Boundedness in a quasilinear fully parabolic Keller-Segel system via maximal Sobolev regularity

Abstract

This paper deals with the quasilinear Keller-Segel system$ \begin{align*} \begin{cases} u_t = \nabla\cdot(D(u)\nabla u)-\nabla\cdot(S(u)\nabla v), &x \in \Omega, \ t>0, \\ \ v_t = \Delta v - v +u, &x \in \Omega, \ t>0 \end{cases} \end{align*} $in $ \Omega = \mathbb{R}^N $ or in a smoothly bounded domain $ \Omega\subset \mathbb{R}^N $, with nonnegative initial data $ u_0\in L^1(\Omega) \cap L^\infty(\Omega) $, and $ v_0\in L^1(\Omega) \cap W^{1, \infty}(\Omega) $; in the case that $ \Omega $ is bounded, it is supplemented with homogeneous Neumann boundary condition. The diffusivity $ D(u) $ and the sensitivity $ S(u) $ are assumed to fulfill $ D(u)\ge u^{m-1}\ (m\geq1) $ and $ S(u)\leq u^{q-1}\ (q\geq 2) $, respectively. This paper derives uniform-in-time boundedness of nonnegative solutions to the system when $ q<m+\frac{2}{N} $. In the case $ \Omega = \mathbb{R}^N $ 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.