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| We consider the one-dimensional fully parabolic Keller-Segel system with nonlinear diffusion. It turns out that it possesses global-in-time solutions, provided the nonlinear diffusion is equal to $\frac{1}{(1+u)^\alpha}$ with $\alpha < 1$, independently on the volume of the initial data. On the other hand, for any nonlinear diffusion integrable at infinity there exist initial data for which a solution blows up in a finite time. Thus the most interesting is the critical case of the diffusion equal to $\frac{1}{1+u}$. There we are able to show the global regularity result for initial masses smaller than a prescribed constant and we conjecture that the bound on the initial mass is of a technical nature. |
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