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

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 &=& Δu - u + \displaystyle{\frac{u^p}{ \left(\mathop{\,\rlap{-}\!\!\int}\nolimits_Ωu^r dr \right)^γ}}\quad\text{in}\quad Ω\times (0,T), \\[0.2cm] \frac{ \partial u}{ \partial ν} & = & 0 \text{ on } Γ= \partial Ω\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. $$u(x_0,t)\sim ({\theta }^{\ast }{)}^{-\frac{1}{p-1}}\left[\kappa (T-t{)}^{-\frac{1}{p-1}}\right],$$ where $${\theta }^{\ast }:=\underset{t\to T}{\lim }{\left({\text{-}\int }_{\Omega }{u}^{r}dr\right)}^{-\gamma }\text{ and }\kappa =(p-1{)}^{-\frac{1}{p-1}}.$$ More precisely, we also give a description on the final asymptotic profile at the blowup point $$u(x,T)\sim ({\theta }^{\ast }{)}^{-\frac{1}{p-1}}{\left[\frac{(p-1{)}^2}{8p}\frac{|x-x_0{|}^2}{|\ln |x-x_0||}\right]}^{-\frac{1}{p-1}}\text{ as }x\to 0,$$ 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}.

31 pages