-
Previous Article
Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent
- CPAA Home
- This Issue
-
Next Article
Well-posedness for the supercritical gKdV equation
Finite speed of propagation and algebraic time decay of solutions to a generalized thin film equation
1. | School of Science, Nanjing University of Science and Technology, Nanjing, 210094, China, China |
References:
[1] |
L. Ansini and L. Giacomelli, Doubly nonlinear thin-film equations in one space dimension,, Arch. Rational Mech. Anal., 173 (2004), 89.
doi: 10.1007/s00205-004-0313-x. |
[2] |
F. Bernis, Finite speed of propagation and continuity of the interface for thin viscous flows,, Adv. Differential Equations, 1 (1996), 337.
|
[3] |
F. Bernis, Finite speed of propagation for thin viscous flows when $ 2 \le n < 3 $,, C. R. Acad. Sci. Paris Sér. I Math., 322 (1996), 1169.
|
[4] |
F. Bernis and A. Friedman, Higher-order nonlinear degenerate parabolic equations,, J. Diff. Equations, 83 (1990), 179.
doi: 10.1016/0022-0396(90)90074-Y. |
[5] |
F. Bernis, L. A. Peletier and S. M. Williams, Source type solutions of a fourth order nonlinear degenerate parabolic equation,, Nonlinear Anal., 18 (1992), 217.
doi: 10.1016/0362-546X(92)90060-R. |
[6] |
A. L. Bertozzi and M. C. Pugh, The lubrication approximation for thin viscous films: the moving contact line with a porous media cutoff of the van der Waals interactions,, Nonlinearity, 7 (1994), 1535.
|
[7] |
A. L. Bertozzi and M. C. Pugh, The lubrication approximation for viscous films: regularity and long time behavior of weak solutions,, Comm. Pure Appl. Math., 49 (1996), 85.
doi: 10.1002/(SICI)1097-0312(199602)49:2<85::AID-CPA1>3.3.CO;2-V. |
[8] |
A. L. Bertozzi and M. C. Pugh, Long-wave instabilities and saturation in thin film equations,, Comm. Pure Appl. Math., 51 (1998), 625.
doi: 10.1002/(SICI)1097-0312(199806)51:6<625::AID-CPA3>3.3.CO;2-2. |
[9] |
M. Bertsch, R. Dal Passo, H. Garcke and G. Grün, The thin viscous flow equation in higher space dimensions,, Adv Differ. Equ., 3 (1998), 417.
|
[10] |
E. Bertta, M. Bertsch and R. Dal Passo, Nonnegative solutions of a fourth-order nonlinear degenerate parabolic equation,, Arch. Ration. Mech. Anal., 129 (1995), 175.
|
[11] |
M. Boutat, S. Hilout, J. E. Rakotoson and J. M. Rakotoson, A generalized thin-film equation in multidimensional space,, Nonlinear Analysis, 69 (2008), 1268.
doi: 10.1016/j.na.2007.06.028. |
[12] |
M. Boutat, S. Hilout, J. E. Rakotoson, J. M. Rakotoson, The generalized thin film equation with periodic-domain conditions,, Applied Mathematics Letters, 21 (2008), 101.
doi: 10.1016/j.aml.2007.02.014. |
[13] |
E. A. Carlen and S. Ulusory, An entropy dissipation-entropy estimate for a thin film type equation,, Comm. Math. Sci., 3 (2005), 171.
|
[14] |
J. A. Carrillo, A. Jüngel, P. A. Markowich, G. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities,, Monatsh. Math., 133 (2001), 1.
doi: 10.1007/s006050170032. |
[15] |
J. A. Carrillo and G. Toscani, Long-time asymptotics for strong solutions of the thin film equation,, Commun. Math. Phys., 225 (2002), 551.
doi: 10.1007/s002200100591. |
[16] |
M. Chugunova, M. Pugh and R. Taranets, Research announcement: Finite-time blow up and long-wave unstable thin film equations,, preprint, (). Google Scholar |
[17] |
R. Dal Passo, H. Garcke and G. Grün, On a fourth order degenerate parabolic equation: global entropy estimates, existence, and qualitative behaviour of solutions,, SIAM J. Math. Anal., 29 (1998), 321.
|
[18] |
R. Dal Passo, L. Giacomelli and G. Grün, A waiting time phenomenon for thin film equations,, Ann. Scuola Norm. Sup. Pisa, 30 (2001), 437.
|
[19] |
R. Dal Passo, L. Giacomelli and A. Shishkov, The thin film equation with nonlinear diffusion,, Comm. Partial Differential Equations, 26 (2001), 1509.
