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

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. Contents 1. Introduction 230 2. The Krasnoselskiȋ-Quittner Formula 231 3. Instability of the Reaction-Diffusion System 241 4. Other Obstacles 246 Appendix A. A Local Lipschitz Condition for a Superposition Operator in L p Spaces 248 References 249

Unilateral regulation breaks regularity of Turing patterns

“…This indicates that our findings about the effects of the unilateral term are not limited to a single kinetics. The generality of these results is also supported by earlier results [10,23,27], which guarantee the existence of bifurcations of spatial patterns and certain instability of the ground state for large d 1 /d 2 for a unilateral source described by variational inequalities, but they are valid for a very large class of kinetics.…”

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

“…The result is based on a refinement of a theoretical tool developed in [4]. In a sense, the result presented in this paper complements the application from [4], since we consider a different range of values (d 1 , d 2 ) than in [4], which leads to a result under somewhat different assumptions on the obstacle/boundary conditions (and allows us also to treat Γ D = ∅).…”

“…The main difference of Theorem 1 to the corresponding result in [4] is that in the latter, points (d 1 , d 2 ) with large values d 1 , d 2 > 0 play a crucial role instead of points (d 1 , d 2 ) close to a certain hyperbola C n (or to C n ∩ C m ). This makes a fundamental difference in the hypotheses of the two results: While in [4] there is apparently no hypothesis similar to (2.12)/(2.13), this hypothesis is implicitly in an requirement about E 0 (corresponding to the eigenvalue κ 0 = 0). In fact, the corresponding result in [4] applies only to the non-Dirichlet case Γ D = ∅ and only if there is no obstacle for u (Ω 1 = Γ 1 = ∅) and the obstacle for v acts in a uniform direction (Γ − 2 = Ω − 2 = ∅ or Γ − 1 = Ω − 1 = ∅).…”

“…This makes a fundamental difference in the hypotheses of the two results: While in [4] there is apparently no hypothesis similar to (2.12)/(2.13), this hypothesis is implicitly in an requirement about E 0 (corresponding to the eigenvalue κ 0 = 0). In fact, the corresponding result in [4] applies only to the non-Dirichlet case Γ D = ∅ and only if there is no obstacle for u (Ω 1 = Γ 1 = ∅) and the obstacle for v acts in a uniform direction (Γ − 2 = Ω − 2 = ∅ or Γ − 1 = Ω − 1 = ∅). It seems likely that none of these additional requirements can be dropped for the result from [4] (since they are crucial for a major tool provided by [5]).…”

“…In fact, the corresponding result in [4] applies only to the non-Dirichlet case Γ D = ∅ and only if there is no obstacle for u (Ω 1 = Γ 1 = ∅) and the obstacle for v acts in a uniform direction (Γ − 2 = Ω − 2 = ∅ or Γ − 1 = Ω − 1 = ∅). It seems likely that none of these additional requirements can be dropped for the result from [4] (since they are crucial for a major tool provided by [5]). In contrast, Theorem 1 needs none of these requirements, but instead (2.12)/(2.13) is needed.…”