On the structure of similarity solutions of a differential equation arising from boundary layer problem

We consider an ordinary differential equation f ‴ − f f ″ − β f ′ 2 = 0 with f ( 0 ) = a , f ′ ( 0 ) = 1 , f ′ ( ∞ ) : = lim t → ∞ f ′ ( t ) = 0 , where β is a real constant. The given problem may arise from the study of steady free convection flow over a vertical semi-infinite flat plate in a porous medium, or the study of a boundary layer flow over a vertical stretching wall. In this paper, the structure of solutions for the cases of β ⩾ − 2 is studied. Combining the results of [B. Brighi, T. Sari, Blowing-up coordinates for a similarity boundary layer equation, Discrete Contin. Dyn. Syst. 5 (2005) 929–948; J.-S. Guo, J.-C. Tsai, The structure of solution for a third order differential equation in boundary layer theory, Japan J. Indust. Appl. Math. 22 (2005) 311–351; J.-C. Tsai, Similarity solutions for boundary layer flows with prescribed surface temperature, Appl. Math. Lett. 21 (1) (2008) 67–73], we conclude that the given problem may possess at most two types solutions for β ∈ R . Moreover, multiple solutions are also verified for various pairs of ( a , β ) .