Home Fractional diffusion-wave equations: Hidden regularity for weak solutions
Article
Licensed
Unlicensed Requires Authentication

Fractional diffusion-wave equations: Hidden regularity for weak solutions

  • Paola Loreti and Daniela Sforza EMAIL logo
Published/Copyright: August 23, 2021
Become an author with De Gruyter Brill
Author Information Explore this Subject
Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 24 Issue 4

Abstract

We prove a “hidden” regularity result for weak solutions of time fractional diffusion-wave equations where the Caputo fractional derivative is of order α ∈ (1, 2). To establish such result we analyse the regularity properties of the weak solutions in suitable interpolation spaces.

MSC 2010: Primary 35R11; Secondary 26A33; 35L05
Key Words and Phrases: Caputo fractional derivative; fractional diffusion-wave equations; hidden regularity; Riemann–Liouville fractional integral

References

[1] L. Bociu, J.-P. Zolésio, A pseudo-extractor approach to hidden boundary regularity for the wave equation with mixed boundary conditions. J. Differential Equations 259, No 11 (2015), 5688–5708.10.1016/j.jde.2015.07.006Search in Google Scholar

[2] H. Brezis, Analyse Fonctionnelle. Théorie et Applications. Masson, Paris (1983).Search in Google Scholar

[3] Y. Fujita, Integrodifferential equation which interpolates the heat equation and the wave equation. Osaka J. Math. 27, No 2 (1990), 309–321.Search in Google Scholar

[4] R. Gorenflo, F. Mainardi, Fractional calculus: Integral and differential equations of fractional order. In: Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, New York (1997), 223–276.10.1007/978-3-7091-2664-6_5Search in Google Scholar

[5] R. Gorenflo, M. Yamamoto, Operator theoretic treatment of linear Abel integral equations of first kind. Japan J. Indust. Appl. Math. 16, No 1 (1999), 137–161.10.1007/BF03167528Search in Google Scholar

[6] R. Gorenflo, Y. Luchko, M. Yamamoto, Time-fractional diffusion equation in the fractional Sobolev spaces. Fract. Calc. Appl. Anal. 18, No 3 (2015), 799–820; DOI: 10.1515/fca-2015-0048; https://www.degruyter.com/journal/key/FCA/18/3/html.Search in Google Scholar

[7] Y. Kian, M. Yamamoto, Well-posedness for weak and strong solutions of non-homogeneous initial boundary value problems for fractional diffusion equations. Fract. Calc. Appl. Anal. 24, No 1 (2021), 168–201; DOI: 10.1515/fca-2021-0008; https://www.degruyter.com/journal/key/FCA/24/1/html.Search in Google Scholar

[8] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006).Search in Google Scholar

[9] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method. Masson, Paris; John Wiley and Sons, Ltd., Chichester (1994).Search in Google Scholar

[10] I. Lasiecka, R. Triggiani, Regularity of hyperbolic equations under L2(0, T; L2(Γ))-Dirichlet boundary terms. Appl. Math. Optim. 10, No 3, (1983), 275–286.10.1007/BF01448390Search in Google Scholar

[11] J.-L. Lions, Hidden regularity in some nonlinear hyperbolic equations. Mat. Apl. Comput. 6, No 1 (1987), 7–15.Search in Google Scholar

[12] J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués I-II. Masson, Paris (1988).Search in Google Scholar

[13] P. Loreti, D. Sforza, Hidden regularity for wave equations with memory. Riv. Math. Univ. Parma (N.S.) 7, No 2 (2016), 391–405.Search in Google Scholar

[14] P. Loreti, D. Sforza, A semilinear integro-differential equation: global existence and hidden regularity. In: Trends in Control Theory and Partial Differential Equations, Springer, Cham (2019), 157–180.10.1007/978-3-030-17949-6_9Search in Google Scholar

[15] A. Lunardi, Interpolation Theory. Edizioni della Normale, Pisa (2018).10.1007/978-88-7642-638-4Search in Google Scholar

[16] Y. Luchko, F. Mainardi, Y. Povstenko, Propagation speed of the maximum of the fundamental solution to the fractional diffusion-wave equation. Comput. Math. Appl. 66, No 5 (2013), 774–784.10.1016/j.camwa.2013.01.005Search in Google Scholar

[17] F. Mainardi, The fundamental solutions for the fractional diffusion-wave equation. Appl. Math. Lett. 9, No 6 (1996), 23–28.10.1016/0893-9659(96)00089-4Search in Google Scholar

