Extensibility criterion ruling out gradient blow-up in a quasilinear degenerate chemotaxis system with flux limitation

Abstract

This paper deals with the quasilinear degenerate chemotaxis system with flux limitation \begin{equation*} \begin{cases} u_t = \nabla\cdot\left(\dfrac{u^p \nabla u}{\sqrt{u^2 + |\nabla u|^2}} \right) -χ\nabla\cdot\left(\dfrac{u^q\nabla v}{\sqrt{1 + |\nabla v|^2}}\right), \\[1mm] 0 = Δv - μ+ u \end{cases}\end{equation*} under no-flux boundary conditions in balls $\Omega \subset {\mathbb{R}}^n$, and the initial condition $u{|}_{t=0}=u_0$ for a radially symmetric and positive initial data $u_0\in C^3(\overline{\Omega })$, where $\chi >0$ and $\mu :=\frac{1}{|\Omega |}{\int }_{\Omega }{u}_0$. Bellomo--Winkler (Comm.\ Partial Differential Equations;2017;42;436--473) proved local existence of unique classical solutions and extensibility criterion ruling out gradient blow-up as well as global existence and boundedness of solutions when $p=q=1$ under some conditions for $\chi$ and ${\int }_{\Omega }{u}_0$. This paper derives local existence and extensibility criterion ruling out gradient blow-up when $p,q\ge 1$, and moreover shows global existence and boundedness of solutions when $p>q+1-\frac{1}{n}$.

44 pages