Global boundedness of radial solutions to a parabolic-elliptic chemotaxis system with flux limitation and nonlinear signal production
Hong Yi, Chunlai Mu, Shuyan Qiu, Lü Xu
Abstract
The following degenerate chemotaxis system with flux limitation and nonlinear signal production $ \begin{equation*} \begin{cases} u_t = \nabla\cdot(\frac{u\nabla u}{\sqrt {u^{2}+|\nabla u|^{2}}})-\chi\nabla\cdot(\frac{u\nabla v}{\sqrt {1+|\nabla v|^{2}}}) \quad &in\quad B_{R}\times(0, +\infty), \\ 0 = \Delta v-\mu (t)+u^{\kappa}, \quad \mu(t): = \frac{1}{|\Omega|}\int_{\Omega}u^{\kappa}(\cdot, t) \quad &in\quad B_{R}\times(0, +\infty) \end{cases} \end{equation*} $ is considered in balls $ B_R = B_R(0)\subset \mathbb{R}^n $ for $ n\geq 1 $ and $ R>0 $ with no-flux boundary conditions, where $ \chi>0, \kappa>0 $. We obtained local existence of unique classical solution and extensibility criterion ruling out gradient blow-up, and moreover proved global existence and boundedness of solutions under some conditions for $ \chi, \kappa $ and $ \int_{B_R}u_{0} $.