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Nonautonomous Dynamical Systems

formerly Nonautonomous and Stochastic Dynamical Systems

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Space-Time Estimates of Mild Solutions of a Class of Higher-Order Semilinear Parabolic Equations in Lp

Albert N. Sandjo / Célestin Wafo Soh
  • Department of Mathematics and Statistical Sciences, Jackson State University, JSU Box 17610, 1400 J R Lynch Str., Jackson, MS 39217, USA
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Published Online: 2014-05-17 | DOI: https://doi.org/10.2478/msds-2014-0003

Abstract

We establish the well-posedness of boundary value problems for a family of nonlinear higherorder parabolic equations which comprises some models of epitaxial growth and thin film theory. In order to achieve this result, we provide a unified framework for constructing local mild solutions in C0([0, T]; Lp(Ω)) by introducing appropriate time-weighted Lebesgue norms inspired by a priori estimates of solutions. This framework allows us to obtain global existence of solutions under the proviso that initial data are reasonably small

Keywords : Epitaxy; Thin-film Equation; Scaling invariance; Lp − Lq Estimates; Analytic Semigroup; Kato’s Method; Mild Solution

References

  • [1] H. Amann, Nonhomogeneous Linear and quasiliear Elliptic and Parabolic Boundary Value Problem, Function Spaces, Differential Operators and Nonlinear Analysis. H. J. Schmeisser, H. Triebel (editors), Teubner, Stuttgart, Leipzig, 1993, 9-126.Google Scholar

  • [2] H. Amann, Dynamic theory of quasilinear parabolic equations- II. Reation-difision systems., Difierential Integral Equations, Vol.3, 1990, 13-75. Google Scholar

  • [3] H. Amann, Linear and quasilinear parabolic problems. Vol. I, Monographs in Mathematics, Vol.89, Abstract linear theory, Birkhäuser Boston Inc.,Boston, MA, 1995.Google Scholar

  • [4] G. Caristi and E. Mitidieri, Existence and nonexistence of globale solutions of higher-order parabolic problems with slow decay initial data, J. Math. Appl. 279 (2003), 710-722.Google Scholar

  • [5] G. Dore, and A. Venni, On the closedness of the sum of two closed operators, Math. Z. 196, 2 (1987), 189-201.Google Scholar

  • [6] C. M. Elliot, S. Zheng, On the Cahn Hilliard equation, Arch. Rational Mech. Anal. 96 (1986), 339-357.Google Scholar

  • [7] H. Fujita, T. Kato, On the Navier-Stokes initial value problem., Arch. Rational Mech. Anal., 16 (1964), 269-315.CrossrefGoogle Scholar

  • [8] Y. Giga and T. Miyakawa , Solution in Lr of Navier-Stokes initial value problem, Arch. Rat. Mech. Anal., 89 (1985), 267-281.CrossrefGoogle Scholar

  • [9] Y. Giga , Solutions for semilinear parabolic equations in Lp and regularity of weak solutions of the Navier-Stokes system, J. Difierential Equations 62 (1986), no. 2, 186-212.Google Scholar

  • [10] L. Golubovic, A. Levandovsky, D. Moldovan, Interface dynamics and far-from equilibrium phase transitions in multilayer epitaxial growth and erosion on crystal surfaces: continuum theory insights, East Asian Journal on Applied Mathematics, 4 (2011), 297-371.Google Scholar

  • [11] T. Halicioglu, P. J. White, Structures of microclusters: an atomistic approach with three-body interactions, Surface Science, 106 (1981), 45-50.Google Scholar

  • [12] T. Kato, Strong Lp solutions of the Navier-Stokes equations in Rm with applications, Math. Z. 187 (1984), 471-480.Google Scholar

  • [13] T. Kato, Strong solutions of the Navier-Stokes equation in Morrey spaces, Bol. Soc. Bras. Mat. (N.S.) 22 (1992), 127-155.CrossrefGoogle Scholar

