Dynamics of Shadow System of a Singular Gierer–Meinhardt System on an Evolving Domain

Abstract

The main purpose of the current paper is to contribute towards the comprehension of the dynamics of the shadow system of a singular Gierer–Meinhardt model on an isotropically evolving domain. In the case where the inhibitor’s response to the activator’s growth is rather weak, then the shadow system of the Gierer–Meinhardt model is reduced to a single though non-local equation whose dynamics is thoroughly investigated throughout the manuscript. The main focus is on the derivation of blow-up results for this non-local equation, which can be interpreted as instability patterns of the shadow system. In particular, a diffusion-driven instability (DDI), or Turing instability, in the neighbourhood of a constant stationary solution, which then is destabilised via diffusion-driven blow-up, is observed. The latter indicates the formation of some unstable patterns, whilst some stability results of global-in-time solutions towards non-constant steady states guarantee the occurrence of some stable patterns. Most of the theoretical results are verified numerically, whilst the numerical approach is also used to exhibit the dynamics of the shadow system when analytical methods fail.

Impacts of noise on quenching of some models arising in MEMS technology

“…Remark 5.7. Remarkably Theorem 5.5 (ii) implies that when the diffusion coefficient g(t) is large, enough ensured by condition (5.40), then quenching behaviour dominates for the case of a big domain D. This looks in the counterintutive to what has been pointed out in Remark 5.4 in the first place, however it is in full agreement with the phenomenon observed in [32] where a strong reaction coefficient, enhanced there by the evolution of underlying domain, fights against the development of a singularity.…”

Diffusion-induced blowup solutions for the shadow limit model of a singular Gierer-Meinhardt system

“…[KS17] and [KS18]. For analogous blowup results for the Gierer-Meinhardt system on an evolving domain and for a non-local Fisher-KPP equation one can see [KBM19] and [KL20] respectively. The main purpose of the current work is to describe the form of the developing Turing instability (blowup) patterns for the solution of problem (1.4) in a region of any blowup point.…”

“…v(x, t) = ξ(t), cf. [KS17], [KS18] and [KBM19] (a rigorous proof for a version of Gierer-Meinhardt system can be found in [MCM17] and [MCHKS18], whilst for the case of general reaction-diffusion systems the interested reader can check [BK19]). Next, integrating the second equation in (1.1), we finally derive the shadow system for u and ξ…”

Blowup solutions for the shadow limit model of singular Gierer-Meinhardt system with critical parameters

“…Its dynamical behavior was first considered in [40], where in particular global-in-time existence as well as blowup results were derived according to the range of the involved parameters p, r and γ. The case of an isotropically evolving domain was also considered in [36], where an analytical and numerical study was delivered. In the aforementioned works, it was pointed out that for the limiting problem (1.1), diffusion-driven (Turing) instability occurs, an intriguing phenomenon which was introduced in the seminal paper [70], for a specific range of the parameters.…”

“…cf. [36,40,41]. However, in those works, only a rough form of the Turing instability (blowup) patterns is presented and only in the case of a sphere; it is based on known results, cf.…”