Steady states of thin-film equations with van der Waals force with mass constraint

• Published in: European journal of applied mathematics Vol. 34; no. 2; pp. 280 - 302
• Main Authors: CHEN, XINFU, JIANG, HUIQIANG, LIU, GUOQING
• Format: Journal Article
• Language: English
• Published: Cambridge Cambridge University Press 01.04.2023
• Subjects: Applied mathematics; Boundary conditions; Nonlinear differential equations; Ordinary differential equations; Steady state; Thin films; Van der Waals forces
• ISSN: 0956-7925; 1469-4425
• DOI: 10.1017/S0956792522000134

We consider steady states with mass constraint of the fourth-order thin-film equation with van der Waals force in a bounded domain which leads to a singular elliptic equation for the thickness with an unknown pressure term. By studying second-order nonlinear ordinary differential equation, \begin{equation*}h_{rr}+\frac{1}{r}h_{r}=\frac{1}{\alpha}h^{-\alpha}-p\end{equation*} we prove the existence of infinitely many radially symmetric solutions. Also, we perform rigorous asymptotic analysis to identify the blow-up limit when the steady state is close to a constant solution and the blow-down limit when the maximum of the steady state goes to the infinity.