|
[20] |
S. D. Èĭdel'man, Parabolic Systems,, Translated from the Russian by Scripta Technica, (1969). Google Scholar |
[21] |
A. Friedman, Partial Differential Equations,, Holt, (). Google Scholar |
[22] |
L. Giacomelli, A fourth-order degenerate parabolic equation describing thin viscous flows over an inclined plane,, Applied Mathematics Letters, 12 (1999), 107.
doi: 10.1016/S0893-9659(99)00130-5. |
[23] |
L. Giacomelli and A. Shishkov, Propagation of support in one-dimensional convected thin-film flow,, Indiana Univ. Math. J., 54 (2005), 1181.
doi: 10.1512/iumj.2005.54.2532. |
[24] |
G. Grün, Degenerate parabolic differential equations of fourth order and a plasticity model with nonlocal harding,, Z. Anal. Anwendungen., 14 (1995), 541.
|
[25] |
G. Grün, "On Free Boundary Problems Arising in Thin Film Flow,", Habilitation thesis, (2001). Google Scholar |
[26] |
G. Grün, Droplet spreading under weak slippage: the waiting time phenomenon,, Ann. I. H. Poincaré–AN, 21 (2004), 255.
doi: 10.1016/j.ahihpc.2003.02.002. |
[27] |
L. M. Hocking, Spreading and instability of a viscous fluid sheet,, Journal of Fluid Mechanics, 211 (1990), 373.
doi: 10.1017/S0022112090001616. |
[28] |
J. R. King, Two generalisations of the thin film equation,, Math. Comput. Modelling, 34 (2001), 737.
doi: 10.1016/S0895-7177(01)00095-4. |
[29] |
J. J. Li, On a fourth order degenerate parabolic equation in higher space dimensions,, Journal of Mathematical Physics, 50 (2009).
doi: 10.1063/1.3272788. |
[30] |
X. Liu and C. Qu, Finite speed of propagation for thin viscous flows over an inclined plane,, Nonlinear Anal. Real World Appl., 13 (2012), 464.
doi: 10.1016/j.nonrwa.2011.08.003. |
[31] |
E. Momoniata, T. G. Myers and S. Abelman, Similarity solutions of thin film flow driven by gravity and surface shear,, Nonlinear Analysis: Real World Applications, 10 (2009), 3443.
doi: 10.1016/j.nonrwa.2008.10.070. |
[32] |
A. Oron, S. H, Davis and S. G. Bankoff, Long-scale evolution of thin liquid films,, Rev. Modern Phys., 69 (1997), 931.
doi: 10.1103/RevModPhys.69.931. |
[33] |
E. O. Tuck and L. W. Schwaxtz, Thin static drops with a free attachment boundary,, Journal of Fluid Mechanics, 223 (1991), 313.
|
[34] |
C. Villani, A review of mathematical topics in collisional kinetic theory,, in, (2002), 71. Google Scholar |
show all references
References:
[1] |
L. Ansini and L. Giacomelli, Doubly nonlinear thin-film equations in one space dimension,, Arch. Rational Mech. Anal., 173 (2004), 89.
doi: 10.1007/s00205-004-0313-x. |
[2] |
F. Bernis, Finite speed of propagation and continuity of the interface for thin viscous flows,, Adv. Differential Equations, 1 (1996), 337.
|
[3] |
F. Bernis, Finite speed of propagation for thin viscous flows when $ 2 \le n < 3 $,, C. R. Acad. Sci. Paris Sér. I Math., 322 (1996), 1169.
|
[4] |
F. Bernis and A. Friedman, Higher-order nonlinear degenerate parabolic equations,, J. Diff. Equations, 83 (1990), 179.
doi: 10.1016/0022-0396(90)90074-Y. |
[5] |
F. Bernis, L. A. Peletier and S. M. Williams, Source type solutions of a fourth order nonlinear degenerate parabolic equation,, Nonlinear Anal., 18 (1992), 217.
doi: 10.1016/0362-546X(92)90060-R. |
[6] |
A. L. Bertozzi and M. C. Pugh, The lubrication approximation for thin viscous films: the moving contact line with a porous media cutoff of the van der Waals interactions,, Nonlinearity, 7 (1994), 1535.
|
[7] |
A. L. Bertozzi and M. C. Pugh, The lubrication approximation for viscous films: regularity and long time behavior of weak solutions,, Comm. Pure Appl. Math., 49 (1996), 85.
doi: 10.1002/(SICI)1097-0312(199602)49:2<85::AID-CPA1>3.3.CO;2-V. |
[8] |
A. L. Bertozzi and M. C. Pugh, Long-wave instabilities and saturation in thin film equations,, Comm. Pure Appl. Math., 51 (1998), 625.
