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Global existence of solutions of the time fractional Cahn–Hilliard equation in \({\mathbb {R}}^3\)

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Abstract

Cauchy problem for the Caputo-type time fractional Cahn–Hilliard equation in \({\mathbb {R}}^3\) is examined. The local existence and uniqueness of mild solutions and strong solutions are obtained for the initial data \(u_0\) satisfying \(u_0-{\bar{u}}\in L^\infty ({\mathbb {R}}^3)\cap L^1({\mathbb {R}}^3)\), where \({\bar{u}}\) is an equilibrium constant. The local solutions are extended globally if \(u_0-{\bar{u}}\) is small in \(L^1({\mathbb {R}}^3)\). These results are consistent with those of the traditional Cahn–Hilliard equation such as the property of mass conservation. However, extra difficulties arise in dealing with the singularity of Mittag-Leffler operators and non-Markovian property in the Caputo-type time fractional problem.

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References

  1. B. Andrade, A.N. Carvalho, P.M. Carvalho-Neto, and P. Marín-Rubio, Semilinear fractional differential equations: global solutions, critical nonlinearities and comparison results, Topol. Methods Nonlinear Anal., 45(2) (2015) 439–467.

    Article  MathSciNet  MATH  Google Scholar 

  2. H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Springer, Berlin 2011.

    MATH  Google Scholar 

  3. J. Bricmont, A. Kupiainen, and J. Taskinen, Stability of Cahn–Hilliard fronts, Commun. Pure Appl. Math., 52 (1999), 839–871.

    Article  MathSciNet  MATH  Google Scholar 

  4. L. A. Caffarelli, and N. E. Muler, An \(L^{\infty }\) bound for solutions of the Cahn–Hilliard equation, Arch. Ration. Mech. Anal., 133 (1995), 129–144.

    MathSciNet  MATH  Google Scholar 

  5. J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258–267.

    Article  MATH  Google Scholar 

  6. R. Caponetto, G. Dongola, L. Fortuna and I. Petrá, Fractional order systems: modeling and control applications, World Scientific, Singapore, 2010.

    Book  Google Scholar 

  7. P. M. Carvalho-Neto, and G. Planas, Mild solutions to the time fractional Navier–Stokes equations in \({\mathbb{R}}^N\), J. Differ. Equ., 259 (2015), 2948–2980.

    MATH  Google Scholar 

  8. P. M. Carvalho-Neto, Fractional differential equations: a novel study of local and global solutions in Banach spaces, PhD thesis, Universidade de Sao Paulo, Sao Carlos, 2013.

    Google Scholar 

  9. S. D. Eidelman, and A. N. Kochubei, Cauchy problem for fractional diffusion equations, J. Differ. Equ., 199 (2004), 211–255.

    Article  MathSciNet  MATH  Google Scholar 

  10. K. J. Engel, R. Nagel, One-Parameter semigroups for linear evolution equations. Springer, Berlin, 2000.

    MATH  Google Scholar 

  11. W. G. Glockle, and T. F. Nonnenmacher, A fractional calculus approach to self-similar protein dynamics, Biophysical Journal, 68 (1995), 46–53.

    Article  Google Scholar 

  12. S. Guo, L. Mei, Z. Zhang, J. Chen, Y. He, and Y. Li, Finite difference/Hermite-Galerkin spectral method for multi-dimensional time-fractional nonlinear reaction-diffusion equation in unbounded domains, Appl. Math. Model., 70 (2019), 246–263.

    Article  MathSciNet  MATH  Google Scholar 

  13. O. S. Iyiola, and F. D. Zaman, A fractional diffusion equation model for cancer tumor, AIP Advances, 4, 107121 (2014), 1–16.

    Google Scholar 

  14. M. Janev, S. Pilipovic, T. Atanackovic, R. Obradovic, and N. Ralevic, Fully fractional anisotropic diffusion for image denoising, Math. Comput. Modell., 54 (2011), 729–741.

    Article  MathSciNet  MATH  Google Scholar 

  15. J. Kemppainen, J. Siljander, R. Zacher, Representation of solutions and large-time behavior for fully nonlocal diffusion equations, J. Differ. Equ., 263 (2017), 149–201.

    Article  MathSciNet  MATH  Google Scholar 

  16. T. A. M. Langlands, B. I. Henry, and S. L. Wearne, Fractional cable equation models for anomalous electrodiffusion in nerve cells: finite domain solutions, SIAM J. Appl. Math., 71 (2011), 1168–1203.

    Article  MathSciNet  MATH  Google Scholar 

  17. T. A. M. Langlands, B. I. Henry, and S. L. Wearne, Fractional cable equation models for anomalous electrodiffusion in nerve cells: infinite domain solutions, J. Math. Biol., 59 (2009), 761–808.

    Article  MathSciNet  MATH  Google Scholar 

  18. L. Li, and J. G. Liu, A generalized definition of caputo derivatives and its application to fractional ODEs, SIAM J. Math. Anal., 50 (2018), 2867–2900.

    Article  MathSciNet  MATH  Google Scholar 

  19. L. Li, and J. G. Liu, Some compactness criteria for weak solutions of time fractional PDEs, SIAM J. Math. Anal., 50 (2018), 3963–3995.

    Article  MathSciNet  MATH  Google Scholar 

  20. Y. Li, F. Liu, I. W. Turner, and T. Li, Time-fractional diffusion equation for signal smoothing, Appl. Math. Comput., 326 (2018), 108–116.

