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On the Limit of Solutions for a Reaction–Diffusion Equation Containing Fractional Laplacians

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Abstract

A kind of nonlocal reaction-diffusion equations on an unbounded domain containing a fractional Laplacian operator is analyzed. To be precise, we prove the convergence of solutions of the equation governed by the fractional Laplacian to the solutions of the classical equation governed by the standard Laplacian, when the fractional parameter grows to 1. The existence of global attractors is investigated as well. The novelty of this paper is concerned with the convergence of solutions when the fractional parameter varies, which, as far as the authors are aware, seems to be the first result of this kind of problems in the literature.

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References

  1. Abe, S., Thurner, S.: Anomalous diffusion in view of Einstein’s: theory of Brownian motion. Physica A 356(2005), 403–407 (1905)

    Google Scholar 

  2. Babin, A.V., Vishik, M.I.: Attractors of partial differential evolution equations in an unbounded domain. Proc. R. Soc. Edinb. Sect. A 116(3–4), 221–243 (1990)

    Article  MathSciNet  Google Scholar 

  3. Bezerra, F.D.M., Carvalho, A.N., Dlotko, T., Nascimento, M.J.D.: Fractional Schrödinger equation; solvability and connection with classical Schrödinger equation. J. Math. Anal. Appl. 457, 336–360 (2018)

    Article  MathSciNet  Google Scholar 

  4. Bezerra, F.D.M., Carvalho, A.N., Nascimento, M.J.D.: Fractional approximations of abstract semilinear parabolic problems. Discret. Contin. Dyn. Syst. Ser. B 25, 4221–4255 (2020)

    MathSciNet  Google Scholar 

  5. Bao, N., Caraballo, T., Tuan, N., Zhou, Y.: Existence and regularity results for terminal value problem for nonlinear fractional wave equations. Nonlinearity 34, 1448–1503 (2021)

    Article  MathSciNet  Google Scholar 

  6. Di Nezza, E., Patatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)

    Article  MathSciNet  Google Scholar 

  7. Dłotko, T.W., Wang, Y.J.: Critical Parabolic-Type problems, De Gruyter Series in Nonlinear Analysis and Applications, 34. De Gruyter, Berlin (2020)

    Google Scholar 

  8. Garroni, A., Müller, S.: A variational model for dislocations in the line tension limit. Arch. Ration. Mech. Anal. 181, 535–578 (2006)

    Article  MathSciNet  Google Scholar 

  9. Gal, C.G., Warma, M.: Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions. Discret. Contin. Dyn. Syst. 36, 1279–1319 (2016)

    Article  MathSciNet  Google Scholar 

  10. Gui, C., Zhao, M.: Traveling wave solutions of Allen-Cahn equation with a fractional Laplacian. Ann. Inst. H. Poincaré Anal. Non. Linéaire 32, 785–812 (2015)

    Article  MathSciNet  Google Scholar 

  11. Jara, M.: Nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps. Commun. Pure Appl. Math. 62, 198–214 (2009)

    Article  MathSciNet  Google Scholar 

  12. Kapustyan, O.V., Melnik, V.S., Valero, J., Yasinsky, V.V.: Global Attractors of Multi-valued Dynamical Systems and Evolution Equations without Uniqueness. Naukova Dumka, Kyiv (2008)

    Google Scholar 

  13. Koslowski, M., Cuitino, A.M., Ortiz, M.: A phasefield theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystal. J. Mech. Phys. Solids 50, 2597–2635 (2002)

    Article  MathSciNet  Google Scholar 

  14. Ladyzhenskaya, O.: Attractors for Semigroups and Evolution Equations. Cambridge University Press, Cambridge (1991)

    Book  Google Scholar 

  15. Lions, J.L.: Quelques Méthodes de Résolution des Problèmes aux Limites Non Lineaires. Dunod, Paris (1969)

    Google Scholar 

  16. Lu, H., Qi, J., Wang, B., Zhang, M.: Random attractors for non-autonomous fractional stochastic parabolic equations on unbounded domains. Discret. Contin. Dyn. Syst. 39, 683–706 (2019)

    Article  MathSciNet  Google Scholar 

  17. Martínez-Carracedo, C., Sanz Alix, M.: The Theory of Fractional Powers of Operators. North-Holland, Amsterdam (2001)

    Google Scholar 

  18. Mellet, A., Mischler, S., Mouhot, C.: Fractional diffusion limit for collisional kinetic equations. Arch. Ration. Mech. Anal. 199, 493–525 (2011)

    Article  MathSciNet  Google Scholar 

  19. Morillas, F., Valero, J.: Attractors for reaction-diffusion equations in \(\mathbb{R} ^{N}\) with continuous nonlinearity. Asymptot. Anal. 44, 111–130 (2005)

    MathSciNet  Google Scholar 

  20. Nec, Y., Nepomnyashchy, A.A., Golovin, A.A.: Front-type solutions of fractional Allen–Cahn equation. Physica D 237, 3237–3251 (2008)

