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Saddle-node bifurcations of multiple homoclinic solutions in ODES
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 |
$h_t+\partial_x (h^n\,\partial_{xxx} h)+\partial_x (h^{n+2}\partial_{x} h)=0,$ |
$n≥q 1$ |
$M_c=\frac{2\sqrt{6}π}{3}$ |
$n=1$ |
$M_c$ |
$n≥q 4$ |
$n=1$ |
$M_c$ |
$n=1$ |
$M_c$ |
$ h(·, t_k)\rightharpoonup 0$ |
$L^1(\mathbb{R})$ |
${t_k} \to \infty $ |
References:
[1] |
P. Álvarez-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). Google Scholar |
[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 Méthodes de Résolution Des Problémes Aux Limites Non Linéaires 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. Savaré,
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, Über Integralungleichungen zwischen einer Funktion und ihrer Ableitung (German), Acta Univ. Szeged. Sect. Sci. Math., 10 (1941), 64-74. Google Scholar |
[33] |
F. Otto,
Lubrication approximation with prescribed nonzero contact angle, Comm. Partial Differential Equations, 23 (1998), 2077-2164.
doi: 10.1080/03605309808821411. |
[34] |
D. SlepÄev 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. Álvarez-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). Google Scholar |
[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 Méthodes de Résolution Des Problémes Aux Limites Non Linéaires 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. Savaré,
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, Über Integralungleichungen zwischen einer Funktion und ihrer Ableitung (German), Acta Univ. Szeged. Sect. Sci. Math., 10 (1941), 64-74. Google Scholar |
[33] |
F. Otto,
Lubrication approximation with prescribed nonzero contact angle, Comm. Partial Differential Equations, 23 (1998), 2077-2164.
doi: 10.1080/03605309808821411. |
[34] |
D. SlepÄev 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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