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

Abstract

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].

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

“…, which is different from the subcritical regime treated in [13], and also from the case of the standard heat equation (1.3), where no | ln(T − t)| correction appears.…”

“…(1.10) Remark 1.2 (Stability). Following the interpretation of N + 1 parameters for the blowup time and the blowup point, originally done in [57], and then applied in [13], we can prove that behaviors (1.9) and (1.10) are stable under perturbation of initial data, the readers can find more details in Remark 2.3 of [13].…”

“…Now, we come back to the (nonlocal) equation (1.1) when γ = 0. Recently, inspired by the Type I construction mentioned above for equation (1.3), the authors of [13] constructed a blowup solution to equation (1.1) via a rigorous analysis, giving the exact form of the blowup profile in some subcritical regime of the parameters, namely when r p − 1 < N 2 and γr = p − 1, (1.5) being in agreement with Turing condition (1.2). Specifically, under (1.5), the authors constructed a blowup solution to (1.1) with the blowup profile (pattern) as follows…”

“…In particular, up to some natural scaling, both equations show the same profile (see (1.4) and (1.6)). Inspired by the analysis in [13], it naturally arises the need for the consideration of another regime where the non-local term has a different limit, in particular it converges towards "zero", leading hopefully to a different blow-up speed. The following is our main result:…”

“…

We consider a nonlocal parabolic PDE, which may be regarded as the standard semilinear heat equation with power nonlinearity, where the nonlinear term is divided by some Sobolev norm of the solution. Unlike the earlier work in [13] where we consider a subcritical regime of parameters, we focus here on the critical regime, which is much more complicated. Our main result concerns the construction of a blow-up solution with the description of its asymptotic behavior.

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