# American Institute of Mathematical Sciences

June 2017, 22(4): 1461-1492. doi: 10.3934/dcdsb.2017070

## Global existence for a thin film equation with subcritical mass

 1 School of Mathematics, Liaoning University, Shenyang 110036, China 2 Department of Physics and Department of Mathematics, Duke University, Durham, NC 27708, USA

Corresponding author: Jinhuan Wang, was supported by National Natural Science Foundation of China (Grant No. 11301243) and Program for Liaoning Excellent Talents in University (Grant No. LJQ2015041)

Jian-Guo Liu was partially supported partially supported by KI-Net NSF RNMS grant No. 1107444, NSF DMS grant No. 1514826

Received  December 2015 Revised  November 2016 Published  February 2017

In this paper, we study existence of global entropy weak solutions to a critical-case unstable thin film equation in one-dimensional case
 $h_t+\partial_x (h^n\,\partial_{xxx} h)+\partial_x (h^{n+2}\partial_{x} h)=0,$
where
 $n≥q 1$
. There exists a critical mass
 $M_c=\frac{2\sqrt{6}π}{3}$
found by Witelski et al.(2004 Euro. J. of Appl. Math. 15,223-256) for
 $n=1$
. We obtain global existence of a non-negative entropy weak solution if initial mass is less than
 $M_c$
. For
 $n≥q 4$
, entropy weak solutions are positive and unique. For
 $n=1$
, a finite time blow-up occurs for solutions with initial mass larger than
 $M_c$
. For the Cauchy problem with
 $n=1$
and initial mass less than
 $M_c$
, we show that at least one of the following long-time behavior holds:the second moment goes to infinity as the time goes to infinity or
 $h(·, t_k)\rightharpoonup 0$
in
 $L^1(\mathbb{R})$
for some subsequence
 ${t_k} \to \infty$
.
Citation: Jian-Guo Liu, Jinhuan Wang. Global existence for a thin film equation with subcritical mass. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1461-1492. doi: 10.3934/dcdsb.2017070
##### References:
 [1] P. Álvarez-Caudevilla and V. A. Galaktionov, Well-posedness of the Cauchy problem for a fourth-order thin film equation via regularization approaches, Nonl. Anal., 121 (2015), 19-35. doi: 10.1016/j.na.2014.08.002. [2] E. F. Beckenbach and R. Bellman, Introduction to Inequalities Random House Inc, 1965. [3] E. Beretta, M. Bertsch and R. Dal Passo, Nonnegative solutions of a fourth order nonlinear degenerate parabolic equation, Arch. Ration. Mech. Anal., 129 (1995), 175-200. doi: 10.1007/BF00379920. [4] F. Bernis, Finite speed of propagation and continuity of the interface for slow viscous flows, Adv. Differential Equations, 1 (1996), 337-368. [5] F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, J. Differential Equations, 83 (1990), 179-206. doi: 10.1016/0022-0396(90)90074-Y. [6] A. L. Bertozzi and M. C. Pugh, The lubrication approximation for thin viscous films, the moving contact line with a porous media cut off of Van der Waals interactions, Nonlinearity, 7 (1994), 1535-1564. doi: 10.1088/0951-7715/7/6/002. [7] A. L. Bertozzi and M. C. Pugh, The lubrication approximation for thin viscous films: Regularity and long time behavior of weak solutions, Comm. Pure Appl. Math., 49 (1996), 85-123. doi: 10.1002/(SICI)1097-0312(199602)49:2<85::AID-CPA1>3.0.CO;2-2. [8] A. L. Bertozzi and M. C. Pugh, Long-wave instabilities and saturation in thin film equations, Comm. Pure Appl. Math., 51 (1998), 625-661. doi: 10.1002/(SICI)1097-0312(199806)51:6<625::AID-CPA3>3.0.CO;2-9. [9] 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 (2000), 1323-1366. doi: 10.1512/iumj.2000.49.1887. [10] M. Bertsch, L. Giacomelli and G. Karali, Thin-film equations with "partial wetting" energy: Existence of weak solutions, Physica D, 209 (2005), 17-27. doi: 10.1016/j.physd.2005.06.012. [11] M. Bertsch, R. Dal Passo, H. Garcke and G. Grün, The thin viscous flow equation in higher space dimensions, Adv. Differential Equations, 3 (1998), 417-440. [12] S. Bian and J.