doi: 10.1002/(SICI)1097-0312(199806)51:6<625::AID-CPA3>3.3.CO;2-2. |
[9] |
M. Bertsch, R. Dal Passo, H. Garcke and G. Grün, The thin viscous flow equation in higher space dimensions,, Adv Differ. Equ., 3 (1998), 417.
|
[10] |
E. Bertta, M. Bertsch and R. Dal Passo, Nonnegative solutions of a fourth-order nonlinear degenerate parabolic equation,, Arch. Ration. Mech. Anal., 129 (1995), 175.
|
[11] |
M. Boutat, S. Hilout, J. E. Rakotoson and J. M. Rakotoson, A generalized thin-film equation in multidimensional space,, Nonlinear Analysis, 69 (2008), 1268.
doi: 10.1016/j.na.2007.06.028. |
[12] |
M. Boutat, S. Hilout, J. E. Rakotoson, J. M. Rakotoson, The generalized thin film equation with periodic-domain conditions,, Applied Mathematics Letters, 21 (2008), 101.
doi: 10.1016/j.aml.2007.02.014. |
[13] |
E. A. Carlen and S. Ulusory, An entropy dissipation-entropy estimate for a thin film type equation,, Comm. Math. Sci., 3 (2005), 171.
|
[14] |
J. A. Carrillo, A. Jüngel, P. A. Markowich, G. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities,, Monatsh. Math., 133 (2001), 1.
doi: 10.1007/s006050170032. |
[15] |
J. A. Carrillo and G. Toscani, Long-time asymptotics for strong solutions of the thin film equation,, Commun. Math. Phys., 225 (2002), 551.
doi: 10.1007/s002200100591. |
[16] |
M. Chugunova, M. Pugh and R. Taranets, Research announcement: Finite-time blow up and long-wave unstable thin film equations,, preprint, (). Google Scholar |
[17] |
R. Dal Passo, H. Garcke and G. Grün, On a fourth order degenerate parabolic equation: global entropy estimates, existence, and qualitative behaviour of solutions,, SIAM J. Math. Anal., 29 (1998), 321.
|
[18] |
R. Dal Passo, L. Giacomelli and G. Grün, A waiting time phenomenon for thin film equations,, Ann. Scuola Norm. Sup. Pisa, 30 (2001), 437.
|
[19] |
R. Dal Passo, L. Giacomelli and A. Shishkov, The thin film equation with nonlinear diffusion,, Comm. Partial Differential Equations, 26 (2001), 1509.
|
[20] |
S. D. Èĭdel'man, Parabolic Systems,, Translated from the Russian by Scripta Technica, (1969). Google Scholar |
[21] |
A. Friedman, Partial Differential Equations,, Holt, (). Google Scholar |
[22] |
L. Giacomelli, A fourth-order degenerate parabolic equation describing thin viscous flows over an inclined plane,, Applied Mathematics Letters, 12 (1999), 107.
doi: 10.1016/S0893-9659(99)00130-5. |
[23] |
L. Giacomelli and A. Shishkov, Propagation of support in one-dimensional convected thin-film flow,, Indiana Univ. Math. J., 54 (2005), 1181.
doi: 10.1512/iumj.2005.54.2532. |
[24] |
G. Grün, Degenerate parabolic differential equations of fourth order and a plasticity model with nonlocal harding,, Z. Anal. Anwendungen., 14 (1995), 541.
|
[25] |
G. Grün, "On Free Boundary Problems Arising in Thin Film Flow,", Habilitation thesis, (2001). Google Scholar |
[26] |
G. Grün, Droplet spreading under weak slippage: the waiting time phenomenon,, Ann. I. H. Poincaré–AN, 21 (2004), 255.
doi: 10.1016/j.ahihpc.2003.02.002. |
[27] |
L. M. Hocking, Spreading and instability of a viscous fluid sheet,, Journal of Fluid Mechanics, 211 (1990), 373.
doi: 10.1017/S0022112090001616. |
[28] |
J. R. King, Two generalisations of the thin film equation,, Math. Comput. Modelling, 34 (2001), 737.
doi: 10.1016/S0895-7177(01)00095-4. |
[29] |
J. J. Li, On a fourth order degenerate parabolic equation in higher space dimensions,, Journal of Mathematical Physics, 50 (2009).
doi: 10.1063/1.3272788. |
[30] |
X. Liu and C. Qu, Finite speed of propagation for thin viscous flows over an inclined plane,, Nonlinear Anal. Real World Appl., 13 (2012), 464.
