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- W3093831521 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 }}quadtext{in}quad Omega times [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}." @default.
- W3093831521 created "2020-10-29" @default.
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- W3093831521 date "2020-10-19" @default.
- W3093831521 modified "2023-09-27" @default.
- W3093831521 title "Diffusion-induced blowup solutions for the shadow limit model of a singular Gierer-Meinhardt system." @default.
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