Skip to main content
SpringerLink
Log in

Singular Traveling Waves and Non-linear Reaction-Diffusion Equations

  • Chapter
  • First Online:
Computational Mathematics, Numerical Analysis and Applications

Part of the book series: SEMA SIMAI Springer Series ((SEMA SIMAI,volume 13))

Abstract

We review some recent results on singular traveling waves arising as solutions to reaction-diffusion equations combining flux saturation mechanisms and porous media type terms. These can be regarded as toy models in connection with some difficulties arising on the mathematical modelization of several scenarios in Developmental Biology, exemplified by pattern formation in the neural tube of chick’s embryo.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 99.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 129.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 129.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Andreu, F., Caselles, V., Mazón, J.M.: A strongly degenerate quasilinear elliptic equation. Nonlinear Anal. 61, 637–669 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Andreu, F., Caselles, V., Mazón, J.M.: The cauchy problem for a strongly degenerate quasilinear equation. J. Eur. Math. Soc. (JEMS) 7, 361–393 (2005)

    Google Scholar 

  3. Andreu, F., Caselles, V., Mazón, J.M., Moll, S.: Finite propagation speed for limited flux diffusion equations. Arch. Ration. Mech. Anal. 182, 269–297 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Andreu, F., Calvo, J., Mazón, J.M., Soler, J.: On a nonlinear flux–limited equation arising in the transport of morphogens. J. Differ. Equ. 252, 5763–5813 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brenier, Y.: Extended Monge-Kantorovich theory. In: Caffarelli, L.A., Salsa, S. (eds.) Lecture Notes in Mathematics, vol. 1813, pp. 91–122. Springer, New York (2003)

    Google Scholar 

  6. Calvo, J.: Analysis of a class of diffusion equations with a saturation mechanism. SIAM J. Math. Anal. 47, 2917–2951 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Calvo, J., Mazón, J.M., Soler, J., Verbeni, M.: Qualitative properties of the solutions of a nonlinear flux-limited equation arising in the transport of morphogens. Math. Models Methods Appl. Sci. 21, 893–937 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Calvo, J., Campos, J., Caselles, V., Sánchez, O., Soler, J.: Qualitative behavior for fluz-saturated mechanisms: traveling waves, waiting times and smoothing effects. EMS Surv. Math. Sci. 2, 2917–2951 (2015)

    Article  Google Scholar 

  9. Calvo, J., Campos, J., Caselles, V., Sánchez, O., Soler, J.: Pattern formation in a flux limited reaction-diffusion equation of porous media type. Invent. Math. 206, 57–108 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. Campos, J., Soler, J.: Qualitative behavior and traveling waves for flux-saturated porous media equations arising in optimal mass transportation. Nonlinear Anal. 137, 266–290 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Campos, J., Guerrero, P., Sánchez, O., Soler, J.: On the analysis of traveling waves to a nonlinear flux limited reaction-diffusion equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 30, 141–155 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Caselles, V.: On the entropy conditions for some flux limited diffusion equations. J. Differ. Equ. 250, 3311–3348 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Crick, F.: Diffusion in embryogenesis. Nature 40, 561–563 (1970)

    Article  Google Scholar 

  14. Dessaud, E., Yang, L.L., Hill, K., Cox., B., Ulloa, F., Ribeiro, A., Mynett, A., Novitch, B.G., Briscoe, J.: Interpretation of the sonic hedgehog morphogen gradient by a temporal adaptation mechanism. Nature 450, 717–720 (2007)

    Article  Google Scholar 

  15. Fisher, R.A.: The wave of advance of advantageous genes. Ann. Eugen. 7, 335–369 (1037)

    Google Scholar 

  16. Gilding, B.H., Kersner, R.: Traveling Waves in Nonlinear Diffusion-Convection-Reaction. Birkhäuser Verlag, Basel (2004)

    Book  MATH  Google Scholar 

  17. Kolmogoroff, A.N., Petrovsky, I.G., Piscounoff, N.S., Étude de l’equation de la diffusion avec croissance de la quantité de matiére et son application á un probléme biologique. Bull. Univ. de Etatá Moscou Ser. Int. A 1, 1–26 (1937)

    MATH  Google Scholar 

  18. Levermore, C.D., Pomraning, G.C.: A flux-limited diffusion theory. Astrophys. J. 248, 321–334 (1981)

    Article  Google Scholar 

  19. Meinhardt, H.: Space-dependent cell determination under the control of a morphogen gradient. J. Theor. Biol. 74, 307–321 (1978)

    Article  Google Scholar 

  20. Newman, W.I.: Some exact solutions to a non-linear diffusion problem in population genetics and combustion. J. Theor. Biol. 85, 325–334 (1980)

    Article  Google Scholar 

  21. Rosenau, P.: Tempered diffusion: a transport process with propagating front and inertial delay. Phys. Rev. A 46, 7371–7374 (1992)

    Article  Google Scholar 

  22. Saha, K., Schaffer, D.V.: Signal dynamics in sonic hedgehog tissue patterning. Development 133, 889–900 (2006)

    Article  Google Scholar 

  23. Sánchez, O., Calvo, J., Ibáñez, C., Guerrero, I., Soler, J.: Modeling hedgehog signaling through flux-saturated mechanisms. In: Riobo, N.A. (ed.) Methods in Molecular Biology, vol. 1322, pp. 19–33. Springer, New York (2015)

    Google Scholar 

  24. Turing, A.M.: The chemical basis of Morphogenesis. Philos. Trans. R. Soc. Lond. Ser. B Biol. Sci. 237, 37–72 (1952)

    Article  MathSciNet  Google Scholar 

  25. Vazquez, J.L.: The Porous Medium Equation: Mathematical Theory. Oxford University Press, Oxford (2007)

    MATH  Google Scholar 

  26. Verbeni, M., Sánchez, O., Mollica, E., Siegl–Cachedenier, I., Carleton, A., Guerrero, I., Ruiz i Altaba, A., Soler, J.: Morphogenetic action through flux-limited spreading. Phys. Life Rev. 10, 457–475 (2013)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Facultad de Ciencias, Universidad de Granada, Campus de Fuentenueva, 18071, Granada, Spain

    Juan Calvo

Authors
  1. Juan Calvo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Juan Calvo .

Editor information

Editors and Affiliations

  1. Matemáticas, Universidad de Oviedo , Gijón, Asturias, Spain

    Mariano Mateos

  2. Matemáticas, Universidad de Oviedo , Gijón, Asturias, Spain

    Pedro Alonso

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Calvo, J. (2017). Singular Traveling Waves and Non-linear Reaction-Diffusion Equations. In: Mateos, M., Alonso, P. (eds) Computational Mathematics, Numerical Analysis and Applications. SEMA SIMAI Springer Series, vol 13. Springer, Cham. https://doi.org/10.1007/978-3-319-49631-3_5

Download citation

Publish with us

Policies and ethics

Search

Navigation