Local existence and global boundedness for a chemotaxis system with gradient dependent flux limitation (Q6610241)

The authors study the parabolic-elliptic chemotaxis system with flux limitation \N\[\N\begin{cases} u_t = \nabla \cdot \left( \frac{u \nabla u}{\sqrt{u^2 + |\nabla u|^2}} \right) - \chi \nabla \cdot \left( \frac{u \nabla v}{(1+ |\nabla v|^2)^\alpha} \right), & \quad (x,t) \in \Omega \times (0,\infty),\\\N0 = \Delta v - \mu +u, & \quad (x,t) \in \Omega \times (0,\infty),\\\N\end{cases}\N\]\Nendowed with homogeneous Neumann boundary conditions and initial data \(u_0 \in C^3 (\overline{\Omega})\). Moreover, it is assumed that \(\Omega = B_R(0) \subset \mathbb{R}^n\), \(n \in \mathbb{N}\), is a ball, \(\chi\) and \(\alpha\) are positive constants, \(\mu := \frac{1}{|\Omega|} \int_\Omega u_0(x) dx >0\), and \(u_0\) is radially symmetric and positive in \(\overline{\Omega}\) with \(\nabla u_0 \cdot \nu =0\) on \(\partial \Omega\), where \(\nu\) denotes the outward unit normal on \(\partial\Omega\).\N\NRequiring these assumptions, the authors prove the existence of a unique classical solution along with an extensibility criterion for the maximal existence time. In addition, the authors prove that if \(\chi < \chi_0\) is satisfied with some explicitly determined \(\chi_0 >0\) depending on \(\alpha\) and the mass \(\int_\Omega u_0(x)dx\), then this classical solution is global in time and bounded. These results extend those for the case \(\alpha = \frac{1}{2}\) from [\textit{N. Bellomo} and \textit{M. Winkler}, Commun. Partial Differ. Equations 42, No. 3, 436--473 (2017; Zbl 1430.35166)].\N\NImportant ingredients of the proofs are comparison arguments and appropriate a priori estimates involving \(u\), the radial derivative \(u_r\), and \(z:=\frac{u_t}{u}\) as well as a parabolic inequality satisfied by \(z\).