[18] M. Milla Miranda, L.A. Medeiros, Hidden regularity for semilinear hyperbolic partial differential equations. Ann. Fac. Sci. Toulouse Math. (5) 9, No 1 (1988), 103–120.10.5802/afst.651Search in Google Scholar

[19] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, Berlin (1983).10.1007/978-1-4612-5561-1Search in Google Scholar

[20] I. Podlubny, Fractional Differential Equations. Academic Press, San Diego (1999).Search in Google Scholar

[21] Y. Povstenko, Linear Fractional Diffusion–Wave Equation for Scientists and Engineers. Birkhäuser, Cham (2015).10.1007/978-3-319-17954-4Search in Google Scholar

[22] K. Sakamoto, M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J. Math. Anal. Appl. 382, No 1 (2011), 426–447.10.1016/j.jmaa.2011.04.058Search in Google Scholar

[23] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives. Gordon and Breach Science Publishers, Philadelphia (1993).Search in Google Scholar

[24] M. Stynes, Too much regularity may force too much uniqueness. Fract. Calc. Appl. Anal. 19, No 6 (2016), 1554–1562; DOI: 10.1515/fca-2016-0080; https://www.degruyter.com/journal/key/FCA/19/6/html.Search in Google Scholar
Received: 2020-09-25
Published Online: 2021-08-23
Published in Print: 2021-08-26

© 2021 Diogenes Co., Sofia

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. FCAA related news, events and books (FCAA–volume 24–4–2021)
  4. Research Paper
  5. Three representations of the fractional p-Laplacian: Semigroup, extension and Balakrishnan formulas
  6. Tutorial paper
  7. The bouncing ball and the Grünwald-Letnikov definition of fractional derivative
  8. Research Paper
  9. Fractional diffusion-wave equations: Hidden regularity for weak solutions
  10. Censored stable subordinators and fractional derivatives
  11. Variational methods to the p-Laplacian type nonlinear fractional order impulsive differential equations with Sturm-Liouville boundary-value problem
  12. Multivariable fractional-order PID tuning by iterative non-smooth static-dynamic H∞ synthesis
  13. Filter regularization method for a nonlinear Riesz-Feller space-fractional backward diffusion problem with temporally dependent thermal conductivity
  14. The rate of convergence on fractional power dissipative operator on compact manifolds
  15. Fractional Langevin type equations for white noise distributions
  16. Local existence and non-existence for a fractional reaction–diffusion equation in Lebesgue spaces
  17. Maximum principles and applications for fractional differential equations with operators involving Mittag-Leffler function
  18. The Crank-Nicolson type compact difference schemes for a loaded time-fractional Hallaire equation
  19. Robust fractional-order perfect control for non-full rank plants described in the Grünwald-Letnikov IMC framework
  20. Properties of the set of admissible “state control” pair for a class of fractional semilinear evolution control systems
https://doi.org/10.1515/fca-2021-0044
Keywords for this article
Caputo fractional derivative; fractional diffusion-wave equations; hidden regularity; Riemann–Liouville fractional integral

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. FCAA related news, events and books (FCAA–volume 24–4–2021)
  4. Research Paper
  5. Three representations of the fractional p-Laplacian: Semigroup, extension and Balakrishnan formulas
  6. Tutorial paper
  7. The bouncing ball and the Grünwald-Letnikov definition of fractional derivative
  8. Research Paper
  9. Fractional diffusion-wave equations: Hidden regularity for weak solutions
  10. Censored stable subordinators and fractional derivatives
  11. Variational methods to the p-Laplacian type nonlinear fractional order impulsive differential equations with Sturm-Liouville boundary-value problem
  12. Multivariable fractional-order PID tuning by iterative non-smooth static-dynamic H∞ synthesis
  13. Filter regularization method for a nonlinear Riesz-Feller space-fractional backward diffusion problem with temporally dependent thermal conductivity
  14. The rate of convergence on fractional power dissipative operator on compact manifolds
  15. Fractional Langevin type equations for white noise distributions
  16. Local existence and non-existence for a fractional reaction–diffusion equation in Lebesgue spaces
  17. Maximum principles and applications for fractional differential equations with operators involving Mittag-Leffler function
  18. The Crank-Nicolson type compact difference schemes for a loaded time-fractional Hallaire equation
  19. Robust fractional-order perfect control for non-full rank plants described in the Grünwald-Letnikov IMC framework
  20. Properties of the set of admissible “state control” pair for a class of fractional semilinear evolution control systems
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 16.11.2025 from https://www.degruyterbrill.com/document/doi/10.1515/fca-2021-0044/html?lang=en
Scroll to top button