Bifurcation in skew-symmetric reaction-diffusion systems with unilateral terms. (English) Zbl 1467.35038
Summary: The paper deals with skew-symmetric reaction-diffusion systems satisfying assumptions guaranteeing Turing’s instability and supplemented by unilateral terms of type \(v^-\) and \(v^+\). Existence of critical and bifurcation points is proved for diffusion rates, for which it is excluded without any unilateral term. These results are achieved by rewriting the skew-symmetric system as an abstract equation with positively homogeneous potential operator. General theorems about a variational characterization of the largest eigenvalue for positively homogeneous operators in a Hilbert space and bifurcation in equations with potentials are proved and subsequently applied to the reaction-diffusion systems, yielding the desired conclusions.
MSC:
| 35B32 | Bifurcations in context of PDEs |
| 35J50 | Variational methods for elliptic systems |
| 35J57 | Boundary value problems for second-order elliptic systems |
| 35K57 | Reaction-diffusion equations |
Keywords:
positively homogeneous operators; skew-symmetric systems; Rayleigh quotient; local bifurcation; jumping nonlinearitiesReferences:
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