Bifurcation points and eigenvalues of inequalities of reaction-diffusion type. (English) Zbl 0617.35053

Quittner, Pavol

J. Reine Angew. Math. 380, 1-13 (1987).

Let us consider an abstract formulation of a semi-linear reaction- diffusion system with unilateral boundary conditions. Let \(\bar U\) be a stationary spatially homogeneous solution of this problem and let d be a parameter (diffusion coefficient). The existence of bifurcations of stationary solutions from the branch of trivial solutions \(\{(d,\bar U); d\in {\mathbb{R}}^+\}\) and the linearized stability of the solution \(\bar U\) are studied. The results imply that we may loose the stability of a stationary solution of a reaction-diffusion system, if the classical boundary conditions are replaced by suitable unilateral boundary conditions.

MSC:

35J85	Unilateral problems; variational inequalities (elliptic type) (MSC2000)
35K85	Unilateral problems for linear parabolic equations and variational inequalities with linear parabolic operators
35K57	Reaction-diffusion equations
35P30	Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs

Keywords:

semi-linear reaction-diffusion system; unilateral boundary conditions; bifurcations; stationary solutions; linearized stability