Skip to main content
SpringerLink
Log in

Non-simultaneous blow-up for a system with local and non-local diffusion

  • Published:
Journal of Evolution Equations Aims and scope Submit manuscript

Abstract

We study the possibility of non-simultaneous blow-up for positive solutions of a coupled system of two semilinear equations,\(u_t = J*u-u+ u^\alpha v^p\), \(v_t =\Delta v+u^qv^\beta \), \(p, q, \alpha , \beta >0\) with homogeneous Dirichlet boundary conditions and positive initial data. We also give the blow-up rates for non-simultaneous blow-up.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data Availability Statement

No data were used for the research described in the article.

References

  1. F. Andreu–Vaillo, J. M. Mazón, J. D. Rossi, and J. J. Toledo-Melero, Nonlocal diffusion problems. Mathematical Surveys and Monographs, 165. American Mathematical Society, Providence, RI; Real Sociedad Matemática Española, Madrid, 2010. xvi+256 pp.

  2. X. Bai, Finite time blow-up for a reaction-diffusion system in bounded domain. Z. Angew. Math. Phys. 65 (2014), no. 1, 135-138.

    Article  MathSciNet  MATH  Google Scholar 

  3. M. Bogoya, and J. D. Rossi, Systems with local and nonlocal diffusions, mixed boundary conditions, and reaction terms. Abstr. Appl. Anal. 2018, Art. ID 3906431, 10 pp.

  4. R. Cantrell, C. Cosner, Y. Lou, and D. Ryan, Daniel, Evolutionary stability of ideal free dispersal strategies: a nonlocal dispersal model. Can. Appl. Math. Q. 20 (2012), no. 1, 15-38.

  5. H. Chen, Global existence and blow-up for a nonlinear reaction-diffusion system. J. Math. Anal. Appl. 212 (1997), no. 2, 481-492.

    Article  MathSciNet  MATH  Google Scholar 

  6. C. Cosner, J. Dávila, and S. Martínez, Evolutionary stability of ideal free nonlocal dispersal. J. Biol. Dyn. 6 (2012), no. 2, 395-405.

    Article  MathSciNet  MATH  Google Scholar 

  7. J. Dockery, V. Hutson, K. Mischaikow, M .Pernarowski The evolution of slow dispersal rates: a reaction diffusion model. J. Math. Biol. 37 (1998), no. 1, 61–83.

  8. A. de Pablo, J.L. Vázquez, Travelling waves and finite propagation in a reaction-diffusion equation. J. Differential Equations 93 (1991), no. 1, 19-61

    Article  MathSciNet  MATH  Google Scholar 

  9. M. Escobedo and M. A. Herrero, A semilinear parabolic system in a bounded domain. Ann. Mat. Pura Appl. 165 (1993), no. 4, 315–336.

    Article  MathSciNet  MATH  Google Scholar 

  10. M. Escobedo and H. A. Levine, Critical blowup and global existence numbers for a weakly coupled system of reaction-diffusion equations. Arch. Rational Mech. Anal. 129 (1995), no. 1, 47-100.

    Article  MathSciNet  MATH  Google Scholar 

  11. C. Y. Kao, Y. Lou, and W. Shen, Random dispersal vs. non-local dispersal. (English summary) Discrete Contin. Dyn. Syst. 26 (2010), no. 2, 551-596.

  12. C. Y. Kao, Y. Lou, and W. Shen, Evolution of mixed dispersal in periodic environments. (English summary) Discrete Contin. Dyn. Syst. Ser. B 17 (2012), no. 6, 2047-2072.

  13. A. Massaccesi and E. Valdinoci Is a nonlocal diffusion strategy convenient for biological populations in competition? J. Math. Biol. 74 (2017), no. 1-2, 113-147.

  14. R. Ferreira, Blow-up for a semilinear non-local diffusion system. Nonlinear Anal. 189 (2019), 111564, 12 pp.

  15. F. Quirósand J. D. Rossi, Non-simultaneous blow-up in a semilinear parabolic system, Z. Angew. Math. Phys. 52 (2001), no. 2, 342-346.

  16. M. Wang Global existence and finite time blow up for a reaction-diffusion system. Z. Angew. Math. Phys. 51 (2000), no. 1, 160-167.

Download references

Acknowledgements

L.M.D.P. was partially supported by the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No 777822, (Agencia Nacional de Promoción de la Investigación, el Desarrollo Tecnológico y la Innovación PICT-2018-3183, and PICT-2019-00985 and UBACYT 20020190100367. R.F. was supported by the Spanish project PID2020-116949GB-I00 and by Grupo de Investigación UCM 920894.

Author information

Authors and Affiliations

  1. CONICET and UTDT, Departamento de Matemáticas y Estadística, Universidad Torcuato Di Tella, Av. Figueroa Alcorta 7350 (C1428BCW), Buenos Aires, Argentina

    Leandro M. Del Pezzo

  2. Departamento de Matemática Aplicada, Fac. de C.C. Químicas, U. Complutense de Madrid, 28040, Madrid, Spain

    Raúl Ferreira

Authors
  1. Leandro M. Del Pezzo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Raúl Ferreira
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Leandro M. Del Pezzo.

Ethics declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pezzo, L.M.D., Ferreira, R. Non-simultaneous blow-up for a system with local and non-local diffusion. J. Evol. Equ. 23, 48 (2023). https://doi.org/10.1007/s00028-023-00896-w

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00028-023-00896-w

Keywords

Search

Navigation