Boundedness in a quasilinear fully parabolic Keller-Segel system via maximal Sobolev regularity
Boundedness in a quasilinear fully parabolic Keller-Segel system via maximal Sobolev regularity
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DOI：
10.3934/dcdss.2020012
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发表时间：
2020
期刊：
影响因子：
--
通讯作者：
Sachiko Ishida;T. Yokota
中科院分区：
文献类型：
--
作者：
Sachiko Ishida;T. Yokota
This paper deals with the quasilinear Keller-Segel system \begin{document}$ \begin{align*} \begin{cases} u_t = \nabla\cdot(D(u)\nabla u)-\nabla\cdot(S(u)\nabla v), x'>in \begin{document}$ \Omega = \mathbb{R}^N $\end{document} or in a smoothly bounded domain \begin{document}$ \Omega\subset \mathbb{R}^N $\end{document} , with nonnegative initial data \begin{document}$ u_0\in L^1(\Omega) \cap L^\infty(\Omega) $\end{document} , and \begin{document}$ v_0\in L^1(\Omega) \cap W^{1, \infty}(\Omega) $\end{document} ; in the case that \begin{document}$ \Omega $\end{document} is bounded, it is supplemented with homogeneous Neumann boundary condition. The diffusivity \begin{document}$ D(u) $\end{document} and the sensitivity \begin{document}$ S(u) $\end{document} are assumed to fulfill \begin{document}$ D(u)\ge u^{m-1}\ (m\geq1) $\end{document} and \begin{document}$ S(u)\leq u^{q-1}\ (q\geq 2) $\end{document} , respectively. This paper derives uniform-in-time boundedness of nonnegative solutions to the system when \begin{document}$ q . In the case \begin{document}$ \Omega = \mathbb{R}^N $\end{document} the result says boundedness which was not attained in a previous paper (J. Differential Equations 2012; 252:1421-1440). The proof is based on the maximal Sobolev regularity for the second equation. This also simplifies a previous proof given by Tao-Winkler (J. Differential Equations 2012; 252:692-715) in the case of bounded domains.
This paper deals with the quasilinear Keller-Segel system \begin{document}$ \begin{align*} \begin{cases} u_t = \nabla\cdot(D(u)\nabla u)-\nabla\cdot(S(u)\nabla v), x'>in \begin{document}$ \Omega = \mathbb{R}^N $\end{document} or in a smoothly bounded domain \begin{document}$ \Omega\subset \mathbb{R}^N $\end{document} , with nonnegative initial data \begin{document}$ u_0\in L^1(\Omega) \cap L^\infty(\Omega) $\end{document} , and \begin{document}$ v_0\in L^1(\Omega) \cap W^{1, \infty}(\Omega) $\end{document} ; in the case that \begin{document}$ \Omega $\end{document} is bounded, it is supplemented with homogeneous Neumann boundary condition. The diffusivity \begin{document}$ D(u) $\end{document} and the sensitivity \begin{document}$ S(u) $\end{document} are assumed to fulfill \begin{document}$ D(u)\ge u^{m-1}\ (m\geq1) $\end{document} and \begin{document}$ S(u)\leq u^{q-1}\ (q\geq 2) $\end{document} , respectively. This paper derives uniform-in-time boundedness of nonnegative solutions to the system when \begin{document}$ q . In the case \begin{document}$ \Omega = \mathbb{R}^N $\end{document} the result says boundedness which was not attained in a previous paper (J. Differential Equations 2012; 252:1421-1440). The proof is based on the maximal Sobolev regularity for the second equation. This also simplifies a previous proof given by Tao-Winkler (J. Differential Equations 2012; 252:692-715) in the case of bounded domains.