A variational approach to bifurcation in reaction-diffusion systems with Signorini type boundary conditions

Jamol I. Baltaev; Milan Kučera; Martin Väth

Applications of Mathematics (2012)

  • Volume: 57, Issue: 2, page 143-165
  • ISSN: 0862-7940

Abstract

top
We consider a simple reaction-diffusion system exhibiting Turing's diffusion driven instability if supplemented with classical homogeneous mixed boundary conditions. We consider the case when the Neumann boundary condition is replaced by a unilateral condition of Signorini type on a part of the boundary and show the existence and location of bifurcation of stationary spatially non-homogeneous solutions. The nonsymmetric problem is reformulated as a single variational inequality with a potential operator, and a variational approach is used in a certain non-direct way.

How to cite

top

Baltaev, Jamol I., Kučera, Milan, and Väth, Martin. "A variational approach to bifurcation in reaction-diffusion systems with Signorini type boundary conditions." Applications of Mathematics 57.2 (2012): 143-165. <http://eudml.org/doc/246145>.

@article{Baltaev2012,
abstract = {We consider a simple reaction-diffusion system exhibiting Turing's diffusion driven instability if supplemented with classical homogeneous mixed boundary conditions. We consider the case when the Neumann boundary condition is replaced by a unilateral condition of Signorini type on a part of the boundary and show the existence and location of bifurcation of stationary spatially non-homogeneous solutions. The nonsymmetric problem is reformulated as a single variational inequality with a potential operator, and a variational approach is used in a certain non-direct way.},
author = {Baltaev, Jamol I., Kučera, Milan, Väth, Martin},
journal = {Applications of Mathematics},
keywords = {reaction-diffusion system; unilateral condition; variational inequality; local bifurcation; variational approach; spatial patterns; reaction-diffusion system; unilateral condition; variational inequality; local bifurcation; variational approach; spatial patterns},
language = {eng},
number = {2},
pages = {143-165},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A variational approach to bifurcation in reaction-diffusion systems with Signorini type boundary conditions},
url = {http://eudml.org/doc/246145},
volume = {57},
year = {2012},
}

TY - JOUR
AU - Baltaev, Jamol I.
AU - Kučera, Milan
AU - Väth, Martin
TI - A variational approach to bifurcation in reaction-diffusion systems with Signorini type boundary conditions
JO - Applications of Mathematics
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 57
IS - 2
SP - 143
EP - 165
AB - We consider a simple reaction-diffusion system exhibiting Turing's diffusion driven instability if supplemented with classical homogeneous mixed boundary conditions. We consider the case when the Neumann boundary condition is replaced by a unilateral condition of Signorini type on a part of the boundary and show the existence and location of bifurcation of stationary spatially non-homogeneous solutions. The nonsymmetric problem is reformulated as a single variational inequality with a potential operator, and a variational approach is used in a certain non-direct way.
LA - eng
KW - reaction-diffusion system; unilateral condition; variational inequality; local bifurcation; variational approach; spatial patterns; reaction-diffusion system; unilateral condition; variational inequality; local bifurcation; variational approach; spatial patterns
UR - http://eudml.org/doc/246145
ER -