doi: 10.1016/j.nonrwa.2011.08.003. |
[31] |
E. Momoniata, T. G. Myers and S. Abelman, Similarity solutions of thin film flow driven by gravity and surface shear,, Nonlinear Analysis: Real World Applications, 10 (2009), 3443.
doi: 10.1016/j.nonrwa.2008.10.070. |
[32] |
A. Oron, S. H, Davis and S. G. Bankoff, Long-scale evolution of thin liquid films,, Rev. Modern Phys., 69 (1997), 931.
doi: 10.1103/RevModPhys.69.931. |
[33] |
E. O. Tuck and L. W. Schwaxtz, Thin static drops with a free attachment boundary,, Journal of Fluid Mechanics, 223 (1991), 313.
|
[34] |
C. Villani, A review of mathematical topics in collisional kinetic theory,, in, (2002), 71. Google Scholar |
[1] |
Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252 |
[2] |
Michiel Bertsch, Danielle Hilhorst, Hirofumi Izuhara, Masayasu Mimura, Tohru Wakasa. A nonlinear parabolic-hyperbolic system for contact inhibition and a degenerate parabolic fisher kpp equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3117-3142. doi: 10.3934/dcds.2019226 |
[3] |
Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247 |
[4] |
Van Duong Dinh. Random data theory for the cubic fourth-order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (2) : 651-680. doi: 10.3934/cpaa.2020284 |
[5] |
Liupeng Wang, Yunqing Huang. Error estimates for second-order SAV finite element method to phase field crystal model. Electronic Research Archive, 2021, 29 (1) : 1735-1752. doi: 10.3934/era.2020089 |
[6] |
Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020319 |
[7] |
Abdollah Borhanifar, Maria Alessandra Ragusa, Sohrab Valizadeh. High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020355 |
[8] |
Thomas Frenzel, Matthias Liero. Effective diffusion in thin structures via generalized gradient systems and EDP-convergence. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 395-425. doi: 10.3934/dcdss.2020345 |
[9] |
Xiaopeng Zhao, Yong Zhou. Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 795-813. doi: 10.3934/dcdsb.2020142 |
[10] |
Ying Liu, Yanping Chen, Yunqing Huang, Yang Wang. Two-grid method for semiconductor device problem by mixed finite element method and characteristics finite element method. Electronic Research Archive, 2021, 29 (1) : 1859-1880. doi: 10.3934/era.2020095 |
[11] |
Yohei Yamazaki. Center stable manifolds around line solitary waves of the Zakharov–Kuznetsov equation with critical speed. Discrete & Continuous Dynamical Systems - A, 2021 doi: 10.3934/dcds.2021008 |
[12] |
Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, , () : -. doi: 10.3934/era.2020120 |
[13] |
Xiu Ye, Shangyou Zhang, Peng Zhu. A weak Galerkin finite element method for nonlinear conservation laws. Electronic Research Archive, 2021, 29 (1) : 1897-1923. doi: 10.3934/era.2020097 |
[14] |
François Dubois. Third order equivalent equation of lattice Boltzmann scheme. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 221-248. doi: 10.3934/dcds.2009.23.221 |
[15] |
Vo Van Au, Hossein Jafari, Zakia Hammouch, Nguyen Huy Tuan. On a final value problem for a nonlinear fractional pseudo-parabolic equation. Electronic Research Archive, 2021, 29 (1) : 1709-1734. doi: 10.3934/era.2020088 |
[16] |
Stanislav Nikolaevich Antontsev, Serik Ersultanovich Aitzhanov, Guzel Rashitkhuzhakyzy Ashurova. An inverse problem for the pseudo-parabolic equation with p-Laplacian. Evolution Equations & Control Theory, 2021 doi: 10.3934/eect.2021005 |
[17] |
Gang Luo, Qingzhi Yang. The point-wise convergence of shifted symmetric higher order power method. Journal of Industrial & Management Optimization, 2021, 17 (1) : 357-368. doi: 10.3934/jimo.2019115 |
[18] |
Ville Salo, Ilkka Törmä. Recoding Lie algebraic subshifts. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 1005-1021. doi: 10.3934/dcds.2020307 |
[19] |
Jing Zhou, Cheng Lu, Ye Tian, Xiaoying Tang. A SOCP relaxation based branch-and-bound method for generalized trust-region subproblem. Journal of Industrial & Management Optimization, 2021, 17 (1) : 151-168. doi: 10.3934/jimo.2019104 |
[20] |
Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079 |
2019 Impact Factor: 1.105
Tools
Metrics
Other articles
by authors
[Back to Top]