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

*(English)*Zbl 1340.35145For a system of two reaction-diffusion equations of activator-inhibitor type with Neumann/Dirichlet boundary conditions on different parts of the boundary and interior regions where some unilateral obstacles are posed (i.e. modeling a unilateral membrane) the stability of the trivial solution with respect to Turing instability is examined.

It is shown that, if some obstacle for the inhibitor (or for both the activator and the inhibitor) is present, the trivial solution fails to be asymptotically stable for values of the diffusion coefficients where it is asymptotically stable without obstacles. More precisely, the trivial solution fails to be spherically stable, a new notion that the author introduces for stationary solutions \(u_0\) and which requires that all solutions starting in an \(\varepsilon\)-ball around \(u_0\) may leave this ball but return and ultimately remain in the same \(\varepsilon\)-ball. The proof is based on a careful weak formulation of the problem and an application of the topological fixed point index. Although the quite general formulation of the problem requires considerable notation, the author does explain the actual meaning of the conditions imposed, in particular, the relationship between the obstacle regions for the different variables.

It is shown that, if some obstacle for the inhibitor (or for both the activator and the inhibitor) is present, the trivial solution fails to be asymptotically stable for values of the diffusion coefficients where it is asymptotically stable without obstacles. More precisely, the trivial solution fails to be spherically stable, a new notion that the author introduces for stationary solutions \(u_0\) and which requires that all solutions starting in an \(\varepsilon\)-ball around \(u_0\) may leave this ball but return and ultimately remain in the same \(\varepsilon\)-ball. The proof is based on a careful weak formulation of the problem and an application of the topological fixed point index. Although the quite general formulation of the problem requires considerable notation, the author does explain the actual meaning of the conditions imposed, in particular, the relationship between the obstacle regions for the different variables.

Reviewer: Jörg Härterich (Bochum)

##### MSC:

35K57 | Reaction-diffusion equations |

35K86 | Unilateral problems for nonlinear parabolic equations and variational inequalities with nonlinear parabolic operators |

35K87 | Unilateral problems for parabolic systems and systems of variational inequalities with parabolic operators |

35B35 | Stability in context of PDEs |

47H11 | Degree theory for nonlinear operators |

47J20 | Variational and other types of inequalities involving nonlinear operators (general) |

34D20 | Stability of solutions to ordinary differential equations |