Geodesic
Instability of Turing type for a reaction-diffusion system with unilateral obstacles modeled by variational inequalities
Mathematica Bohemica, Tome 139 (2014) no. 2, pp. 195-211

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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.
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.
DOI : 10.21136/MB.2014.143849
Classification : 34D20, 35B35, 35K51, 35K57, 35K86, 35K87, 47H05, 47H11, 47J20, 47J35
Keywords: reaction-diffusion system; Signorini condition; unilateral obstacle; instability; asymptotic stability; parabolic obstacle equation 
Väth, Martin. Instability of Turing type for a reaction-diffusion system with unilateral obstacles modeled by variational inequalities. Mathematica Bohemica, Tome 139 (2014) no. 2, pp. 195-211. doi: 10.21136/MB.2014.143849
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[1] Drábek, P., Kučera, M., Míková, M.: Bifurcation points of reaction-diffusion systems with unilateral conditions. Czech. Math. J. 35 (1985), 639-660. | MR | Zbl

[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. | DOI | MR | Zbl

[3] Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics 840 Springer, Berlin (1981). | DOI | MR | Zbl

[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.: Bifurcation for a reaction-diffusion system with unilateral and {Neumann} boundary conditions. J. Differ. Equations 252 (2012), 2951-2982. | DOI | MR | Zbl

[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. | DOI | MR | Zbl

[7] Turing, A. M.: The chemical basis of morphogenesis. Phil. Trans. R. Soc. London Ser. B 237 (1952), 37-72. | DOI

[8] Väth, M.: Ideal Spaces. Lecture Notes in Mathematics 1664 Springer, Berlin (1997). | DOI | MR | Zbl

[9] Väth, M.: Continuity and differentiability of multivalued superposition operators with atoms and parameters. I. Z. Anal. Anwend. 31 (2012), 93-124. | DOI | MR | Zbl

[10] Ziemer, W. P.: Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation. Graduate Texts in Mathematics 120 Springer, Berlin (1989). | DOI | MR | Zbl

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