# American Institute of Mathematical Sciences

March  2014, 13(2): 543-566. doi: 10.3934/cpaa.2014.13.543

## 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

Received  October 2012 Revised  July 2013 Published  October 2013

We consider a fourth order degenerate equation describing thin films over an inclined plane in this paper. A new approximating problem is introduced in order to obtain the local energy estimate of the solution. Based on combined use of local entropy estimate, local energy estimate and the suitable extensions of Stampacchia's Lemma to systems, we obtain the finite speed of propagation property of strong solutions, which has been known for the case of strong slippage $n<2,$ in the case of weak slippage $2 \leq n < 3.$ The long time behavior of positive classical solutions is also discussed. We apply the entropy dissipation method to quantify the explicit rate of convergence in the $L^\infty$ norm of the solution, and this improves and extends the previous results.
Citation: Lihua Min, Xiaoping Yang. Finite speed of propagation and algebraic time decay of solutions to a generalized thin film equation. Communications on Pure and Applied Analysis, 2014, 13 (2) : 543-566. doi: 10.3934/cpaa.2014.13.543
##### References:
 [1] L. Ansini and L. Giacomelli, Doubly nonlinear thin-film equations in one space dimension, Arch. Rational Mech. Anal., 173 (2004), 89-131. 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), 337-368. [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), 1169-1174. [4] F. Bernis and A. Friedman, Higher-order nonlinear degenerate parabolic equations, J. Diff. Equations, 83 (1990), 179-206. 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), 217-234. 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), 1535-1564. [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), 85-123. 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), 625-661. 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), 417-440. [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), 175-200. [11] M. Boutat, S. Hilout, J. E. Rakotoson and J. M. Rakotoson, A generalized thin-film equation in multidimensional space, Nonlinear Analysis, 69 (2008), 1268-1286. 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), 101-104. 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), 171-178. [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), 1-82. 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), 551-571. 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, (). [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), 321-342. [18] R. Dal Passo, L. Giacomelli and G. Grün, A waiting time phenomenon for thin film equations, Ann. Scuola Norm. Sup. Pisa, 30 (2001), 437-463. [19] R. Dal Passo, L. Giacomelli and A. Shishkov, The thin film equation with nonlinear diffusion, Comm. Partial Differential Equations, 26 (2001), 1509-1557. [20] S. D. Èĭdel'man, Parabolic Systems, Translated from the Russian by Scripta Technica, London, North-Holland Publishing Co., Amsterdam, 1969. [21] A. Friedman, Partial Differential Equations,, Holt, (). [22] L. Giacomelli, A fourth-order degenerate parabolic equation describing thin viscous flows over an inclined plane, Applied Mathematics Letters, 12 (1999), 107-111. 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), 1181-1215. 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), 541-574. [25] G. Grün, "On Free Boundary Problems Arising in Thin Film Flow," Habilitation thesis, University of Bonn, 2001. [26] G. Grün, Droplet spreading under weak slippage: the waiting time phenomenon, Ann. I. H. Poincaré–AN, 21 (2004), 255-269. 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), 373-392. doi: 10.1017/S0022112090001616. [28] J. R. King, Two generalisations of the thin film equation, Math. Comput. Modelling, 34 (2001), 737-756. 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), 123524, 26 pp. Available from: http://dx.doi.org/10.1063/1.3272788. 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), 464-475. 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), 3443-3450. 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), 931-980. 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), 313-324. [34] C. Villani, A review of mathematical topics in collisional kinetic theory, in "Handbook of Mathematical Fluid Dynamics" (Vol. I), Amsterdam: North-Holland, (2002), 71-305.

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##### References:
 [1] L. Ansini and L. Giacomelli, Doubly nonlinear thin-film equations in one space dimension, Arch. Rational Mech. Anal., 173 (2004), 89-131. 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), 337-368. [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), 1169-1174. [4] F. Bernis and A. Friedman, Higher-order nonlinear degenerate parabolic equations, J. Diff. Equations, 83 (1990), 179-206. 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), 217-234. 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), 1535-1564. [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), 85-123. 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), 625-661. 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), 417-440. [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), 175-200. [11] M. Boutat, S. Hilout, J. E. Rakotoson and J. M. Rakotoson, A generalized thin-film equation in multidimensional space, Nonlinear Analysis, 69 (2008), 1268-1286. 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), 101-104. 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), 171-178. [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), 1-82. 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), 551-571. 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, (). [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), 321-342. [18] R. Dal Passo, L. Giacomelli and G. Grün, A waiting time phenomenon for thin film equations, Ann. Scuola Norm. Sup. Pisa, 30 (2001), 437-463. [19] R. Dal Passo, L. Giacomelli and A. Shishkov, The thin film equation with nonlinear diffusion, Comm. Partial Differential Equations, 26 (2001), 1509-1557. [20] S. D. Èĭdel'man, Parabolic Systems, Translated from the Russian by Scripta Technica, London, North-Holland Publishing Co., Amsterdam, 1969. [21] A. Friedman, Partial Differential Equations,, Holt, (). [22] L. Giacomelli, A fourth-order degenerate parabolic equation describing thin viscous flows over an inclined plane, Applied Mathematics Letters, 12 (1999), 107-111. 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), 1181-1215. 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), 541-574. [25] G. Grün, "On Free Boundary Problems Arising in Thin Film Flow," Habilitation thesis, University of Bonn, 2001. [26] G. Grün, Droplet spreading under weak slippage: the waiting time phenomenon, Ann. I. H. Poincaré–AN, 21 (2004), 255-269. 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), 373-392. doi: 10.1017/S0022112090001616. [28] J. R. King, Two generalisations of the thin film equation, Math. Comput. Modelling, 34 (2001), 737-756. 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), 123524, 26 pp. Available from: http://dx.doi.org/10.1063/1.3272788. 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), 464-475. 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), 3443-3450. 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), 931-980. 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), 313-324. [34] C. Villani, A review of mathematical topics in collisional kinetic theory, in "Handbook of Mathematical Fluid Dynamics" (Vol. I), Amsterdam: North-Holland, (2002), 71-305.
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