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On the longtime behavior of solutions to a model for epitaxial growth

On the longtime behavior of solutions to a model for epitaxial growth. Atsushi Yagi Hideaki Fujimura Jian Luca Mola Maurizio Grasselli. Model Equation. Johnson-Orme-Hunt-Graff-Sudijono-Sauder-Orr, Phys. Rev. Lett. (1994). Surface Diffusion. Simplification due to W.W. Mullins (1957).

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On the longtime behavior of solutions to a model for epitaxial growth

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  1. On the longtime behavior of solutions to a model forepitaxial growth Atsushi Yagi Hideaki Fujimura Jian Luca Mola Maurizio Grasselli

  2. Model Equation Johnson-Orme-Hunt-Graff-Sudijono-Sauder-Orr, Phys. Rev. Lett.(1994)

  3. Surface Diffusion Simplification due to W.W. Mullins (1957)

  4. Schwoebel Effect Schwoebel障壁 テラス 原子 ステップ Negative diffusion due to M.D.Johnson et al. (1994)

  5. Epitaxial Growth Model Our problem is:

  6. Analytical Results • Fundamentals • Convergence as t   • Finite dimensional Attractors • Homogeneous Stationary Solutions

  7. Abstract Formulation Abstract parabolic semi-linear evolution equation:

  8. Global Solutions and Dynamical System Global existence:

  9. Lyapunov Function Therefore, (u) becomes aLyapunov function

  10. S(t)u0 ū as t  Simon-Łojasiewicz theory:

  11. Finite Dimensional Attractors Eden, Foias, Nicolaenko,Temam (1994) introduced: B1 h B2

  12. Fractal Dimension Mañé-Hölder type embedding:

  13. Exponential attractors for (E) (S(t),Hm1(Ω),L2,m(Ω)) has exponential attractors.

  14. Stationary Solutions Linearized principle:

  15. Stability-Instability of 0 We have the estimate dF(M) N.

  16. Summary • (EGM) generates a dynamical system (S(t),H1(),L2()). • u0H1(), S(t)u0 ū as t , where ū is a stationary solution of (EGM). • The set {stationary solutions}  M (finite dimensional attractor). • If m < a1, the homogeneous stationary solutions are stable; otherwise, unstable.

  17. Problems • How is the structure of stationary solutions? • What is a profile of stationary solution?

  18. Thank You for Your Kind Attention

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