Global Bifurcation for a Reaction-Diffusion System with Inclusions

Abstract

We consider a reaction-diffusion system exhibiting diffusion driven instability if supplemented by Dirichlet–Neumann boundary conditions. We impose unilateral conditions given by inclusions on this system and prove that global bifurcation of spatially non-homogeneous stationary solutions occurs in the domain of parameters where bifurcation is excluded for the original mixed boundary value problem. Inclusions can be considered in one of the equations itself as well as in boundary conditions. The proof is based on the degree theory for multivalued mappings (jump of the degree implies bifurcation). We show how the degree for a class of multivalued maps including those corresponding to a weak formulation of our problem can be calculated.