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Liouville Property for Solutions of the Linearized Degenerate Thin Film Equation of Fourth Order in a Halfspace

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Abstract

We consider a boundary value problem in a half-space for a linear parabolic equation of fourth order with a degeneration on the boundary of the half-space. The equation under consideration is substantially a linearized thin film equation. We prove that, if the right hand side of the equation and the boundary condition are polynomials in the tangential variables and time, the same property has any solution of a power growth. It is shown also that the specified property does not apply to the normal variable. As an application, we present a theorem of uniqueness for the problem in the class of functions of power growth.

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References

  1. Edmunds D.E., Peletier L.A.: A Liouville theorem for degenerate elliptic equations. J. Lond. Math. Soc. 7(2), 95–100 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  2. Tianling J., Jingang X.: A Liouville theorem for solutions of degenerate Monge–Ampe’re equations. Commun. Partial Differ. Equ. 39(2), 306–320 (2014)

    Article  MATH  Google Scholar 

  3. Genggeng H.: A Liouville theorem of degenerate elliptic equation and its application. Discrete Contin. Dyn. Syst. 33(10), 4549–4566 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Diaz G.: A note on the Liouville method applied to elliptic eventually degenerate fully nonlinear equations governed by the Pucci operators and the Keller–Osserman condition. Math. Ann. 353(1), 145–159 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Eidelman, S.D., Malickaja, A.P.: Liouville theorems for a certain class of degenerate parabolic equations. In: Mathematics Collection, pp. 250–253. “Naukova Dumka”, Kiev (1976)

  6. Shapoval A.B.: Liouville’s theorem for a second-order elliptic equation with degenerate coefficients. Moscow Univ. Math. Bull. 53(2), 22–27 (1998)

    MathSciNet  Google Scholar 

  7. DiBenedetto E., Gianazza U., Vespri V.: Liouville-type theorems for certain degenerate and singular parabolic equations. C. R. Math. Acad. Sci. Paris 348(15–16), 873–877 (1998)

    MathSciNet  MATH  Google Scholar 

  8. Moschini L.: New Liouville theorems for linear second order degenerate elliptic equations in divergence form. Ann. Inst. H. Poincare’ Anal. Non Line’aire 22(1), 11–23 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kuz’menko Yu.T.: The Liouville theorem for degenerate elliptic and parabolic equations. Mat. Zametki. 29(3), 397–408 (1981)

    MathSciNet  MATH  Google Scholar 

  10. Kolodii I.M.: The Liouville theorem for generalized solutions of degenerate uasilinear parabolic equations. Differentsial’nye Uravneniya 21(5), 841–854 (1985)

    MathSciNet  Google Scholar 

  11. Degtyarev, S.P.: Classical solvability of multidimensional two-phase Stefan problem for degenerate parabolic equations and Schauder’s estimates for a degenerate parabolic problem with dynamic boundary conditions. Nonlinear Differ. Equ. Appl. (2007). doi:10.1007/s00030-014-0280-3

  12. Dominik, J.: On Uniqueness of weak solutions for the thin-film equation (2013). arXiv:1310.6222

  13. Knüpfer H.: Well-posedness for the Navier slip thin-film equation in the case of partial wetting. Commun. Pure Appl. Math. 64(9), 1263–1296 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Giacomelli L., Knüpfer H., Otto F.: Smooth zero-contact-angle solutions to a thin-film equation around the steady state. J. Differ. Equ. 245(6), 1454–1506 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Giacomelli L., Knüpfer H.: A free boundary problem of fourth order: classical solutions in weighted Hölder spaces. Commun. Partial Differ. Equ. 35(10–12), 2059–2091 (2010)

