Asymptotic behaviour of similar profiles (Q1090189)

The equation studied is \[ u'''+\alpha \quad u\quad u''+\beta (1- u^{'2})+\gamma \quad u''=0. \] Here \(u=u(t)\), together with the boundary conditions \(u=u'=0\) at \(t=0\), u'\(\to 1\) as \(t\to \infty\). The value of \(\alpha\) is considered to be unity, whereas \(\gamma\) has been chosen arbitrarily. Along with the above boundary conditions, the side condition \(0<u'<1\) on (0,\(\infty)\) has also been taken into consideration. The present paper deals with asymptotic behaviour, as \(t\to \infty\), of the solutions of the above nonlinear, autonomous third order, differential equation. The results are based on the asymptotic integrations of the second order, linear differential equations.