Journal of Advances in Applied Mathematics

Download PDF (533.5 KB) PP. 270 - 279 Pub. Date: October 1, 2016

DOI: 10.22606/jaam.2016.14007

• Attou Miloua^*
  Department of Mathematics & information Systems CALU, California, PA, 15419

We consider the elliptic equation
u = f (u) in a region  RN, N  3, where f is a positive continuous function satisfying lim u!0+ f (u) = 1. Motivated by the thin film equations, a solution u is said to be a point rupture solution if for some p 2 , u (p) = 0 and u (p) > 0 in \ {p}. Solving the associated ordinary differential equations confirm our main results of sufficient conditions on f for the existence uniquness of radial point rupture solution and its asymptotic behavior. Furthermore, we can prove that our results can be applied to the point rupture solutions for a class of quasi-linear elliptic equations of the form div (a (u)
u) = a0 (u) 2 |ru|2 + f (u)

Thin film; point rupture; radial solution; singular equation; quasi-linear elliptic equation

[1] Robert Almgren, Andrea Bertozzi, and Michael P. Brenner. Stable and unstable singularities in the unforced Hele-Shaw cell. Phys. Fluids, 8(6):1356–1370, 1996.

[2] Francisco Bernis and Avner Friedman. Higher order nonlinear degenerate parabolic equations. J. Differential Equations, 83(1):179–206, 1990.

[3] Andrew J. Bernoff and Andrea L. Bertozzi. Singularities in a modified Kuramoto-Sivashinsky equation describing interface motion for phase transition. Phys. D, 85(3):375–404, 1995.

[4] A. L. Bertozzi and M. C. Pugh. Finite-time blow-up of solutions of some long-wave unstable thin film equations. Indiana Univ. Math. J., 49(4):1323–1366, 2000.

[5] Andrea L. Bertozzi, Michael P. Brenner, Todd F. Dupont, and Leo P. Kadanoff. Singularities and similarities in interface flows. In Trends and perspectives in applied mathematics, volume 100 of Appl. Math. Sci., page.155–208. Springer, New York, 1994.

[6] Francesca Gladiali and Marco Squassina. On Explosive Solutions for a class of Quasi-linear Elliptic Equations. Advanced Nonlinear Studies, 13:663–698, 2013.

[7] Xinfu Chen and Huiqiang Jiang. Singular limit of an energy minimizer arising from dewetting thin film model with van der waal, born repulsion and surface tension forces. Calc. Var. Partial Differential Equations,44(1-2):221–246, 2012.

[8] Peter Constantin, Todd F. Dupont, Raymond E. Goldstein, Leo P. Kadanoff, Michael J. Shelley, and Su-Min Zhou. Droplet breakup in a model of the Hele-Shaw cell. Phys. Rev. E (3), 47(6):4169–4181, 1993.

[9] Todd F. Dupont, Raymond E. Goldstein, Leo P. Kadanoff, and Su-Min Zhou. Finite-time singularity formation in Hele-Shaw systems. Phys. Rev. E (3), 47(6):4182–4196, 1993.

[10] P. Ehrhard. The spreading of hanging drops. J. Colloid & Interface, 168(1):242–246, 1994.

[11] Zongming Guo, Dong Ye, and Feng Zhou. Existence of singular positive solutions for some semilinear elliptic equations. Pacific J. Math., 236(1):57–71, 2008.

[12] Huiqiang Jiang. Energy minimizers of a thin film equation with Born repulsion force. Commun. Pure Appl. Anal., 10(2):803–815, 2011.

[13] Huiqiang Jiang and Fanghua Lin. Zero set of soblev functions with negative power of integrability. Chinese Ann. Math. Ser. B, 25(1):65–72, 2004.

[14] Huiqiang Jiang and Attou Miloua. Point rupture solutions of a singular elliptic equation. Electronic Journal of Differential Equations, 2013(70):1–8,2013.

[15] Huiqiang Jiang and Wei-Ming Ni. On steady states of van der Waals force driven thin film equations. European J. Appl. Math., 18(2):153–180, 2007.

[16] R. S. Laugesen and M. C. Pugh. Properties of steady states for thin film equations. European J. Appl. Math., 11(3):293–351, 2000.

[17] Michael J. Shelley, Raymond E. Goldstein, and Adriana I. Pesci. Topological transitions in Hele-Shaw flow. In Singularities in fluids, plasmas and optics (Heraklion, 1992), volume 404 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., pages 167–188. Kluwer Acad. Publ., Dordrecht, 1993.

[18] B. Williams and S. H. Davis. Nonlinear theory of film rupture. J. Colloid Interface Sci., 90:220–228, 1982.

[19] Thomas P. Witelski and Andrew J. Bernoff. Stability of self-similar solutions for van der Waals driven thin film rupture. Phys. Fluids, 11(9):2443–2445, 1999.

[20] Wendy W. Zhang and John R. Lister. Similarity solutions for van der Waals rupture of a thin film on a solid substrate. Phys. Fluids, 11(9):2454–2462, 1999.