Diffusion-induced blowup solutions for the shadow limit model of a singular Gierer–Meinhardt system - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Mathematical Models and Methods in Applied Sciences Année : 2021

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

G. Duong
  • Fonction : Auteur
N. Kavallaris
  • Fonction : Auteur
Hatem Zaag

Résumé

In this paper, we provide a thorough investigation of the blowing up behavior induced via diffusion of the solution of the following non-local problem: [Formula: see text] where [Formula: see text] is a bounded domain in [Formula: see text] with smooth boundary [Formula: see text] such problem is derived as the shadow limit of a singular Gierer–Meinhardt system, Kavallaris and Suzuki [On the dynamics of a non-local parabolic equation arising from the Gierer–Meinhardt system, Nonlinearity (2017) 1734–1761; Non-Local Partial Differential Equations for Engineering and Biology: Mathematical Modeling and Analysis, Mathematics for Industry, Vol. 31 (Springer, 2018)]. Under the Turing type condition [Formula: see text] we construct a solution which blows up in finite time and only at an interior point [Formula: see text] of [Formula: see text] i.e. [Formula: see text] where [Formula: see text] More precisely, we also give a description on the final asymptotic profile at the blowup point [Formula: see text] and thus we unveil the form of the Turing patterns occurring in that case due to driven-diffusion instability. The applied technique for the construction of the preceding blowing up solution mainly relies on the approach developed in [F. Merle and H. Zaag, Reconnection of vortex with the boundary and finite time quenching, Nonlinearity 10 (1997) 1497–1550] and [G. K. Duong and H. Zaag, Profile of a touch-down solution to a nonlocal MEMS model, Math. Models Methods Appl. Sci. 29 (2019) 1279–1348].

Dates et versions

hal-03418340 , version 1 (07-11-2021)

Identifiants

Citer

G. Duong, N. Kavallaris, Hatem Zaag. Diffusion-induced blowup solutions for the shadow limit model of a singular Gierer–Meinhardt system. Mathematical Models and Methods in Applied Sciences, 2021, 31 (07), pp.1469-1503. ⟨10.1142/S0218202521500305⟩. ⟨hal-03418340⟩
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