizing effect. Some of their results are generalizations and complement of the earlier work, on the porous media equation, i.e. \(\phi (u)=u^ m\) which are due to Brezis and Friedmann, Herrero and Pierre, and other authors including themselves. But their results are not merely simple generalizations from the technical viewpoit \[ \partial u/\partial t=\Delta \phi (u),\quad x\in {\mathbb{R}}^ n,\quad 0<t<T\leq \infty,\quad u(0,)=u(),\quad x\in {\mathbb{R}}^ n. \] Here \(\phi\) is a continuous, increasing function with \(\phi (0)=0\) on \([0,\infty)\). They deal with a fast diffusion type nonlinearity \(\phi\), which is given by the condition \[ a\leq u\phi '(u)/\phi (u)\leq 1/a,\quad (n-2)/n+a\leq u\phi '(u)/\phi (u)\leq 1-a,\quad u>u_ 0. \] They prove the existence and uniqueness of solution for initial value \(\mu\) which is a locally finite Borel measure and they also prove the \(L^ 1_{loc}-L^{\infty}_{loc}\) regularizing effect. In particular, they give a complete characterization of the set of the initial values \(\mu\) such that the equation has a non- negative continuous weak solution.