The existence of similarity solutions to a quasilinear parabolic equation
The author studies the existence of non-constant positive solutions of the differential equation \[ \triangle w+ w^p- (0, 5y\nabla w+ \lambda w) w^{- q}= 0,\quad y\in \mathbb{R}^n,\quad 0< q< 1,\quad p> 1. \] In the radially symmetric case, this problem is reduced to the investigation of the existence of positive non-constant solutions of the initial value problem \[ w''+ {n- 1\over r} w'+ w^p- \Biggl({r\over 2} w'+ \lambda w\Biggr) w^{- q}= 0,\quad w'(0)= 0,\quad w(0)= \eta> 0.\tag{\(*\)} \] New criteria for the existence (non-existence) of such solutions to the equation \((*)\) are proved.
self-similar solution of the porous media equation
scientific article; zbMATH DE number 856940