Analysis of oscillations in a drainage equation (Q2902732)

The drainage equation NEWLINE\[NEWLINE \left(\dfrac{d^3\Phi}{d\tau^3}+1\right)\Phi^3=1 NEWLINE\]NEWLINE is considered. It arises in the model of thin-film free surface flow on a vertical wall that is moving downwards with constant velocity. There exist solutions that oscillate with increasing amplitude as \(\tau\to\infty\). First, the authors prove the existence of solutions. Then they analyze the local behavior near the critical point \(P_s\) (when \(\Phi=1\)), i.e., on the stable manifold. It is proved that either \(\Phi\to\infty\) as \(\tau\to -\infty\) or there exists a finite \(\tau^{*}\) such that \(\Phi\to 0\) as \(\tau\to (r^{*})^{+}\) with \(\Phi(\tau)>0\) for all \(\tau>\tau^{*}\). The solutions on the stable manifold are those solutions satisfying \(\Phi(\tau)\to 1\) as \(\tau\to\infty\). Then they study the behavior outside the stable manifold. The corresponding lemma deals with such solutions. It is shown that there exists two-dimensional positive solutions of the system, defined for all \(-\infty<\tau<\infty\), satisfying \(\lim_{\tau\to -\infty}\Phi(\tau)=1\), \(\lim\sup_{\tau\to\infty}\Phi(\tau)=\infty\), \(\lim\inf_{\tau\to \infty}\Phi(\tau)=0\). The authors prove that every solution such that \(\Phi(\tau)\) does not converge to 1 as \(\tau\to\infty\) oscillates with increasing amplitude as \(\tau\to\infty\) in the form described by the formal asymptotics. The asymptotics and auxiliary estimates are found. The case \(\Phi\sim 0\) is fully studied in the appendix.