References

top
  1. Baltaev, J. I., Kučera, M., 10.1016/j.jmaa.2008.04.056, J. Math. Anal. Appl. 345 (2008), 917-928. (2008) Zbl1145.49004MR2429191DOI10.1016/j.jmaa.2008.04.056
  2. Drábek, P., Kučera, M., Míková, M., Bifurcation points of reaction-diffusion systems with unilateral conditions, Czech. Math. J. 35 (1985), 639-660. (1985) Zbl0604.35042MR0809047
  3. Drábek, P., Kufner, A., Nicolosi, F., Quasilinear Elliptic Equations with Degenerations and Singularities, Walter de Gruyter Berlin (1997). (1997) Zbl0894.35002MR1460729
  4. Edelstein-Keshet, L., Mathematical Models in Biology, McGraw-Hill Boston (1988). (1988) Zbl0674.92001MR1010228
  5. Eisner, J., 10.1016/S0362-546X(99)00446-0, Nonlinear Anal., Theory Methods Appl. 46 (2001), 69-90. (2001) Zbl0980.35029MR1845578DOI10.1016/S0362-546X(99)00446-0
  6. Eisner, J., Kučera, M., Spatial patterning in reaction-diffusion systems with nonstandard boundary conditions, Fields Institute Communications 25 (2000), 239-256. (2000) Zbl0969.35019MR1759546
  7. Eisner, J., Kučera, M., Recke, L., 10.1016/S0022-247X(02)00273-1, J. Math. Anal. Appl. 274 (2002), 159-180. (2002) Zbl1040.49006MR1936692DOI10.1016/S0022-247X(02)00273-1
  8. Eisner, J., Kučera, M., Recke, L., 10.1016/j.jmaa.2004.07.021, J. Math. Anal. Appl. 301 (2005), 276-294. (2005) MR2105671DOI10.1016/j.jmaa.2004.07.021
  9. Eisner, J., Kučera, M., Väth, M., 10.4171/ZAA/1390, J. Anal. Anwend. 28 (2009), 373-409. (2009) Zbl1182.35025MR2550696DOI10.4171/ZAA/1390
  10. Fučík, S., Kufner, A., Nonlinear Differential Equations, Elsevier Amsterdam-Oxford-New York (1980). (1980) MR0558764
  11. Jones, D. S., Sleeman, B. D., Differential Equations and Mathematical Biology, Chapman & Hall/CRC Boca Raton (2003). (2003) Zbl1020.92001MR1967145
  12. Kučera, M., 10.1023/A:1022411501260, Czech. Math. J. 47 (1997), 469-486. (1997) Zbl0898.35010MR1461426DOI10.1023/A:1022411501260
  13. Kučera, M., Recke, L., Eisner, J., Smooth bifurcation for variational inequalities and reaction-diffusion systems, Progresses in Analysis J. G. W. Begehr, R. P. Gilbert, M. W. Wong World Scientific Singapore-New Jersey-London-Hong Kong (2001), 1125-1133. (2001) MR2032793
  14. Miersemann, E., 10.1002/mana.19750650118, Math. Nachr. 65 (1975), 187-209 German. (1975) Zbl0324.49035MR0387843DOI10.1002/mana.19750650118
  15. Mimura, M., Nishiura, Y., Yamaguti, M., 10.1111/j.1749-6632.1979.tb29492.x, Ann. New York Acad. Sci. 316 (1979), 490-510. (1979) Zbl0437.92027MR0556853DOI10.1111/j.1749-6632.1979.tb29492.x
  16. Murray, J. D., Mathematical Biology, 2nd ed, Springer Berlin (1993). (1993) Zbl0779.92001MR1007836
  17. Nishiura, Y., Global structure of bifurcating solutions of some reaction-diffusion systems and their stability problem, Proceedings of the 5th Int. Symp. Computing Methods in Applied Sciences and Engineering, Versailles, France, 1981 R. Glowinski, J. L. Lions North-Holland Amsterdam-New York-Oxford (1982). (1982) Zbl0505.76103MR0784643
  18. Quittner, P., Bifurcation points and eigenvalues of inequalities of reaction-diffusion type, J. Reine Angew. Math. 380 (1987), 1-13. (1987) Zbl0617.35053MR0916198
  19. Recke, L., Eisner, J., Kučera, M ., 10.1016/S0022-247X(02)00272-X, J. Math. Anal. Appl. 275 (2002), 615-641. (2002) Zbl1018.34042MR1943769DOI10.1016/S0022-247X(02)00272-X
  20. Smoller, J., Shock Waves and Reaction-Diffusion Equations, Springer New York-Heidelberg-Berlin (1983). (1983) Zbl0508.35002MR0688146
  21. Zeidler, E., Nonlinear Functional Analysis and Its Applications, vol. II/A, Springer New York-Berlin-Heidelberg (1990). (1990) 
  22. Zeidler, E., Nonlinear Functional Analysis and Its Applications, vol. III, Springer New York-Berlin-Heidelberg (1985). (1985) MR0768749
  23. Zeidler, E., Nonlinear Functional Analysis and Its Applications, vol. IV, Springer New York-Berlin-Heidelberg (1988). (1988) MR0932255

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.