    MathSciNet  MATH  Google Scholar 

  21. L. Li, J. G. Liu, and L. Z. Wang, Cauchy problems for Keller–Segel type time-space fractional diffusion equation, J. Differ. Equ., 265 (2018), 1044–1096.

    Article  MathSciNet  MATH  Google Scholar 

  22. H. Liu, A. Cheng, H. Wang, and J. Zhao, Time-fractional Allen–Cahn and Cahn–Hilliard phase-field models and their numerical investigation, Comput. Math. Appl., 76 (2018), 1876–1892.

    Article  MathSciNet  MATH  Google Scholar 

  23. S. Q. Liu, F. Wang, and H. J. Zhao, Global existence and asymptotics of solutions of the Cahn–Hilliard equation, J. Differ. Equ., 238 (2007), 426–469.

    Article  MathSciNet  MATH  Google Scholar 

  24. F. Mainardi, On the initial value problem for the fractional diffusion-wave equation, Ser. Adv. Math. Appl. Sci., 23 (1994) 246–251.

    MathSciNet  Google Scholar 

  25. R. Metzler, E. Barkai, and J. Klafter, Anomalous diffusion and relaxation close to thermal equilibrium: A fractional Fokker-Planck equation approach, Physical Review Letters, 82 (1999), 3563–3567.

    Article  Google Scholar 

  26. R. Metzler, and J. Klafter, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A Math. Gen., 37 (2004), R161–R208.

    Article  MathSciNet  MATH  Google Scholar 

  27. C. X. Miao, B. Q. Yuan, B. Zhang, Well-posedness of the Cauchy problem for the fractional power dissipative equations, Nonlinear Analysis, 68 (2008), 461–484.

    Article  MathSciNet  MATH  Google Scholar 

  28. I. Podlubny, Fractional-order systems and \(PI^\lambda D^\mu \)-controllers, IEEE Trans. Automat. Control, 44 (1999), 208–214.

    MathSciNet  MATH  Google Scholar 

  29. J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics 87, Birkhäuser, Basel, 1993.

    Book  Google Scholar 

  30. J. Prüss, V. Vergara and R. Zacher, Well-posedness and long-time behaviour for the non-isothermal Cahn-Hilliard equation with memory, Discrete Contin. Dyn. Syst., 26 (2010), 625–647.

    Article  MathSciNet  MATH  Google Scholar 

  31. M. Raberto, E. Scalas, and F. Mainardi, Waiting-times and returns in high-frequency financial data: an empirical study, Phys. A, 314 (2002), 749–755.

    Article  MATH  Google Scholar 

  32. M. G. Sakar, O. Saldir, and F. Erdogan, An iterative approximation for time-fractional Cahn-Allen equation with reproducing kernel method, Comput. Appl. Math., 37 (2018), 5951–5964.

    Article  MathSciNet  MATH  Google Scholar 

  33. S. G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, 1993.

    MATH  Google Scholar 

  34. W. A. Strauss, Decay and asymptotic for \(u_{tt}-\Delta u=F(u)\), J. Funct. Anal., 2 (1968), 409–457.

    Google Scholar 

  35. T. Tang, H.J. Yu, and T. Zhou, On Energy Dissipation Theory and Numerical Stability for Time-Fractional Phase-Field Equations, SIAM J. Sci. Comput., 41(6) (2019), A3757–A3778.

    Article  MathSciNet  MATH  Google Scholar 

  36. N. K. Tripathi, S. Das, S. H. Ong, H. Jafari, and M. M. Al Qurashi, Solution of time-fractional Cahn-Hilliard equation with reaction term using homotopy analysis method, Advances in Mechanical Engineering, 9 (2017).

  37. R. N. Wang, D. H. Chen, T.J. Xiao, Abstract fractional Cauchy problems with almost sectorial operators, J. Differ. Equ., 252 (2012), 202–235.

    Article  MathSciNet  MATH  Google Scholar 

  38. H. C. Yaslan, New analytic solutions of the space-time fractional Cahn–Hilliard equations, Optik, 130 (2017), 990–995.

    Article  Google Scholar 

  39. J. Zhao, L. Chen, and H. Wang, On power law scaling dynamics for time-fractional phase field models during coarsening, Commun. Nonlinear Sci. Numer. Simul., 70 (2019), 257–270.

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The correspondence author thanks Prof. Keith Promislow for careful reading of the manuscript and insightful suggestions. All authors would like to express sincere thanks to the referee for patient and careful reading and give valuable suggestions and comments to improve the quality of this manuscript greatly.

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Authors and Affiliations

  1. College of Mathematics and Statistics, Shenzhen University, Shenzhen, 518060, China

    Hailong Ye, Qiang Liu & Zhi-Min Chen

  2. Shenzhen Key Laboratory of Advanced Machine Learing and Applications, Shenzhen University, Shenzhen, 518060, China

    Hailong Ye

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  1. Hailong Ye
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Correspondence to Qiang Liu.

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This work partially supported by research Grants: NSFC (Nos. 11701384, 11671155), NSF of GD (2020A1515010554) and Natural Science Foundation of SZU (No. 2017057)

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Ye, H., Liu, Q. & Chen, ZM. Global existence of solutions of the time fractional Cahn–Hilliard equation in \({\mathbb {R}}^3\). J. Evol. Equ. 21, 2377–2411 (2021). https://doi.org/10.1007/s00028-021-00687-1

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