Existence and nonexistence of global solutions to fast diffusions with source
The authors consider positive solutions of the Cauchy problem \[ \partial_t u= \Delta u^m+ u^p,\quad (x, t)\in \mathbb{R}^N\times (0, T),\tag{1} \] \[ u(x, 0)= u_0(x),\quad x\in \mathbb{R}^N,\tag{2} \] where \(\max\{0,1- 2/N\}< m< 1< p\) and \(u_0(x)\geq 0\). The main result of the authors reflects the fact that if \(p\leq m+ 2/N\) then all positive solutions of (1), (2) blow up in a finite time, and if \(p> m+ 2/N\) for sufficiently small \(u_0(x)\), the Cauchy problem (1), (2) has a global solution. The proofs are based on the comparison principle for parabolic equations, construction of supersolutions and some results of A. Friedman and S. Kamin.
scientific article; zbMATH DE number 798946