Abstract
We prove the existence of the effect of time delay of propagation of the support of “strong” solutions of the Cauchy problem for an equation of thin films and establish exact conditions on the behavior of an initial function near the free boundary that guarantee the appearance of this effect.
Similar content being viewed by others
REFERENCES
C. Elliot and H. Garcke, “On the Cahn - Hilliard equation with degenerate mobility,” SIAM J. Math. Anal., 27, 404–423 (1996).
G. Grün, “Degenerate parabolic differential equations of fourth order and plasticity model with nonlocal hardening,” Z. Anal. Anwendungen., 14, 541–574 (1995).
F. Bernis, “Viscous flows, fourth order nonlinear degenerate parabolic equations and singular elliptic problems,” in: J. I. Diaz, M. A. Herrero, A. Linan, and J. L. Vasquez (editors), Free Boundary Problems: Theory and Applications, Pitman Res. Notes Math., 323, 40–56 (1995).
F. Bernis and A. Friedman, “Higher order nonlinear degenerate parabolic equations,” J. Different. Equat., 83, 179–206 (1990).
F. Bernis, “Finite rate of propagation and continuity of the interface for thin viscous flows,” Adv. Different. Equat., 1, No. 3, 337–368 (1996).
F. Bernis, “Finite rate of propagation for thin viscous flows when 2 ≤n < 3,” C. R. Acad. Sci., Ser. Math., 322, 1169–1174 (1996).
E. Beretta, M. Bertsch, and R. Dal Passo, “Nonnegative solutions of a fourth-order nonlinear degenerate parabolic equation,” Arch. Ration. Mech. Anal., 129, No. 2, 175–200 (1995).
A. L. Bertozzi and M. Pugh, “Long-wave instabilities and saturation in thin film equations,” Commun Pure App. Math., 51, 625–661 (1998).
R. Kersner and A. Shishkov, Existence of Free-Boundaries in Thin-Film Theory, Preprint No. 6, Institute of Applied Mathematics and Mechanics, Ukrainian Academy of Sciences, Donetsk (1996).
J. Hulshof and A. Shishkov, “The thin film equation with 2 < n < 3: Finite rate of propagation in terms of the L1-norm,” Adv. Different. Equat., 3, 625–642 (1998).
M. Bertsch, R. Dal Passo, H. Garcke, and G. Grün, “The thin viscous flow equation in higher space dimension,” Adv. Different. Equat., 3, 417–440 (1998).
R. Dal Passo, L. Giacomelli, and A. Shishkov, “The thin film equation with nonlinear diffusion,” Communs Part. Different. Equat., 26, No. 9, 10, 1509–1557 (2001).
A. L. Bertozzi and M. 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, 1535–1564 (1994).
L. K. Martinson and K. B. Pavlov, “On the problem of space localization of heat perturbations in the theory of nonlinear heat conduction,” Zh. Vych. Mat. Mat. Fiz., 12, 261–268 (1972).
A. S. Kalashnikov, “Propagation of perturbations in nonlinear media with absorption,” Zh. Vych. Mat. Mat. Fiz., 14, 70–85 (1974).
A. S. Kalashnikov, “On the differential properties of generalized solutions of equations of the type of nonlinear filtration,” Vestn. Moscow Univ., 29, 48–53 (1974).
J. I. Diaz and M. A. Herrero, “Estimates on the support of the solutions of some nonlinear elliptic and parabolic problems,” Proc. Roy. Soc. Edinburgh A, 89, 249–258 (1981).
M. Bertsch, T. Nambu, and L. A. Peletier, “Decay of solutions of a degenerate nonlinear diffusion equations,” Nonlin. Anal., 6, 539–554 (1982).
B. F. Knerr, “The porous medium equation in one dimension,” Trans. Amer. Math. Soc., 234, 381–415 (1977).
S. N. Antontsev, “On the localization of solutions of nonlinear degenerate elliptic and parabolic equations,” Dokl. Akad. Nauk SSSR, 260, No. 6, 1289–1293 (1981).
J. I. Diaz and L. Veron, “Local vanishing properties of solutions of elliptic and parabolic equations,” Trans. Amer. Math. Soc., 290, No. 2, 787–814 (1985).
S. N. Antontsev and S. I. Shmarev, “Local energy method and vanishing of weak solutions of nonlinear parabolic equations,” Dokl. Akad. Nauk SSSR, 318, No.4, 777–781 (1991).
S. N. Antontsev, J. I. Diaz, and S. I. Shmarev, “New results of the character of localization of solutions of elliptic and parabolic equations,”Int. Ser. Numer. Math., 106, 59–66 (1992).
S. N. Antontsev, J. I. Diaz, and S. I. Shmarev, “The support shrinking properties for local solutions of quasilinear parabolic equations with strong absorption term,” Ann. Fac. Sci. Toulouse, 4, No. 1, 3–19 (1995).
A. E. Shishkov, “Dynamics of the geometry of a support of a generalized solution of a quasilinear high-order differential parabolic equation,” Differents. Uravn., 29, No. 3, 537–547 (1993).
A. E. Shishkov and A. G. Shchelkov, “Dynamics of supports of energy solutions of mixed problems for quasilinear differential parabolic equations of arbitrary order,” Izv. Ros. Akad. Nauk, 62, No. 3, 175–200 (1998).
G. Stampacchia, “Equations elliptiques du secont order a coefficients discontinues,” in: Sem. Math. Super. (Ete, 1965), 0Vol. 16, Les Presses de L'Universite de Montreal, Que (1966), p. 326.
R. Dal Passo, L. Giacomelli, and G. Grün, “A waiting time phenomenon for thin film equations,” Ann. Scuola Norm. Super. Pisa, Ser. IV, 30, No. 2, 437–464 (2001).
J. Simon, “Compact sets in the space L p0, T; B,” Ann. Math. Pura Appl., 146, 65–96 (1988).
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, No. 2, 321–342 (1998).
D. A. Sapronov and A. E. Shishkov, “Propagation of perturbations in quasilinear multidimensional parabolic equations with convective term,” Ukr. Mat. Zh., 53, No. 7, 953–969(2001).
L. Nirenberg, “On elliptic partial differential equations,” Ann. Scuola Norm. Super., 20, 733–737 (1996).
F. Bernis, “Finite rate of propagation and asymptotic rates for some nonlinear higher order parabolic equation with absorption,”Proc. Roy. Soc. Edinburgh A, 104, 1–19 (1986).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Taranets, R.M., Shishkov, A.E. Effect of Time Delay of Support Propagation in Equations of Thin Films. Ukrainian Mathematical Journal 55, 1131–1152 (2003). https://doi.org/10.1023/B:UKMA.0000010611.77537.3c
Issue Date:
DOI: https://doi.org/10.1023/B:UKMA.0000010611.77537.3c