On a Similarity Boundary Layer Equation

The purpose of this paper is to study the autonomous third order nonlinear differential equation f ′′′+ m+1 2 ff ′′−mf ′2 = 0 on (0,∞), subject to the boundary conditions f(0) = a ∈ R, f ′(0) = 1 and f ′(t) → 0 as t → ∞. This problem arises when looking for similarity solutions to problems of boundary-layer theory in some contexts of fluids mechanics, as free convection in porous medium or flow adjacent to a stretching wall. Our goal here is to investigate by a direct approach this boundary value problem as completely as possible, say studying existence or non-existence and uniqueness or non-uniqueness of solutions according to the values of the real parameter m. In particular, we will emphasize similarities and differences between the cases a = 0 and a 6= 0 in the boundary condition f(0) = a.