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Uniform $L^{∞}$ boundedness for a degenerate parabolic-parabolic Keller-Segel model

  • * Corresponding author: Wenting Cong

    * Corresponding author: Wenting Cong 
The first author is supported by NSFC grant 11271154.
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  • This paper investigates the existence of a uniform in time $L^{∞}$ bounded weak entropy solution for the quasilinear parabolic-parabolic Keller-Segel model with the supercritical diffusion exponent $0<m<2-\frac{2}{d}$ in the multi-dimensional space ${\mathbb{R}}^d$ under the condition that the $L^{\frac{d(2-m)}{2}}$ norm of initial data is smaller than a universal constant. Moreover, the weak entropy solution $u(x,t)$ satisfies mass conservation when $m>1-\frac{2}{d}$ . We also prove the local existence of weak entropy solutions and a blow-up criterion for general $L^1\cap L^{∞}$ initial data.

    Mathematics Subject Classification: Primary:35K65, 35K59, 92C17.


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