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

TL;DR

This paper investigates finite-time blowup solutions in a nonlocal diffusion model derived from a singular Gierer-Meinhardt system, revealing detailed asymptotic behavior and pattern formation due to diffusion-driven instability.

Contribution

It constructs and analyzes finite-time blowup solutions for a shadow limit model, providing detailed asymptotic profiles and insights into Turing pattern formation.

Findings

Blowup occurs at an interior point in finite time.

Asymptotic profile of the solution near blowup point is characterized.

The pattern formation due to diffusion instability is described in detail.

Abstract

In the current paper, we provide a thorough investigation of the blowing up behaviour induced via diffusion of the solution of the following non local problem \begin{equation*} \left\{\begin{array}{rcl} \partial_t u &=& \Delta u - u + \displaystyle{\frac{u^p}{ \left(\mathop{\,\rlap{-}\!\!\int}\nolimits_\Omega u^r dr \right)^\gamma }}\quad\text{in}\quad \Omega \times (0,T), \\[0.2cm] \frac{ \partial u}{ \partial \nu} & = & 0 \text{ on } \Gamma = \partial \Omega \times (0,T),\\ u(0) & = & u_0, \end{array} \right. \end{equation*} where $\Omega$ is a bounded domain in $\mathbb{R}^N$ with smooth boundary $\partial \Omega;$ such problem is derived as the shadow limit of a singular Gierer-Meinhardt system, cf. \cite{KSN17, NKMI2018}. Under the Turing type condition $$ \frac{r}{p-1} < \frac{N}{2}, \gamma r \ne p-1, $$ we construct a solution which blows up in finite time and only at an interior point $x_0$ of $\Omega,$ i.e. $$ u(x_0, t) \sim (\theta^*)^{-\frac{1}{p-1}} \left[\kappa (T-t)^{-\frac{1}{p-1}} \right], $$ where $$ \theta^* := \lim_{t \to T} \left(\mathop{\,\rlap{-}\!\!\int}\nolimits_\Omega u^r dr \right)^{- \gamma} \text{ and } \kappa = (p-1)^{-\frac{1}{p-1}}. $$ More precisely, we also give a description on the final asymptotic profile at the blowup point $$ u(x,T) \sim ( \theta^* )^{-\frac{1}{p-1}} \left[ \frac{(p-1)^2}{8p} \frac{|x-x_0|^2}{ |\ln|x-x_0||} \right]^{ -\frac{1}{p-1}} \text{ as } x \to 0, $$ 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 \cite{MZnon97} and \cite{DZM3AS19}.