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

Part of: Thin bodies, structures Boundary value problems Elliptic equations and systems

Published online by Cambridge University Press:  30 May 2022

XINFU CHEN ,
HUIQIANG JIANG  and
GUOQING LIU Open the ORCID record for GUOQING LIU [Opens in a new window]
XINFU CHEN
Affiliation:
Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA emails: xinfu@pitt.edu; hqjiang@pitt.edu; gul8@pitt.edu
HUIQIANG JIANG
Affiliation:
Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA emails: xinfu@pitt.edu; hqjiang@pitt.edu; gul8@pitt.edu
GUOQING LIU
Affiliation:
Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA emails: xinfu@pitt.edu; hqjiang@pitt.edu; gul8@pitt.edu
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Abstract

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.

Keywords

Singular elliptic equationvan der Waals forceThin film

MSC classification

Primary: 35J20: Variational methods for second-order elliptic equations 74K35: Thin films
Secondary: 34B18: Positive solutions of nonlinear boundary value problems
Type
Papers
Information
European Journal of Applied Mathematics , Volume 34 , Issue 2 , April 2023 , pp. 280 - 302
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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