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$ L^γ$-measure criteria for boundedness in a quasilinear parabolic-elliptic Keller-Segel system with supercritical sensitivity

  • * Corresponding author: Mengyao Ding

    * Corresponding author: Mengyao Ding 
The first author is supported by the National Natural Science Foundation of China (11571020, 11671021, 11171048).
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  • This paper studies the parabolic-elliptic Keller-Segel system with supercritical sensitivity: $u_{t} = \nabla·(D(u)\nabla u)-\nabla ·(S(u)\nabla v)$ , $0 = Δ v -v+u$ in $Ω× (0,T)$ , where the bounded domain $Ω\subset\mathbb{R}^n$ , $n≥2$ , subject to the non-flux boundary conditions, $D(u)≥ a_0(u+1)^{-q}$ , $0≤ S(u)≤ b_0u(u+1)^{α-q-1}$ with $q \in \mathbb{R}$ , $α>\frac{2}{n}$ , and $a_0, b_0>0$ . It is proved that the problem possesses a unique globally bounded solution for $α>\frac{2}{n}$ whenever $\|u_0\|_{L^{\frac{nα}{2}}}$ is sufficiently small. In addition, we establish the large-time behavior of solutions when $q = 0$ .

    Mathematics Subject Classification: Primary: 35B35, 35K55; Secondary: 92C17.


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