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Existence of Weak Solutions to Nonlocal PDEs With a Generalized Definition of Caputo Derivative

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Abstract

Some compactness criteria that are analogies of the Aubin–Lions lemma for the existence of weak solutions of nonlinear evolutionary PDEs play crucial roles for the existence of weak solutions to time-fractional PDEs. Based on this fact, in this paper, we consider the existence of weak solutions to a kind of partial differential equations with Caputo time-fractional differential operator of order \(\gamma \in (0,1)\) and fractional Laplacian operator \((-\Delta )^\alpha \), \(\alpha \in (0,1)\).

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References

  1. Ambrosetti, A., Prodi, G.: A Primer of Nonlinear Analysis, Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1993)

    Google Scholar 

  2. 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  MATH  Google Scholar 

  3. Caffarelli, L., Roquejoffre, J., Sire, Y.: Variational problems for free boundaries for the fractional Laplacian. J. Eur. Math. Soc. 12, 1151–1179 (2009)

    MathSciNet  MATH  Google Scholar 

  4. Coleman, B.D., Noll, W.: Foundations of linear viscoelasticity. Rev. Mod. Phys. 33, 239 (1961)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Diethelm, K.: The Analysis of Fractional Differential Equations. An Application-Oriented Exposition Using Differential Operators of Caputo Type Lecture Notes in Mathematics. Springer-Verlag, Berlin (2010)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Giorgi, C., Pata, V., Marzocchi, A.: Asymptotic behavior of a semilinear problem in heat conduction with memory. Nonlinear Differ. Equ. Appl. 5, 333–354 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gu, A.H., Li, D.S., Wang, B.X., Yang, H.: Regularity of random attractors for fractional stochastic reaction-diffusion equations on \({\mathbb{R} }^n\). J. Differ. Eqs. 264, 7094–7137 (2018)

    Article  MATH  Google Scholar 

  10. Guan, Q.Y.: Integration by parts formula for regional fractional Laplacian. Comm. Math. Phys. 266, 289–329 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, vol. 204. Elsevier Science, Amsterdam (2006)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Kubo, R.: The fluctuation-dissipation theorem. Rep. Progr. Phys. 29, 255 (1966)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Li, L., Liu, J.-G.: A generalized definition of Caputo derivatives and its application to fractional ODEs. SIAM J. Math. Anal. 50, 2867–2990 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  16. Li, Y.J., Wang, Y.J.: The existence and asymptotic behavior of solutions to fractional stochastic evolution equations with infinite delay. J. Differ. Eqs. 266, 3514–3558 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  17. Li, Y.J., Wang, Y.J., Deng, W.H.: Galerkin finite element approximations for stochastic space-time fractional wave equations. SIAM. J. Numer. Anal. 55, 3173–3202 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Liu, W., Röckner, M., Da Silva, J.L.: Quasi-linear (stochastic) partial differential equations with time-fractional derivatives. SIAM. J. Math. Anal. 50, 2588–2607 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  20. Piero, G., Deseri, L.: On the concepts of state and free energy in linear viscoelasticity. Arch. Ration. Mech. Anal. 138, 1–35 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ros-Oton, X., Serra, J.: The Dirichlet problem for the fractional Laplacian: regularity up to the boundary. J. Math. Pures Appl. 101, 275–302 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  23. Servadei, R., Valdinoci, E.: Variational methods for non-local operators of elliptic type. Discrete Contin. Dyn. Syst. 33, 2105–2137 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Servadei, R., Valdinoci, E.: On the spectrum of two different fractional operators. Proc. Roy. Soc. Edinburgh Sect. A 144, 831–855 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  29. Xu, J.H., Caraballo, T.: Long time behavior of stochastic nonlocal partial differential equations and Wong-Zakai approximations. SIAM J. Math. Anal. 54, 2792–2844 (2022)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  31. Xu, J.H., Zhang, Z.C., Caraballo, T.: Mild solutions to time fractional 2D-Stokes equations with bounded and unbounded delay. J. Dynam. Differ. Eqs. 34, 583–603 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  32. Xu, J.H., Caraballo, T., Valero, J.: Asymptotic behavior of nonlocal partial differential equations with long time memory. Discrete Cont. Dyn. Syst. Ser. S. 15, 3059–3078 (2022)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  34. Zwanzig, R.: Nonequilibrium Statistical Mechanics. Oxford University Press, Oxford (2001)

    MATH  Google Scholar 

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Acknowledgements

This research was supported by FEDER and the Spanish Ministerio de Ciencia e Innovación under projects PID2021-122991 NB-C21 and PGC2018-096540-B-I00, Junta de Andalucía (Spain) under project P18-FR-4509, and the Nature Science Foundation of Jiangsu Province (Grant No. BK20220233). The authors express their sincere thanks to the editors for their kind help and the anonymous reviewer for the careful reading of the paper, giving valuable comments and suggestions.

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Authors and Affiliations

  1. Center for Nonlinear Studies, School of Mathematics, Northwest University, Xi’an, 710127, P. R. 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, P. R. China

    Tomás Caraballo

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  1. Jiaohui Xu
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  2. Tomás Caraballo
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Correspondence to Jiaohui Xu.

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Xu, J., Caraballo, T. Existence of Weak Solutions to Nonlocal PDEs With a Generalized Definition of Caputo Derivative. Mediterr. J. Math. 20, 254 (2023). https://doi.org/10.1007/s00009-023-02429-8

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  • DOI: https://doi.org/10.1007/s00009-023-02429-8

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