  • [14] B. King, O. Stein, M. Winkler, A fourth-order parabolic eqaution modeling epitaxial thin film growth, J. Math. Anal. Appl. 286(2003), 459-490.Google Scholar

  • [15] R. Lam, D. G. Vlachos, Multiscale model for epitaxial growth of films: growth mode transition, Phys. Rev. B, 64 (2001), 035401.CrossrefGoogle Scholar

  • [16] N. H. de Leeuw, S. C. Parker, Surface structure and morphology of calcium carbonate polymorphs calcite, aragonite and vaterite: an atomistic approach, J. Phys. Chem. B, 102 (1998), 2914-2922.Google Scholar

  • [17] T. S. Lo, R. Kohn, A new approach to continuum modeling of epitaxial growth: slope selection, coarsening, and the role of the uphill current, Phisica D: Nonlinear Phenomena, 161 (2002), 237-257.Google Scholar

  • [18] C. Melcher, Well-posedness for a class of nonlinear fourth-order difiusion equations, Preprint.Google Scholar

  • [19] C. Miao, Weak Solution of class of nonlinear heat equation systems and application to the Navier-Stokes system, J. Difierential Equations, 61 (1986), 141-151.Google Scholar

  • [20] C. Miao, Time-Space Estimates of Solutions to General Semilinear Parabolic Equations, Tokio J. Math., Vol.24. No.1, (2001), 246-276.Google Scholar

  • [21] C. Miao, B. Zhang, Cauchy Problem for semilinear parabolic Equations in Besov spaces, Houston Journal of Mathematics, Vol.30, No.3 (2004), 829-878.Google Scholar

  • [22] C. Miao, B. Yuan, B. Zhang, Strong solution to the nonlinear heat equation in homogeneous Besov spaces, J. Nonlinear Analysis, 67 (2007), 1329-1343.Google Scholar

  • [23] C. Miao, B. Yuan, B. Zhang, Well-posedness of the Cauchy problem for the fractional power dissipative equations, J. Nonlinear Analysis, 68 (2008), 461-484.Google Scholar

  • [24] A. Nana Sandjo, C. Wafo Soh and M. Wiegner, Solutions of a fourth-order parabolic equation modeling epitaxial thin film growth, Preprint.Google Scholar

  • [25] A. Nana Sandjo, Solutions for fourth-order parabolic equation modeling epitaxial thin film growth, Dissertation, Fakultät für Mathematik, Informatik und Naturwissenschaften der RWTH Aachen University, Germany, (August 2011).Google Scholar

  • [26] L. Nirenberg, On elliptic partial difierential equations, Annali della Scoula Norm. Sup. Pisa, 13 (1959), 115-162.Google Scholar

  • [27] A. Pazy, Semigroups of linear operators and applications to partial difierential equations, Apllied Mathematical Sciences, 44. Springer-Verlag, New York-Berlin, 1983.Google Scholar

  • [28] H. B. Steward, Generation of analytic semigroup by strongly elliptic operators under general boundary conditions, Trans. Amer. Math. soc. 259 (1980), 299-310.Google Scholar

  • [29] F. B. Weisler, Local existence and nonexistence for semilinear parabolic equation in Lp, Indiana Univ. Math. J. 29 (1980), 219-230.Google Scholar

  • [30] F. B. Weisler, Semilinear evolution equations in Banach spaces, J. of Funct. Anal. 32 (1979), 277-296.Google Scholar

  • [31] M. Wiegner, The Navier-Stokes Equations-a Neverending Challenge?, Jber. d. Dt. Math.-Verein., 101 (1999), 1-25. Google Scholar

About the article

Received: 2014-01-16

Accepted: 2014-03-21

Published Online: 2014-05-17

Published in Print: 2014-01-01


Citation Information: Nonautonomous Dynamical Systems, ISSN (Online) 2353-0626, DOI: https://doi.org/10.2478/msds-2014-0003.

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© 2014 Albert N. Sandjo, Célestin Wafo Soh . This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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