-G. Liu, Dynamic and steady states for multi-dimensional Keller-Segel model with diffusion exponent $m > 0$, Comm. Math. Phys., 323 (2013), 1017-1070. doi: 10.1007/s00220-013-1777-z. [13] M. Chugunova, M. C. Pugh and R. M. Taranets, Research Announcement: Finite-time blow up and long-wave unstable thin film equations, arXiv1008.0385v1, (2010). [14] M. Chugunova and R. M. Taranets, Blow-up with mass concentration for the long-wave unstable thin-film equation, Appl. Anal., 95 (2016), 944-962. doi: 10.1080/00036811.2015.1047829. [15] R. Dal Passo and H. Garcke, Solutions of a fourth order degenerate parabolic equation with weak initial trace, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 28 (1999), 153-181. [16] R. Dal Passo, H. Garcke and G. Grün, On a fourth-order degenerate parabolic equation: Global entropy estimates, existence, and qualitative behavior of solutions, SIAM J. Math. Anal., 29 (1998), 321-342. doi: 10.1137/S0036141096306170. [17] L. Giacomelli, M. V. Gnann and F. Otto, Regularity of source-type solutions to the thin-film equation with zero contact angle and mobility exponent between 3/2 and 3, European J. Appl. Math., 24 (2013), 735-760. doi: 10.1017/S0956792513000156. [18] L. Giacomelli, H. Knüpfer and F. Otto, Smooth zero-contact-angle solutions to a thin-film equation around the steady state, J. Differential Equations, 245 (2008), 1454-1506. doi: 10.1016/j.jde.2008.06.005. [19] M. V. Gnann, Well-posedness and self-similar asymptotics for a thin-film equation, SIAM J. Math. Anal., 47 (2015), 2868--2902. doi: 10.1137/14099190X. [20] G. Grün, Droplet spreading under weak slippage: The optimal asymptotic propagation rate in the multi-dimensional case, Interfaces Free Bound., 4 (2002), 309-323. doi: 10.4171/IFB/63. [21] G. Grün, Droplet spreading under weak slippage: A basic result on nite speed of propagation, SIAM J. Math. Anal., 34 (2003), 992-1006. doi: 10.1137/S0036141002403298. [22] G. Grün, Droplet spreading under weak slippage-existence for the Cauchy problem, Comm. Partial Differential Equations, 29 (2004), 1697-1744. doi: 10.1081/PDE-200040193. [23] D. John, On uniqueness of weak solutions for the thin-film equation, J. Differential Equations, 259 (2015), 4122-4171. doi: 10.1016/j.jde.2015.05.013. [24] H. Knüpfer, Well-posedness for the Navier slip thin film equation in the case of partial wetting, Comm. Pure Appl. Math., 64 (2011), 1263-1296. doi: 10.1002/cpa.20376. [25] H. Knüpfer and N. Masmoudi, Darcy flow on a plate with prescribed contact angle well-posedness and lubrication approximation, Arch. Rational Mech. Anal., 218 (2015), 589-646. doi: 10.1007/s00205-015-0868-8. [26] R. S. Laugesen and M. C. Pugh, Properties of steady states for thin film equations, European J. Appl. Math., 11 (2000), 293-351. doi: 10.1017/S0956792599003794. [27] J. -L. Lions, Quelques MÃ©thodes de RÃ©solution Des ProblÃ©mes Aux Limites Non LinÃ©aires Paris, Dunod, 1969. [28] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow Vol. 27, Cambridge University Press, 2002. [29] D. Matthes, R. J. McCann and G. SavarÃ©, A family of nonlinear fourth order equations of gradient flow type, Comm. Partial Differential Equations, 34 (2009), 1352-1397. doi: 10.1080/03605300903296256. [30] A. Mellet, The thin film equation with non zero contact angle: A singular perturbation approach, Comm. Partial Differential Equations, 40 (2015), 1-39. doi: 10.1080/03605302.2014.895380. [31] T. G. Myers, Thin films with high surface tension, SIAM Rev., 40 (1998), 441-462. doi: 10.1137/S003614459529284X. [32] B. V. Sz. Nagy, Ãœber Integralungleichungen zwischen einer Funktion und ihrer Ableitung (German), Acta Univ. Szeged. Sect. Sci. Math., 10 (1941), 64-74. [33] F. Otto, Lubrication approximation with prescribed nonzero contact angle, Comm. Partial Differential Equations, 23 (1998), 2077-2164. doi: 10.1080/03605309808821411. [34] D. SlepÄev and M. C. Pugh, Self-similar blow-up of unstable thin-film equations, Indiana Univ. Math. J., 54 (2005), 1697-1738. doi: 10.1512/iumj.2005.54.2569. [35] R. M. Taranets and J. R. King, On an unstable thin-film equation in multi-dimensional domains, NoDEA Nonlinear Differential Equations Appl., 21 (2014), 105-128. doi: 10.1007/s00030-013-0240-3. [36] T. P. Witelski, A. J. xBernoff and A. L. Bertozzi, Blow-up and dissipation in a critical-case unstable thin film equation, European J. Appl. Math., 15 (2004), 223-256. doi: 10.1017/S0956792504005418. [37] Z. Q. Wu, J. N. Zhao, J. X. Yin and H. L. Li, Nonlinear Diffusion Equations 2nd edition, Singapore, World Scientific, 2001. doi: 10.1142/9789812799791.

