Asymptotic behavior of the unbounded solutions to some degenerate boundary layer equations revisited (Q2458390)

The focus of this paper are solutions of the third order ordinary differential equation \[ \left( \left| f^{\prime\prime}\right|^{n-1} f^{\prime\prime} \right)^\prime + f f^{\prime\prime} - \beta {f^\prime}^2 = 0\tag{1} \] satisfying the asymptotic condition \[ \lim_{\eta \to \infty} \left| f(\eta) \right| = \infty.\tag{2} \] Here \(\eta \in (0, \infty)\), \(\beta \in {\mathbb R}\), and \(n \geq 1\). The equation (1) arises in the study of the boundary-layer flow of a non-Newtonian fluid (for \(n=1\) it is Newtonian). The authors show that for \(\beta \in [-1,0)\), a solution of (1) and (2) necessarily satisfies \[ f^\prime(\eta) \to 0, \quad f^{\prime\prime}(\eta) \to 0, \quad\text{ and }\quad f(\eta) = \eta^{1/(1-\beta)} (A + o(\eta)) \quad\text{ as } \eta \to \infty, \] for some \(A > 0\). It is also outlined how to prove that, for \(-1 \leq \beta < 0\), there are always solutions satisfying (2). The proofs are based on standard theory for ordinary differential equations.