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 &=& \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}.