On a class of Keller-Segel chemotaxis systems with cross-diffusion
Global existence and the structure of the set of steady states are studied for a chemotaxis system with cross-diffusion \[ \begin{aligned} \partial_t u &= \text{div}\left( d_1 \nabla u - \chi \phi_1(u) \phi_2(v) \nabla v \right), \quad x\in\Omega, \, t>0, \\ \partial_t v &= \delta \Delta u + d_2 \Delta v - \alpha v + f(u), \quad x\in\Omega, \, t>0,\end{aligned} \] supplemented with homogeneous Neumann boundary conditions and nonnegative initial data \((u_0,v_0)\). Here, \(\Omega\) is a bounded domain of \(\mathbb{R}^n\), \(1\leq n\leq 3\), \((\delta,d_1,d_2,\chi,\alpha)\) are positive parameters, and \(f\) is a sublinear increasing function satisfying \(f(0)=0\). The chemotactic sensitivity is taken to be \(\chi \phi_1(u) \phi_2(v)\) with \(c_1 (1+|s|)^{-\beta} \leq \phi_2(s)\leq c_2\) for some positive \(c_1, c_2\) and \(\beta\in (0,1)\) and the assumptions on \(\phi_1\) include in particular the classical Keller-Segel model \(\phi_1(r)=r\) and the volume-filling model \(\phi_1(r) = r(1-r/K)\), \(K>0\). The fact that the addition of the cross-diffusion term \(\delta \Delta u\) prevents the finite time blow-up, which occurs when \(\delta=0\), and ensures global existence was noticed in [\textit{S. Hittmeir} and \textit{A. Jüngel}, SIAM J. Math. Anal. 43, No. 2, 997--1022 (2011; Zbl 1259.35114)] when \(\phi_1=\text{id}\), \(\phi_2=1\), and \(f=\text{id}\). This result is extended here to a wider setting with a similar proof. Concerning the stationary problem, it is clear that \((u^*,f(u^*)/\alpha)\) is a constant steady state which is unstable if \(\delta\) is small enough and \(\chi\) is large enough. Local and global bifurcation techniques are then used to show the existence of non-constant stationary solutions for sufficiently small values of \(\delta\). The stability of these solutions is also studied. More precise results on this issue are obtained in one space dimension.
scientific article