Spatial Patterns for reaction-diffusion systems with conditions described by inclusions
Jan Eisner,Milan Kučera +1 more
TL;DR: In this article, a reaction-diffusion system of the activator-inhibitor type with boundary conditions given by inclusions is considered and it is shown that there exists a bifurcation point at which stationary but spatially nonconstant solutions (spatial patterns) bifurocate from the branch of trivial solutions.
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Abstract: We consider a reaction-diffusion system of the activator-inhibitor type with boundary conditions given by inclusions. We show that there exists a bifurcation point at which stationary but spatially nonconstant solutions (spatial patterns) bifurcate from the branch of trivial solutions. This bifurcation point lies in the domain of stability of the trivial solution to the same system with Dirichlet and Neumann boundary conditions, where a bifurcation of this classical problem is excluded.
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Citations
Eigenvalues and bifurcation for problems with positively homogeneous operators and reaction–diffusion systems with unilateral terms
TL;DR: In this article, a variational characterization of the largest eigenvalue for positively homogeneous operators in a Hilbert space satisfying a condition related to potentiality, and existence of critical points and sometimes also bifurcation of stationary spatially nonhomogeneous solutions are proved for rates of diffusions for which it is excluded without any unilateral term.
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Bifurcation for a reaction–diffusion system with unilateral obstacles with pointwise and integral conditions
TL;DR: In this article, a reaction-diffusion system of activator-inhibitor or substrate-depletion type is considered which is subject to diffusion driven instability, where obstacles (e.g., a unilateral membrane) for one or both quantities introduce a new bifurcation of spatially nonhomogeneous steady states in a parameter domain where the trivial branch is exponentially stable without obstacles.
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Bifurcation of Solutions to Reaction-Diffusion Systems with Jumping Nonlinearities
Jan Eisner,Milan Kučera +1 more
- 01 Jan 2002
TL;DR: Bifurcation of stationary solutions to reaction-diffusion systems of activator-inhibitor type with jumping nonlinearities is located in this article, where the result can be understood as a certain destabilizing effect of jumping terms.
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Reaction-diffusion systems: Destabilizing effect of conditions given by inclusions II, Examples
Jan Eisner
- 01 Jan 2001
TL;DR: In this paper, sufficient conditions for destabilizing effects of certain unilateral boundary conditions and for the existence of bifurcation points for spatial patterns to reaction-diffusion systems of the activator-inhibitor type are proved.
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