Startseite Local existence and non-existence for a fractional reaction–diffusion equation in Lebesgue spaces
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Local existence and non-existence for a fractional reaction–diffusion equation in Lebesgue spaces

  • Ricardo Castillo , Miguel Loayza und Arlúcio Viana EMAIL logo
Veröffentlicht/Copyright: 23. August 2021
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Fractional Calculus and Applied Analysis
Aus der Zeitschrift Fractional Calculus and Applied Analysis Band 24 Heft 4

Abstract

We consider the following fractional reaction-diffusion equation

ut(t)+∂t∫0tgα(s)Au(t−s)ds=tγf(u),

where gα(t) = tα−1/Γ(α) (0 < α < 1), f ∈ C([0, ∞)) is a non-decreasing function, γ > −1, and A is an elliptic operator whose fundamental solution of its associated parabolic equation has Gaussian lower and upper bounds. We characterize the behavior of the functions f so that the above fractional reaction-diffusion equation has a bounded local solution in Lr(Ω), for non-negative initial data u0 ∈ Lr(Ω), when r > 1 and Ω ⊂ ℝN is either a smooth bounded domain or the whole space ℝN. The case r = 1 is also studied.

MSC 2010: Primary 35R11; 35A01; Secondary 35K57; 35K58; 35B09; 35B33
Key Words and Phrases: semilinear fractional partial differential equations; fractional reaction–diffusion equations; local existence; non-existence

Acknowledgments

We are grateful for the reviewers’ time and suggestions. Parts of this work were developed while the authors had opportunities to make short visits one to another and are grateful for the hospitality of the hosting institution. Viana and Castillo visited Loayza at UFPE by October and November 2019, respectively; Loayza visited Viana at UFS by March 2020. This work was partially supported by CAPES-PRINT, 88887.311962/2018-00. Viana is partially supported CNPq under grant 408194/2018-9. Castillo is supported by Bío-Bío University under grant 2020139IF/R.

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Received: 2020-06-12
Revised: 2021-06-19
Published Online: 2021-08-23
Published in Print: 2021-08-26

© 2021 Diogenes Co., Sofia

Artikel in diesem Heft

  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-2021-0051
Schlagwörter für diesen Artikel
semilinear fractional partial differential equations; fractional reaction–diffusion equations; local existence; non-existence

Artikel in diesem Heft

  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
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