A very singular solution for the slow diffusion equation with nonlinear convection

Abstract

We here investigate an existence and uniqueness of the nontrivial, nonnegative solution of a nonlinear ordinary differential equation: ( f m ) ″ + β r f ′ + α f + σ ( f q ) ′ = 0 satisfying a specific decay rate: lim r → ∞ r α / β f ( r ) = 0 with α = 1 / ( 2 q − m − 1 ) and β = ( q − m ) / ( 2 q − m − 1 ) . Here m > 1 , m < q < m + 1 and σ = 1 or −1. Such a solution arises naturally when we study a very singular solution for a slow diffusion equation with nonlinear convection: u t = ( u m ) x x + ( u q ) x defined either on the whole real line or on the half line ( − ∞ , 0 ] .