Spatial Patterns for reaction-diffusion systems with conditions described by inclusions

Bifurcation points for a reaction–diffusion system with two inequalities

“…However, for the case that we consider only the pointwise Signorini condition (1.5) the sign-condition is more restrictive as it might appear at a first glance: Indeed, if the boundary is so smooth that the unique continuation property for the eigenvalue problem of the Laplacian holds then it can be shown that the sign-condition actually becomes the (apparently much more restrictive) second hypothesis of Theorem 2.1. The latter is exactly the hypothesis used in [7,17].…”

“…[7,18].) Let us now formulate (in a rather vague way) a special case of the main result of Section 3.…”

“…The sign-condition is not only a transfer of the main hypothesis of the local bifurcation results from [7,17] to our setting but actually is even much more general in the case that our problem contains integrals as in (1.6). However, for the case that we consider only the pointwise Signorini condition (1.5) the sign-condition is more restrictive as it might appear at a first glance: Indeed, if the boundary is so smooth that the unique continuation property for the eigenvalue problem of the Laplacian holds then it can be shown that the sign-condition actually becomes the (apparently much more restrictive) second hypothesis of Theorem 2.1.…”

“…The existence of bifurcations for (1.2) with unilateral boundary conditions for both u and v with respect to a curve γ was already proved in [7,17], but only local bifurcation was obtained and the hypotheses were essentially stronger. Bifurcation along horizontal paths d 2 = const was obtained under similar hypotheses in [22].…”

Destabilization for quasivariational inequalities of reaction-diffusion type

“…Most recently it can be seen in [7] for boundary conditions leading to a system of differential inclusions but the proof uses a completely different method. The destabilization for inclusions has been also already proved (see [8], [9]). …”