Reaction-diffusion systems: destabilizing effect of unilateral conditions

Reaction-diffusion systems: stabilizing effect of conditions described by quasivariational inequalities

“…An analogous result for the case of variational inequalities was proved in [1], [13] and generalized to inclusions in [7]. A certain destabilizing effect of unilateral conditions given by variational inequalities for the inhibitor in terms of loss of stability of the trivial solution of the linearized unilateral problem was proved in [2], [3], [13].…”

“…For the special case fc = 1 see e.g. [10], for the general case see [3]. See also Weak Formulation 1.1.…”

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

“…for unilateral conditions only for the inhibitor v) with int K 2 = ∅ and for the special curve σ 2 (s) = 1. In this case, the condition (2.25) has the form U 0 ∈ int K. A destabilizing effect of such unilateral conditions in terms of instability of the trivial solution of the corresponding linearized inequality was shown in [4] (already for cones with empty interior). The idea of the proofs in these papers (analogously to the proof of our Theorem 3.1) consists in a certain homotopical joining of the inequality with the corresponding linearized equation.…”

“…ÈÖÓÓ can be done by the same considerations as in Observations 4.1, 4.2 where we will study a slightly different and more complicated eigenvalue problem (see also [17] for the special case N = 1 and Γ D = ∅ or [4] for the general case).…”

“…It was studied in the papers [3], [4], [20], [10], [12], [13], [14], [5] how the situation changes if the classical boundary conditions (1.4) are replaced (or supplemented) by some unilateral conditions, e.g. by boundary conditions (1.2) (some additional conditions in the interior of Ω can be also considered).…”