Osaka Journal of Mathematics

Longtime convergence for epitaxial growth model under Dirichlet conditions

Somayyeh Azizi, Gianluca Mola, and Atsushi Yagi

Full-text: Open access


This paper continues our study on the initial-boundary value problem for a semilinear parabolic equation of fourth order which has been presented by Johnson-Orme-Hunt-Graff-Sudijono-Sauder-Orr [12] to describe the large-scale features of a growing crystal surface under molecular beam epitaxy. In the preceding paper [1], we already constructed a dynamical system generated by the problem and verified that the dynamical system has a finite-dimensional attractor (especially, every trajectory has nonempty $\omega$-limit set) and admits a Lyapunov function (of the form (3.1)). This paper is then devoted to showing longtime convergence of trajectory. We shall prove that every trajectory converges to some stationary solution as $t \to \infty$. As a matter of fact, we have obtained in [10] the similar result for the equation but under the Neumann like boundary conditions $\frac{\partial u}{\partial n}=\frac\partial{\partial n}\varDelta u=0$ on the unknown function $u$. In this paper, we want as in [1] to handle the Dirichlet boundary conditions $u=\frac{\partial u}{\partial n}=0$, maybe physically more natural conditions than before.

Article information

Osaka J. Math., Volume 54, Number 4 (2017), 689-706.

First available in Project Euclid: 20 October 2017

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35B40: Asymptotic behavior of solutions 35K55: Nonlinear parabolic equations 74E15: Crystalline structure


Azizi, Somayyeh; Mola, Gianluca; Yagi, Atsushi. Longtime convergence for epitaxial growth model under Dirichlet conditions. Osaka J. Math. 54 (2017), no. 4, 689--706.

Export citation


  • S. Azizi and A. Yagi: Dynamical system for epitaxial growth model under Dirichlet conditions, Sci. Math. Jpn., accepted for publication.
  • A.V. Babin and M.I. Vishik: Attractors of Evolution Equations, North-Holland, Amsterdam, 1992.
  • H. Cartan: Cours de Calcul Différentiel, Hermann, Paris, 1967.
  • R. Chill: The Łojasiewicz-Simon gradient inequality on Hilbert spaces, Proc. 5th European-Maghrebian Workshop on Semigroup Theory, Evolution Equations and Applications, 2006, 25–36.
  • R. Dautray and J.L. Lions: Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 2, Springer-Verlag, Berlin, 1988.
  • G. Ehrlich and F.G. Hudda: Atomic view of surface self-diffusion: tungsten on tungsten, J. Chem. Phys. 44 (1966), 1039–1049.
  • H. Fujimura and A. Yagi: Dynamical system for BCF model describing crystal surface growth, Vestnik Chelyabinsk Univ. Ser. 3 Mat. Mekh. Inform. 10 (2008), 75–88.
  • H. Fujimura and A. Yagi: Asymptotic behavior of solutions for BCF model describing crystal surface growth , Int. Math. Forum 3 (2008), 1803–1812.
  • H. Fujimura and A. Yagi: Homogeneous stationary solution for BCF model describing crystal surface growth, Sci. Math. Jpn. 69 (2009), 295–302.
  • M. Graselli, G. Mola and A. Yagi: On the longtime behavior of solutions to a model for epitaxial growth, Osaka J. Math. 48 (2011), 987–1004.
  • P. Grisvard: Elliptic Problems in Nonsmooth Domains, Pitman, London, 1985.
  • M.D. Johnson, C. Orme, A.W. Hunt, D. Graff, J. Sudijono, L.M. Sauder and B.G. Orr: Stable and unstable growth in molecular beam epitaxy, Phys. Rev. Lett. 72 (1994), 116–119.
  • S. Łojasiewicz: Une propriété topologique des sous-ensembles analytiques réels, Colloques internationaux du C. N. R. S: Les équations aux dérivées partielles, Paris(1962), Editions du C. N. R. S., Paris 1963, pp. 87–89.
  • W.W. Mullins: Theory of thermal grooving, J. Applied Phys. 28 (1957), 333–339.
  • K. Osaki and A. Yagi: Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkc. Ekvac. 44 (2001), 441–469.
  • R.L. Schwoebel and E.J. Shipsey: Step motion on crystal surfaces, J. Appl. Phys. 37 (1966), 3682–3686.
  • H. Tanabe: Equations of Evolution, Iwanami Shoten, Tokyo, 1975 (in Japanese); English translation: Pitman, London, 1979.
  • H. Tanabe: Functional Analytic Methods for Partial Differential Equations, Marcel Dekker, New York, 1997.
  • R. Temam: Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd ed., Springer-Verlag, Berlin, 1997.
  • M. Uwaha: Study on Mechanism of Crystal Growth, Kyoritsu Publisher, Tokyo, 2002 (in Japanese).
  • A. Yagi: Abstract Parabolic Evolution Equations and their Applications, Springer, Berlin, 2010.