    Article  MATH  Google Scholar 

  16. Giacomelli L., Gnann M.V., Knüpfer H., Otto F.: Well-posedness for the Navier-slip thin-film equation in the case of complete wetting. J. Differ. Equ. 257(1), 15–81 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Giacomelli L., Gnann M.V., Otto F.: Regularity of source-type solutions to the thin-film equation with zero contact angle and mobility exponent between 3/2 and 3. Eur. J. Appl. Math. 24(5), 735–760 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Boutat M., Hilout S., Rakotoson J.-E., Rakotoson J.-M.: A generalized thin-film equation in multidimensional space. Nonlinear Anal. 69(4), 1268–1286 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Bertsch M., Giacomelli L., Karali G.: Thin-film equations with “partial wetting” energy: existence of weak solutions. Phys. D. 209(1–4), 17–27 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Dal Passo R., Garcke H., Grün G.: On a fourth-order degenerate parabolic equation: global entropy estimates, existence, and qualitative behavior of solutions. SIAM J. Math. Anal. 29(2), 321–342 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Liang B.: Mathematical analysis to a nonlinear fourth-order partial differential equation. Nonlinear Anal. 74(11), 3815–3828 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Bazalii B.V., Degtyarev S.P.: On classical solvability of the multidimensional Stefan problem for convective motion of a viscous incompressible fluid. Math. USSR Sb. 60(1), 1–17 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  23. Bizhanova G.I., Solonnikov V.A.: On problems with free boundaries for second-order parabolic equations. St. Petersburg Math. J. 12(6), 949–981 (2001)

    MathSciNet  MATH  Google Scholar 

  24. Giaquinta M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Mathematics Studies, vol. 105. Princeton University Press, Princeton (1983)

    Google Scholar 

  25. Andreucci D., Tedeev A.: Finite speed of propagation for the thin-film equation and other higher-order parabolic equations with general nonlinearity. Interfaces Free Bound 3(3), 233–264 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  26. Andreucci D., Tedeev A.: Universal bounds at the blow-up time for nonlinear parabolic equations. Adv. Differ. Equ. 10(1), 89–120 (2005)

    MathSciNet  MATH  Google Scholar 

  27. Degtyarev S.P., Tedeev A.F.: \({L_{1}-L_{\infty}}\) estimates for the solution of the Cauchy problem for an anisotropic degenerate parabolic equation with double nonlinearity and growing initial data. Sb. Math. 198(5–6), 639–660 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  28. Degtyarev S.P., Tedeev A.F.: Two-sided estimates for the support of a solution of the Cauchy problem for an anisotropic quasilinear degenerate equation. Ukr. Math. J. 58(11), 1673–1684 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  29. Degtyarev S.P., Tedeev A.F.: Estimates for the solution of the Cauchy problem with increasing initial data for a parabolic equation with anisotropic degeneration and double nonlinearity. Dokl. Math. 76(3), 824–827 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  30. Degtyarev S.P.: On conditions for the instantaneous compactification of the support of the solution and on sharp estimates for the support in the Cauchy problem for a parabolic equation with double nonlinearity and absorption. Sb. Math. 199(3–4), 511–538 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  31. Degtyarev S.P.: Instantaneous support shrinking phenomenon in the case of fast diffusion for a doubly nonlinear parabolic equation with absorption. Adv. Differ. Equ. 13(11–12), 1031–1050 (2008)

    MathSciNet  MATH  Google Scholar 

  32. Degtyarev S.P.: On the instantaneous shrinking of the support of a solution in the Cauchy problem for an anisotropic parabolic equation. Ukr. Math. J. 61(5), 747–763 (2009)

    Article  MathSciNet  Google Scholar 

  33. Degtyarev S.P.: The effect of nonhomogeneous absorption on the instantaneous shrinking of the support in the Cauchy problem for a quasilinear degenerate equation. Ukr. Math. Bull. 6(3), 335–366 (2009)

    MathSciNet  Google Scholar 

  34. Maz’ja V.G.: Sobolev Spaces. Springer Series in Soviet Mathematics. Springer, Berlin (1985)

    Google Scholar 

  35. Adams R., Fournier J.J.F.: Sobolev Spaces, 2nd edn. Pure and Applied Mathematics (Amsterdam), vol. 140. Elsevier Academic Press, Amsterdam (2003)

    Google Scholar 

  36. Kufner A., Persson L.-E.: Weighted Inequalities of Hardy Type. World Scientific Publishing Co. Inc., River Edge (2003)

    Book  MATH  Google Scholar 

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  1. Institute for Applied Mathematics and Mechanics, Ukrainian National, Academy of Sciences, Donetsk R. Luxemburg Str. 74, Donetsk, 83114, Ukraine

    S. P. Degtyarev

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Degtyarev, S.P. Liouville Property for Solutions of the Linearized Degenerate Thin Film Equation of Fourth Order in a Halfspace. Results. Math. 70, 137–161 (2016). https://doi.org/10.1007/s00025-015-0467-x

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  • DOI: https://doi.org/10.1007/s00025-015-0467-x

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