Dynamics of Partial Differential Equations

Volume 11 (2014)

Number 3

The Krasnosel’skiĭ-Quittner formula and instability of a reaction-diffusion system with unilateral obstacles

Pages: 229 – 250

DOI: https://dx.doi.org/10.4310/DPDE.2014.v11.n3.a2

Authors

In-Sook Kim (Department of Mathematics, Sungkyunkwan University, Suwon, Korea)

Martin Väth (Math. Institute, Free University of Berlin, Germany)

Abstract

We prove a formula which relates the fixed point index of a parabolic obstacle equation to a fixed point index related to the right-hand side of the equation. The result is applied to a reaction-diffusion system at a constant equilibrium which is subject to Turing’s diffusion-driven instability. It is shown that if a unilateral obstacle is added, the system becomes unstable in a parameter domain where the system without obstacle is stable.

Keywords

reaction-diffusion system, Signorini condition, unilateral obstacle, instability, asymptotic stability, Krasnoselskij formula, parabolic obstacle equation

2010 Mathematics Subject Classification

Primary 35K87. Secondary 34D20, 35K57, 35K86, 47H05, 47H11, 47J20, 47J35.

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Published 19 September 2014