On similarity solutions for boundary layer flows with prescribed heat flux

This paper is concerned with existence, uniqueness and behavior of the solutions of the autonomous third order nonlinear differential equation $f^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}+(m+2)f{f}^{\mathrm{\prime }\mathrm{\prime }}-(2m+1)f^{\mathrm{\prime }2}=0$ on ${\mathbb{R}}^{+}$ with the boundary conditions $f(0)=-\gamma$, $f^{\mathrm{\prime }}(\mathrm{\infty })=0$ and $f^{\mathrm{\prime }\mathrm{\prime }}(0)=-1$. This problem arises when looking for similarity solutions for boundary layer flows with prescribed heat flux. To study solutions we use some direct approach as well as blowing-up coordinates to obtain a plane dynamical system.