Nonlinear PDE’s, Dynamics and Continuum Physics

Stability Results for Bounded Stationary Solutions of Reaction-Diffusion-ODE Systems

“…Since in the case of reaction-diffusion-ODE systems the spectrum of the linear operator might not solely consist of eigenvalues, such a spectral gap condition is not always fulfilled. Moreover, classical nonlinear stability results as in [40,49] only apply in the case where the linear and the nonlinear operator is studied on the same function space. Consequently, in the case of a continuous steady state, the choice of the space C(Ω) m+k is expedient for stability and instability results.…”

“…As shown in this paper, the growth of the semigroup can still be controlled by the spectral bound of the linear operator. We adapt the works [40,49], using continuity of solutions to the nonlinear problem for t > 0, to finally accomplish the transfer from linear to nonlinear stability and instability.…”

“…In the nonlinear case, there are mainly two ways showing instability via linearization. Superlinear decay of the nonlinear reaction term in combination with a non-empty intersection of the spectrum with the complex right half-plane is sufficient for instability, see [40,Theorem 1] or [13, Ch. VII, Theorem 2.3] for bounded operators.…”

“…In this section, we aim for a generalization of the results from [34] to bounded, discontinuous and continuous, steady states of the reaction-diffusion-ODE system (1.1) without requiring a spectral gap. We use the semigroup approach of [40] combined with the notion of a mild solution from [39] to show instability. First, we aim to verify semigroup estimates from [40, Lemma 2, 3] on L ∞ (Ω) m+k in the case of a discontinuous stationary solution.…”

“…Proof. The proof is basically analog to the proof of [40,Theorem 1], where a linearized equation of the form (2.1) is considered. In the following we depict only main adaptions of the proof.…”