Reaction-diffusion-ODE systems: de-novo formation of irregular patterns and model reduction

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

“…We have v r > 0 since m 1 > 2 √ k. It holds u − (v)u + (v) = 1 k , for v < v r . This yields 0 < u − (v) ≤ for 0 ≤ v ≤ v r and u − (v) < for 0 ≤ v < v r .We obtain the following instability result, compare[21, Theorem 3.21].…”

“…The model nonlinearities f : R m+k × Ω → R m and g : R m+k × Ω → R k are given functions which may depend on the unknown solution and the space variable, further details are given in Assumption 2 below. Recall that solutions to reaction-diffusion-ODE system (1.1) may feature low regularity in space, even in homogeneous environments [21,33]. Hence, mild solutions are considered in this work, similar to [39].…”

“…By [21,33], the ODE component of the solution of the stationary problem (1.2) may exhibit jumpdiscontinuouities in space. Thus, we consider a steady state (u, v) ∈ L ∞ (Ω) m × C(Ω) k .…”

“…The DDI-hysteresis model originates from [22], for details see also [21,Section 3.6.1]. In the remainder of this subsection, we explore the underlying pattern formation phenomena applying the developed theory.…”

“…We aim to apply Lemma 4.1 in this case. The following theorem can be compared to the stability results obtained in [21,Theorem 3.21].…”