    Article  MathSciNet  Google Scholar 

  21. Robinson, J.: Infinite-Dimensional Dynamical Systems. Cambridge University Press, Cambridge (2001)

    Book  Google Scholar 

  22. Schertzer, D., Larcheveque, M., Duan, J., Yanovsky, V.V., Lovejoy, S.: Fractional Fokker-Planck equation for nonlinear stochastic differential equations driven by non-Gaussian Lévy stable noises. J. Math. Phys. 42, 200–212 (2001)

    Article  MathSciNet  Google Scholar 

  23. Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York (1988)

    Book  Google Scholar 

  24. Tuan, N., Caraballo, T.: On initial and terminal value problems for fractional nonclassical diffusion equations. Proc. Am. Math. Soc. 149, 143–161 (2021)

    Article  MathSciNet  Google Scholar 

  25. Tuan, N., Caraballo, T.: On the initial value problem for a class of nonlinear biharmonic equation with time-fractional derivative. Proc. R. Soc. Edinb. Sect. A 152, 989–1031 (2022)

    Article  MathSciNet  Google Scholar 

  26. Vlahos, L., Isliker, H., Kominis, Y., Hizonidis, K.: Normal and Anomalous Diffusion: A tutorial. In: Order and Chaos (ed. T. Bountis), Vol. 10. Patras University Press, Patras (2008)

  27. Wang, B.: Attractors for reaction-diffusion equations in unbounded domains. Physica D 128, 41–52 (1999)

    Article  MathSciNet  Google Scholar 

  28. Wang, B.: Pullback attractors for non-autonomous reaction-diffusion equations on \(\mathbb{R} ^{n}\). Front. Math. China 4, 563–583 (2009)

    Article  MathSciNet  Google Scholar 

  29. Wang, B.: Asymptotic behavior of non-autonomous fractional stochastic reaction-diffusion equations. Nonlinear Anal. 158, 60–82 (2017)

    Article  MathSciNet  Google Scholar 

  30. Wang, B.: Dynamics of fractional stochastic reaction-diffusion equations on unbounded domains driven by nonlinear noise. J. Differ. Equ. 268, 1–59 (2019)

    Article  MathSciNet  Google Scholar 

  31. Wang, R., Li, Y., Wang, B.: Random dynamics of fractional nonclassical diffusion equations driven by colored noise. Discret. Contin. Dyn. Syst. 39, 4091–4126 (2019)

    Article  MathSciNet  Google Scholar 

  32. Wang, R., Shi, L., Wang, B.: Asymptotic behavior of fractional nonclassical diffusion equations driven by nonlinear colored noise on \(R^{N}\). Nonlinearity 32, 4524–4556 (2019)

    Article  MathSciNet  Google Scholar 

  33. Wang, R., Li, Y., Wang, B.: Bi-spatial pullback attractors of fractional nonclassical diffusion equations on unbounded domains with \((p, q)\)-growth nonlinearities. Appl. Math. Optim. 84, 425–461 (2021)

    Article  MathSciNet  Google Scholar 

  34. Xu, J., Zhang, Z., Caraballo, T.: Non-autonomous nonlocal partial differential equations with delay and memory. J. Differ. Equ. 270, 505–546 (2021)

    Article  MathSciNet  Google Scholar 

  35. Xu, J., Zhang, Z., Caraballo, T.: Mild solutions to time fractional stochastic \(2\)D-Stokes equations with bounded and unbounded delay. J. Dyn. Differ. Equ. 34, 583–603 (2022)

    Article  MathSciNet  Google Scholar 

  36. Xu, J., Caraballo, T., Valero, J.: Asymptotic behavior of a semilinear problem in heat conduction with long time memory and non-local diffusion. J. Differ. Equ. 327, 418–447 (2022)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We would like to thank the referee for the helpful remarks and suggestions which allow us to improve the presentation of our paper. This research was supported by the National Natural Science Foundation of China (No. 12301234), FEDER and the Spanish Ministerio de Ciencia e Innovación under projects PID2021-122991-NB-C21 and PID2019-108654GB-I00, and by the Generalitat Valenciana, project PROMETEO/2021/063.

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Authors and Affiliations

  1. Center for Nonlinear Studies, School of Mathematics, Northwest University, Xi’an, 710127, People’s Republic of China

    Jiaohui Xu

  2. Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, c/ Tarfia s/n, 41012, Sevilla, Spain

    Tomás Caraballo

  3. Department of Mathematics, Wenzhou University, Wenzhou, Zhejiang Province, 325035, People’s Republic of China

    Tomás Caraballo

  4. Centro de Investigación Operativa, Universidad Miguel Hernández de Elche, Avenida de la Universidad s/n, 03202, Elche, Spain

    José Valero

Authors
  1. Jiaohui Xu
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  2. Tomás Caraballo
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  3. José Valero
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Correspondence to Tomás Caraballo.

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Xu, J., Caraballo, T. & Valero, J. On the Limit of Solutions for a Reaction–Diffusion Equation Containing Fractional Laplacians. Appl Math Optim 89, 22 (2024). https://doi.org/10.1007/s00245-023-10090-6

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