Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Solvability for a reaction-diffusion system modeling biological transportation network

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

The aim of this paper is to investigate the initial-boundary value problem of a possibly degenerate reaction-diffusion system over \(\Omega \subset \mathbb {R}^n\) with \(n\ge 1\) of the following form

$$\begin{aligned} \left\{ \begin{aligned}&\partial _tm_i-\kappa \Delta m_i+|m_i|^{\gamma -2}m_i=(\partial _{x_i}p)^2,\\&-\nabla \cdot [\textbf{m}\nabla p]=S, \end{aligned} \right. \end{aligned}$$

with \(\textbf{m}=\textrm{diag} (m_1,\cdots ,m_n)\), the diffusivity \(\kappa >0\), the metabolic exponent \(\gamma \ge 2\) and the given function S. When \(\kappa =0\), this system was introduced by Haskovec, Kreusser and Markowich as a continuous version of the discrete Hu-Cai model for biological transport networks. In this work, our result asserts that whenever the random fluctuations of the conductance in the medium were considered, i.e., \(\kappa >0\), then for general large data the corresponding initial-boundary value problem possesses a global weak solution.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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 new data were created in this article.

References

  1. Aceves-Sanchez, P., Aymard, B., Peurichard, D., Kennel, P., Lorsignol, A., Plouraboué, F., Casteilla, L., Degond, P.: A new model for the emergence of blood capillary networks. Netw. Heterog. Media 16(1), 91–138 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  2. Albi, G., Artina, M., Foransier, M., Markowich, P.: Biological transportation networks: modeling and simulation. Anal. Appl. (Singap.) 14(1), 185–206 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Albi, G., Burger, M., Haskovec, J., Markowich, P., Schlottbom, M.: Continuum modeling of biological network formation. In: Bellomo, N., Degond, P., Tamdor, T. (eds.) Active Particles. Modeling and Simulation in Science and Technology, vol. I, pp. 1–48. Birkhäuser, Boston (2017)

    Google Scholar 

  4. Haskovec, J., Markowich, P., Perthame, B.: Mathematical analysis of a PDE system for biological network formation. Comm. Partial Diffr. Equ. 40(5), 918–956 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Haskovec, J., Kreusser, L., Markowich, P.: ODE and PDE based modeling of biological transportation networks. Commun. Math. Sci. 17(5), 1235–1256 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  6. Haskovec, J., Kreusser, L., Markowich, P.: Rigorous continuum limit for the discrete network formation problem. Comm. Partial Diffr. Equ. 44(11), 1159–1185 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  7. J. Haskovec, H., Jönsson, L., Kreusser, P., Markowich.: Auxin transport model for leaf venation, Proc. A, 475(2231) : paper20190015 (2019)

  8. Hu, D., Cai, D.: Adaptation and optimization of biological transport networks. Phys. Rev. Lett. 111(13), 138701 (2013)

    Article  MATH  Google Scholar 

  9. Hu, D., Cai, D.: An optimization principle for initiation and adaptation of biological transport networks. Commun. Math. Sci. 17(5), 1427–1436 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  10. Li, B.: Boundedness and long time behavior in a parabolic-elliptic system arising from biological transport networks. Adv. Nonlinear Anal. 13, 20240041 (2024)

    Article  MathSciNet  MATH  Google Scholar 

  11. Li, B., Li, X.: A cross-diffusive evolution system arising from biological transport networks. Commun. Nonlinear Sci. Numer. Simulat. 92(1), 105465 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  12. Li, B., Li, Y.: Blow-up criterion of classical solutions for a parabolic-elliptic system in space dimension 3. Proc. Am. Math. Soc. 148, 2565–2578 (2020)

    Article  MATH  Google Scholar 

  13. Li, Y., Hu, D.: Optimisation of biological transport networks. East Asian J. Appl. Math. 12(1), 72–95 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  14. Simon, J.: Compact sets in the space \(L^p(0; T;B)\). Ann. Mat. Pura Appl. 146(1), 65–96 (1996)

    Article  MATH  Google Scholar 

  15. Winkler, M.: Large-data global generalized solutions in a chemotaxis system with tensor-valued sensitivities. SIAM J. Math. Anal. 47(4), 3092–3115 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Winkler, M.: A quantitative strong parabolic maximum principle and application to a taxis-type migration-consumption model involving signal-dependent degenerate diffusion. Ann. Inst. H. Poincaré Anal. Non Linéaire 41, 95–127 (2024)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are very grateful to the referee for the detailed comments and valuable suggestions, which greatly improved the manuscript.

Funding

The research of BL is supported by Natural Science Foundation of Ningbo Municipality (No. 2022J147).

Author information

Authors and Affiliations

  1. School of Statistics and Data Science, Ningbo University of Technology, Ningbo, 315211, People’s Republic of China

    Bin Li & Zhi Wang

Authors
  1. Bin Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhi Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed equally to this work.

Corresponding author

Correspondence to Zhi Wang.

Ethics declarations

Conflict of interest

The authors declare that they have no Conflict of interest.

Consent for publication

The authors agree to publish.

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

Li, B., Wang, Z. Solvability for a reaction-diffusion system modeling biological transportation network. Z. Angew. Math. Phys. 75, 204 (2024). https://doi.org/10.1007/s00033-024-02349-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-024-02349-x

Keywords

MSC:

Search

Navigation