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Uniqueness and long time asymptotic for the Keller-Segel equation: the parabolic-elliptic case. (English) Zbl 1334.35358
Summary: The present paper deals with the parabolic-elliptic Keller-Segel equation in the plane in the general framework of weak (or “free energy”) solutions associated to initial datum with finite mass $$M$$, finite second moment and finite entropy. The aim of the paper is threefold:
(1)
We prove the uniqueness of the “free energy” solution on the maximal interval of existence $$[0,T\ast)$$ with $$T^\ast =\infty$$ in the case when $$M\leqq 8\pi$$ and $$T^\ast <\infty$$ in the case when $$M>8\pi$$. The proof uses a DiPerna-Lions renormalizing argument which makes it possible to get the “optimal regularity” as well as an estimate of the difference of two possible solutions in the critical $$L^{4/3}$$ Lebesgue norm similarly to the $$2d$$ vorticity Navier-Stokes equation.
(2)
We prove the immediate smoothing effect and, in the case $$M< 8\pi$$, we prove the Sobolev norm bound uniformly in time for the rescaled solution (corresponding to the self-similar variables).
(3)
In the case $$M<8\pi$$, we also prove the weighted $$L^{4/3}$$ linearized stability of the self-similar profile and then the universal optimal rate of convergence of the solution to the self-similar profile. The proof is mainly based on an argument of enlargement of the functional space for semigroup spectral gap.

##### MSC:
 35Q92 PDEs in connection with biology, chemistry and other natural sciences 92C17 Cell movement (chemotaxis, etc.) 35D30 Weak solutions to PDEs
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