Asymptotic behavior of the unbounded solutions of some boundary layer equations

We give an asymptotic equivalent at infinity of the unbounded solutions of some boundary layer equations arising in fluid mechanics. Let us consider the following boundary layer differential equation f ′′′ + ff ′′ − βf ′2 = 0 (1) where β < 0. We are interested in non constant solutions (that we will simply call solutions) of (1) defined on some interval [t0,∞) and such that f (∞) := lim t→∞ f (t) = 0. (2) Equation (1) can be obtained from similarity boundary layer equations as those introduced by numerous authors in [1], [2], [11], [12], [13], [14], [17] and [18], and studied from mathematical point of view in [3], [4], [6], [7], [9], [10] and [15]. In these papers, the corresponding differential equation is considered on [0,∞) with the boundary conditions f(0) = a, f (0) = 1 and (2), or f(0) = a, f (0) = −1 and (2). Here, we will be concerned by unbounded solutions of these problems, and to be as general as possible we will consider all the unbounded solutions of (1)-(2) defined on some interval [t0,∞). The restriction to β < 0 is due to the fact that for β ≥ 0 none of the solutions of (1)-(2) are unbounded (see Remark 6 below). For β = 0, equation (1) reduces to the Blasius equation and a lot of papers have been published about it. To have a survey, we refer to [16], [5], [8] and the references therein. Concerning the existence of unbounded solutions of (1)-(2), elementary direct methods give it for −2 ≤ β < 0 (see for example [7] and [15]). It seems more difficult to get such existence results for β < −2 and the best way to overcome this difficulty should consist in introducing appropriate blow-up coordinates. Precisely, if f is a solution of (1) which does not vanish on some interval I, we set