Asymptotic behaviour of a solution to a nonlinear equation modelling capillary rise (Q2115508)

In the paper under review, the authors prove that the exact solution of the equation \[ u'' (s) + \frac{1}{\sqrt{\omega}}\, u' (s) + \sqrt{2u(s)} = 1 \tag{1} \] with the initial conditions \[ u(0)=0, \; u'(0)=0 \] converges uniformly as \(\epsilon=\frac{1}{\sqrt{\omega}}\to 0^+\), where \(\omega\) is a single nondimensional parameter (see Theorem 1). In fact, this equation models the capillary rise process in a narrow vertical tube. Furthermore, the authors provide a new method for analysing the long-time behaviour of the solution of the above equation in the oscillatory regime (Section 3).