Czasopismo
Tytuł artykułu
Autorzy
Warianty tytułu
Języki publikacji
Abstrakty
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
Wydawca
Czasopismo
Rocznik
Tom
Numer
Opis fizyczny
Daty
wydano
2014-01-01
otrzymano
2014-01-16
zaakceptowano
2014-03-21
online
2014-05-17
Twórcy
autor
- RWTH Aachen University, Templergraben 55, 52056 Aachen, Germany, nanasandjo@iram.rwthaachen.de
autor
- Department of Mathematics and Statistical Sciences, Jackson State University, JSU Box 17610, 1400 J R Lynch Str., Jackson, MS 39217, USA, celestin.wafo_soh@jsums.edu
Bibliografia
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- [2] H. Amann, Dynamic theory of quasilinear parabolic equations- II. Reation-difision systems., Difierential Integral Equations, Vol.3, 1990, 13-75.
- [3] H. Amann, Linear and quasilinear parabolic problems. Vol. I, Monographs in Mathematics, Vol.89, Abstract linear theory, Birkhäuser Boston Inc.,Boston, MA, 1995.
- [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.
- [5] G. Dore, and A. Venni, On the closedness of the sum of two closed operators, Math. Z. 196, 2 (1987), 189-201.
- [6] C. M. Elliot, S. Zheng, On the Cahn Hilliard equation, Arch. Rational Mech. Anal. 96 (1986), 339-357.
- [7] H. Fujita, T. Kato, On the Navier-Stokes initial value problem., Arch. Rational Mech. Anal., 16 (1964), 269-315.[Crossref]
- [8] Y. Giga and T. Miyakawa , Solution in Lr of Navier-Stokes initial value problem, Arch. Rat. Mech. Anal., 89 (1985), 267-281.[Crossref]
- [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.
- [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.
- [11] T. Halicioglu, P. J. White, Structures of microclusters: an atomistic approach with three-body interactions, Surface Science, 106 (1981), 45-50.
- [12] T. Kato, Strong Lp solutions of the Navier-Stokes equations in Rm with applications, Math. Z. 187 (1984), 471-480.
- [13] T. Kato, Strong solutions of the Navier-Stokes equation in Morrey spaces, Bol. Soc. Bras. Mat. (N.S.) 22 (1992), 127-155.[Crossref]
- [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.
- [15] R. Lam, D. G. Vlachos, Multiscale model for epitaxial growth of films: growth mode transition, Phys. Rev. B, 64 (2001), 035401.
- [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.
- [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.
- [18] C. Melcher, Well-posedness for a class of nonlinear fourth-order difiusion equations, Preprint.
- [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.
- [20] C. Miao, Time-Space Estimates of Solutions to General Semilinear Parabolic Equations, Tokio J. Math., Vol.24. No.1, (2001), 246-276.
- [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.
- [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.
- [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.
- [24] A. Nana Sandjo, C. Wafo Soh and M. Wiegner, Solutions of a fourth-order parabolic equation modeling epitaxial thin film growth, Preprint.
- [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).
- [26] L. Nirenberg, On elliptic partial difierential equations, Annali della Scoula Norm. Sup. Pisa, 13 (1959), 115-162.
- [27] A. Pazy, Semigroups of linear operators and applications to partial difierential equations, Apllied Mathematical Sciences, 44. Springer-Verlag, New York-Berlin, 1983.
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- [29] F. B. Weisler, Local existence and nonexistence for semilinear parabolic equation in Lp, Indiana Univ. Math. J. 29 (1980), 219-230.
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Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_msds-2014-0003