show all references

##### References:
 [1] P. Álvarez-Caudevilla and V. A. Galaktionov, Well-posedness of the Cauchy problem for a fourth-order thin film equation via regularization approaches, Nonl. Anal., 121 (2015), 19-35. doi: 10.1016/j.na.2014.08.002. [2] E. F. Beckenbach and R. Bellman, Introduction to Inequalities Random House Inc, 1965. [3] E. Beretta, M. Bertsch and R. Dal Passo, Nonnegative solutions of a fourth order nonlinear degenerate parabolic equation, Arch. Ration. Mech. Anal., 129 (1995), 175-200. doi: 10.1007/BF00379920. [4] F. Bernis, Finite speed of propagation and continuity of the interface for slow viscous flows, Adv. Differential Equations, 1 (1996), 337-368. [5] F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, J. Differential Equations, 83 (1990), 179-206. doi: 10.1016/0022-0396(90)90074-Y. [6] A. L. Bertozzi and M. C. Pugh, The lubrication approximation for thin viscous films, the moving contact line with a porous media cut off of Van der Waals interactions, Nonlinearity, 7 (1994), 1535-1564. doi: 10.1088/0951-7715/7/6/002. [7] A. L. Bertozzi and M. C. Pugh, The lubrication approximation for thin viscous films: Regularity and long time behavior of weak solutions, Comm. Pure Appl. Math., 49 (1996), 85-123. doi: 10.1002/(SICI)1097-0312(199602)49:2<85::AID-CPA1>3.0.CO;2-2. [8] A. L. Bertozzi and M. C. Pugh, Long-wave instabilities and saturation in thin film equations, Comm. Pure Appl. Math., 51 (1998), 625-661. doi: 10.1002/(SICI)1097-0312(199806)51:6<625::AID-CPA3>3.0.CO;2-9. [9] 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 (2000), 1323-1366. doi: 10.1512/iumj.2000.49.1887. [10] M. Bertsch, L. Giacomelli and G. Karali, Thin-film equations with "partial wetting" energy: Existence of weak solutions, Physica D, 209 (2005), 17-27. doi: 10.1016/j.physd.2005.06.012. [11] M. Bertsch, R. Dal Passo, H. Garcke and G. Grün, The thin viscous flow equation in higher space dimensions, Adv. Differential Equations, 3 (1998), 417-440. [12] S. Bian and J.-G. Liu, Dynamic and steady states for multi-dimensional Keller-Segel model with diffusion exponent $m > 0$, Comm. Math. Phys., 323 (2013), 1017-1070. doi: 10.1007/s00220-013-1777-z. [13] M. Chugunova, M. C. Pugh and R. M. Taranets, Research Announcement: Finite-time blow up and long-wave unstable thin film equations, arXiv1008.0385v1, (2010). [14] M. Chugunova and R. M. Taranets, Blow-up with mass concentration for the long-wave unstable thin-film equation, Appl. Anal., 95 (2016), 944-962. doi: 10.1080/00036811.2015.1047829. [15] R. Dal Passo and H. Garcke, Solutions of a fourth order degenerate parabolic equation with weak initial trace, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 28 (1999), 153-181. [16] R. Dal Passo, H. Garcke and G. Grün, On a fourth-order degenerate parabolic equation: Global entropy estimates, existence, and qualitative behavior of solutions, SIAM J. Math. Anal., 29 (1998), 321-342. doi: 10.1137/S0036141096306170. [17] L. Giacomelli, M. V. Gnann and F. Otto, Regularity of source-type solutions to the thin-film equation with zero contact angle and mobility exponent between 3/2 and 3, European J. Appl. Math., 24 (2013), 735-760. doi: 10.1017/S0956792513000156. [18] L. Giacomelli, H. Knüpfer and F. Otto, Smooth zero-contact-angle solutions to a thin-film equation around