| Title:
|
Instability of Turing type for a reaction-diffusion system with unilateral obstacles modeled by variational inequalities (English) |
| Author:
|
Väth, Martin |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
139 |
| Issue:
|
2 |
| Year:
|
2014 |
| Pages:
|
195-211 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
reaction-diffusion system |
| Keyword:
|
Signorini condition |
| Keyword:
|
unilateral obstacle |
| Keyword:
|
instability |
| Keyword:
|
asymptotic stability |
| Keyword:
|
parabolic obstacle equation |
| MSC:
|
34D20 |
| MSC:
|
35B35 |
| MSC:
|
35K51 |
| MSC:
|
35K57 |
| MSC:
|
35K86 |
| MSC:
|
35K87 |
| MSC:
|
47H05 |
| MSC:
|
47H11 |
| MSC:
|
47J20 |
| MSC:
|
47J35 |
| idZBL:
|
Zbl 06362253 |
| idMR:
|
MR3238834 |
| DOI:
|
10.21136/MB.2014.143849 |
| . |
| Date available:
|
2014-07-14T08:14:19Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143849 |
| . |
| Reference:
|
[1] Drábek, P., Kučera, M., Míková, M.: Bifurcation points of reaction-diffusion systems with unilateral conditions.Czech. Math. J. 35 (1985), 639-660. Zbl 0604.35042, MR 0809047 |
| Reference:
|
[2] Eisner, J., Kučera, M., Väth, M.: Bifurcation points for a reaction-diffusion system with two inequalities.J. Math. Anal. Appl. 365 (2010), 176-194. Zbl 1185.35074, MR 2585089, 10.1016/j.jmaa.2009.10.037 |
| Reference:
|
[3] Henry, D.: Geometric Theory of Semilinear Parabolic Equations.Lecture Notes in Mathematics 840 Springer, Berlin (1981). Zbl 0456.35001, MR 0610244, 10.1007/BFb0089647 |
| Reference:
|
[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. |
| Reference:
|
[5] Kučera, M., Väth, M.: Bifurcation for a reaction-diffusion system with unilateral and {Neumann} boundary conditions.J. Differ. Equations 252 (2012), 2951-2982. Zbl 1237.35013, MR 2871789, 10.1016/j.jde.2011.10.016 |
| Reference:
|
[6] Mimura, M., Nishiura, Y., Yamaguti, M.: Some diffusive prey and predator systems and their bifurcation problems.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. Zbl 0437.92027, MR 0556853, 10.1111/j.1749-6632.1979.tb29492.x |
| Reference:
|
[7] Turing, A. M.: The chemical basis of morphogenesis.Phil. Trans. R. Soc. London Ser. B 237 (1952), 37-72. 10.1098/rstb.1952.0012 |
| Reference:
|
[8] Väth, M.: Ideal Spaces.Lecture Notes in Mathematics 1664 Springer, Berlin (1997). Zbl 0896.46018, MR 1463946, 10.1007/BFb0093548 |
| Reference:
|
[9] Väth, M.: Continuity and differentiability of multivalued superposition operators with atoms and parameters. I.Z. Anal. Anwend. 31 (2012), 93-124. Zbl 1237.47065, MR 2899873, 10.4171/ZAA/1450 |
| Reference:
|
[10] Ziemer, W. P.: Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation.Graduate Texts in Mathematics 120 Springer, Berlin (1989). Zbl 0692.46022, MR 1014685, 10.1007/978-1-4612-1015-3 |
| . |
