Instability of Turing type for a reaction-diffusion system with unilateral obstacles modeled by variational inequalities

Martin Väth

Mathematica Bohemica (2014)

  • Volume: 139, Issue: 2, page 195-211
  • ISSN: 0862-7959

Abstract

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We consider a reaction-diffusion system of activator-inhibitor type which is subject to Turing's diffusion-driven instability. It is shown that unilateral obstacles of various type for the inhibitor, modeled by variational inequalities, lead to instability of the trivial solution in a parameter domain where it would be stable otherwise. The result is based on a previous joint work with I.-S. Kim, but a refinement of the underlying theoretical tool is developed. Moreover, a different regime of parameters is considered for which instability is shown also when there are simultaneously obstacles for the activator and inhibitor, obstacles of opposite direction for the inhibitor, or in the presence of Dirichlet conditions.

How to cite

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Väth, Martin. "Instability of Turing type for a reaction-diffusion system with unilateral obstacles modeled by variational inequalities." Mathematica Bohemica 139.2 (2014): 195-211. <http://eudml.org/doc/261888>.

@article{Väth2014,
abstract = {We consider a reaction-diffusion system of activator-inhibitor type which is subject to Turing's diffusion-driven instability. It is shown that unilateral obstacles of various type for the inhibitor, modeled by variational inequalities, lead to instability of the trivial solution in a parameter domain where it would be stable otherwise. The result is based on a previous joint work with I.-S. Kim, but a refinement of the underlying theoretical tool is developed. Moreover, a different regime of parameters is considered for which instability is shown also when there are simultaneously obstacles for the activator and inhibitor, obstacles of opposite direction for the inhibitor, or in the presence of Dirichlet conditions.},
author = {Väth, Martin},
journal = {Mathematica Bohemica},
keywords = {reaction-diffusion system; Signorini condition; unilateral obstacle; instability; asymptotic stability; parabolic obstacle equation; obstacle problem; unilateral obstacle; topological fixed point index; Turing instability; spherical stability; reaction-diffusion system; Signorini condition; parabolic obstacle equation},
language = {eng},
number = {2},
pages = {195-211},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Instability of Turing type for a reaction-diffusion system with unilateral obstacles modeled by variational inequalities},
url = {http://eudml.org/doc/261888},
volume = {139},
year = {2014},
}

TY - JOUR
AU - Väth, Martin
TI - Instability of Turing type for a reaction-diffusion system with unilateral obstacles modeled by variational inequalities
JO - Mathematica Bohemica
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 139
IS - 2
SP - 195
EP - 211
AB - We consider a reaction-diffusion system of activator-inhibitor type which is subject to Turing's diffusion-driven instability. It is shown that unilateral obstacles of various type for the inhibitor, modeled by variational inequalities, lead to instability of the trivial solution in a parameter domain where it would be stable otherwise. The result is based on a previous joint work with I.-S. Kim, but a refinement of the underlying theoretical tool is developed. Moreover, a different regime of parameters is considered for which instability is shown also when there are simultaneously obstacles for the activator and inhibitor, obstacles of opposite direction for the inhibitor, or in the presence of Dirichlet conditions.
LA - eng
KW - reaction-diffusion system; Signorini condition; unilateral obstacle; instability; asymptotic stability; parabolic obstacle equation; obstacle problem; unilateral obstacle; topological fixed point index; Turing instability; spherical stability; reaction-diffusion system; Signorini condition; parabolic obstacle equation
UR - http://eudml.org/doc/261888
ER -

References

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  2. Eisner, J., Kučera, M., Väth, M., 10.1016/j.jmaa.2009.10.037, J. Math. Anal. Appl. 365 (2010), 176-194. (2010) Zbl1185.35074MR2585089DOI10.1016/j.jmaa.2009.10.037
  3. Henry, D., 10.1007/BFb0089647, Lecture Notes in Mathematics 840 Springer, Berlin (1981). (1981) Zbl0456.35001MR0610244DOI10.1007/BFb0089647
  4. Kim, I.-S., Väth, M., The Krasnosel'skii-Quittner formula and instability of a reaction-diffusion system with unilateral obstacles, Submitted to Dyn. Partial Differ. Equ. 20 pages. 
  5. Kučera, M., Väth, M., 10.1016/j.jde.2011.10.016, J. Differ. Equations 252 (2012), 2951-2982. (2012) Zbl1237.35013MR2871789DOI10.1016/j.jde.2011.10.016
  6. Mimura, M., Nishiura, Y., Yamaguti, M., 10.1111/j.1749-6632.1979.tb29492.x, Bifurcation Theory and Applications in Scientific Disciplines (Papers, Conf., New York, 1977) O. Gurel, O. E. Rössler Ann. New York Acad. Sci. 316 (1979), 490-510. (1979) Zbl0437.92027MR0556853DOI10.1111/j.1749-6632.1979.tb29492.x
  7. Turing, A. M., 10.1098/rstb.1952.0012, Phil. Trans. R. Soc. London Ser. B 237 (1952), 37-72. (1952) DOI10.1098/rstb.1952.0012
  8. Väth, M., 10.1007/BFb0093548, Lecture Notes in Mathematics 1664 Springer, Berlin (1997). (1997) Zbl0896.46018MR1463946DOI10.1007/BFb0093548
  9. Väth, M., 10.4171/ZAA/1450, Z. Anal. Anwend. 31 (2012), 93-124. (2012) Zbl1237.47065MR2899873DOI10.4171/ZAA/1450
  10. Ziemer, W. P., 10.1007/978-1-4612-1015-3, Graduate Texts in Mathematics 120 Springer, Berlin (1989). (1989) Zbl0692.46022MR1014685DOI10.1007/978-1-4612-1015-3

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