On the solutions of a boundary value problem arising in free convection with prescribed heat flux

For given , c < 0, we are concerned with the solution f _b of the differential equation f ′′′ + ff ′′ + g(f ′) = 0 satisfying the initial conditions f(0) = a, f ′ (0) = b, f ′′ (0) = c, where g is some nonnegative subquadratic locally Lipschitz function. It is proven that there exists b _* > 0 such that f _b exists on 0, + ∞) and is such that as t → + ∞, if and only if b ≥ b _*. This allows to answer questions about existence, uniqueness and boundedness of solutions to a boundary value problem arising in fluid mechanics, and especially in boundary layer theory.