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The twodimensional KellerSegel model after blowup
On the longtime limit of positive solutions to the degenerate logistic equation
1.  Department of Mathematics, School of Science and Technology, University of New England, Armidale, NSW2351, Australia 
2.  Department of Mathematics, Waseda University, 341 Ohkubo, Shinjukuku 1698555, Tokyo 
$u_t\Delta u=a ub(x)u^p \mbox{ in } (0,\infty)\times \Omega, Bu=0 \mbox{ on } (0,\infty)\times \partial \Omega, $
where $a$ is a real parameter, $b\geq 0$ is in $C^\mu(\bar{\Omega})$ and $p>1$ is a constant, $\Omega$ is a $C^{2+\mu}$ bounded domain in $R^N$ ($N\geq 2$), the boundary operator $B$ is of the standard Dirichlet, Neumann or Robyn type. Under the assumption that $\overline\Omega_0$:=$\{x\in\Omega: b(x)=0\}$ has nonempty interior, is connected, has smooth boundary and is contained in $\Omega$, it is shown in [8] that when $a\geq \lambda_1^D(\Omega_0)$, for any fixed $x\in \overline{\Omega}_0$, $\overline{\lim}_{t\to\infty}u(t,x)$=$\infty$, and for any fixed $x\in \overline{\Omega}\setminus \overline{\Omega}_0$,
$\overline{\lim}_{t\to\infty}u(t,x)\leq \overline{U}_a(x), \underline{\lim}_{t\to\infty}u(t,x)\geq \underline{U}_a(x),
where $\underline{U}_a$ and $\overline{U}_a$ denote respectively the minimal and maximal positive solutions of the boundary blowup problem
$\Delta u=aub(x)u^p \mbox{ in} \ \Omega\setminus\overline{\Omega}_0,\ Bu=0 \mbox{ on}\ \partial \Omega,\ \ u=\infty \mbox{ on}\ \partial \Omega_0.$
The main purpose of this paper is to show that, under the above assumptions,
$\lim_{t\to\infty} u(t,x)=\underline U_a(x),\forall x\in \overline\Omega\setminus \overline\Omega_0.$
This proves a conjecture stated in [8]. Some extensions of this result are also discussed.
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