Global bifurcation for quasivariational inequalities of reaction–diffusion type

Abstract

We consider a reaction-diffusion system with implicit unilateral boundary conditions introduced by U. Mosco. We show that global continua of stationary spatially nonhomogeneous solutions bifurcate in the domain of parameters where bifurcation in the case of classical boundary conditions is excluded. The problem is formulated as a quasivariational inequality and the proof is based on the Leray-Schauder degree.

A variational approach to bifurcation in reaction-diffusion systems with Signorini type boundary conditions

“…The boundary conditions from Example 5.1 can describe unilateral membranes, see e.g. [1]. R e m a r k 5.4.…”

“…As we already mentioned, for the proof it was always essential that certain eigenfunctions of the Laplacian satisfy a certain sign condition (in general, it must be in the interior or in a certain pseudo-interior of the cone related to the corresponding variational inequality), see e.g. [1], [9]. In the present variational approach we need no such assumption, cf.…”

“…[1], [2], [6], [9], [12], [18]), an influence of unilateral conditions to this bifurcation was studied. (Usually only systems of activator-inhibitor type were discussed but in fact only the assumption (1.4) was used, i.e.…”

“…Interpretation of the boundary conditions (1.3) (even in a more general form) and of unilateral conditions from Examples 5.2, 5.3 (the last section) is described e.g. in [1] and in Remark 5.3, respectively.…”