Traveling wave solutions of the porous medium equation with degenerate interfaces (Q1937737)

Consider the partial differential equation \[ {\partial v\over\partial t}= (m- 1)v\,\Delta v+ |\nabla v|^2.\tag{\(*\)} \] The authors look for a solution of \((*)\) of the form \[ v(x,y,z,t)= \sqrt{x^2+ y^2+(z-ct)^2}\,F\Biggl(\tan^{-1}\Biggl({\sqrt{x^2 +y^2}\over z- ct}\Biggr)\Biggr),\tag{\(**\)} \] where \(c> 0\) is the speed of the corresponding traveling wave. Substituting \((**)\) into \((*)\) one get a differential equation for \(F\). The authors study the existence of a positive solution to appropriate boundary value problems.