the steady state, J. Differential Equations, 245 (2008), 1454-1506. doi: 10.1016/j.jde.2008.06.005. [19] M. V. Gnann, Well-posedness and self-similar asymptotics for a thin-film equation, SIAM J. Math. Anal., 47 (2015), 2868--2902. doi: 10.1137/14099190X. [20] G. Grün, Droplet spreading under weak slippage: The optimal asymptotic propagation rate in the multi-dimensional case, Interfaces Free Bound., 4 (2002), 309-323. doi: 10.4171/IFB/63. [21] G. Grün, Droplet spreading under weak slippage: A basic result on nite speed of propagation, SIAM J. Math. Anal., 34 (2003), 992-1006. doi: 10.1137/S0036141002403298. [22] G. Grün, Droplet spreading under weak slippage-existence for the Cauchy problem, Comm. Partial Differential Equations, 29 (2004), 1697-1744. doi: 10.1081/PDE-200040193. [23] D. John, On uniqueness of weak solutions for the thin-film equation, J. Differential Equations, 259 (2015), 4122-4171. doi: 10.1016/j.jde.2015.05.013. [24] H. Knüpfer, Well-posedness for the Navier slip thin film equation in the case of partial wetting, Comm. Pure Appl. Math., 64 (2011), 1263-1296. doi: 10.1002/cpa.20376. [25] H. Knüpfer and N. Masmoudi, Darcy flow on a plate with prescribed contact angle well-posedness and lubrication approximation, Arch. Rational Mech. Anal., 218 (2015), 589-646. doi: 10.1007/s00205-015-0868-8. [26] R. S. Laugesen and M. C. Pugh, Properties of steady states for thin film equations, European J. Appl. Math., 11 (2000), 293-351. doi: 10.1017/S0956792599003794. [27] J. -L. Lions, Quelques MÃ©thodes de RÃ©solution Des ProblÃ©mes Aux Limites Non LinÃ©aires Paris, Dunod, 1969. [28] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow Vol. 27, Cambridge University Press, 2002. [29] D. Matthes, R. J. McCann and G. SavarÃ©, A family of nonlinear fourth order equations of gradient flow type, Comm. Partial Differential Equations, 34 (2009), 1352-1397. doi: 10.1080/03605300903296256. [30] A. Mellet, The thin film equation with non zero contact angle: A singular perturbation approach, Comm. Partial Differential Equations, 40 (2015), 1-39. doi: 10.1080/03605302.2014.895380. [31] T. G. Myers, Thin films with high surface tension, SIAM Rev., 40 (1998), 441-462. doi: 10.1137/S003614459529284X. [32] B. V. Sz. Nagy, Ãœber Integralungleichungen zwischen einer Funktion und ihrer Ableitung (German), Acta Univ. Szeged. Sect. Sci. Math., 10 (1941), 64-74. [33] F. Otto, Lubrication approximation with prescribed nonzero contact angle, Comm. Partial Differential Equations, 23 (1998), 2077-2164. doi: 10.1080/03605309808821411. [34] D. SlepÄev and M. C. Pugh, Self-similar blow-up of unstable thin-film equations, Indiana Univ. Math. J., 54 (2005), 1697-1738. doi: 10.1512/iumj.2005.54.2569. [35] R. M. Taranets and J. R. King, On an unstable thin-film equation in multi-dimensional domains, NoDEA Nonlinear Differential Equations Appl., 21 (2014), 105-128. doi: 10.1007/s00030-013-0240-3. [36] T. P. Witelski, A. J. xBernoff and A. L. Bertozzi, Blow-up and dissipation in a critical-case unstable thin film equation, European J. Appl. Math., 15 (2004), 223-256. doi: 10.1017/S0956792504005418. [37] Z. Q. Wu, J. N. Zhao, J. X. Yin and H. L. Li, Nonlinear Diffusion Equations 2nd edition, Singapore, World Scientific, 2001. doi: 10.1142/9789812799791.
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