Forced waves of parabolic-elliptic Keller-Segel models in shifting environments

The current paper is concerned with the forced waves of Keller-Segel chemoattraction systems in shifting environments of the form, \begin{equation} \begin{cases} u_t=u_{xx}-\chi(uv_x)_x +u(r(x-ct)-bu),\quad x\in\mathbb{R}\cr 0=v_{xx}- \nu v+\mu u,\quad x\in \mathbb{R} \end{cases} (1) \end{equation} where $\chi$, $b$, $\nu$, and $\mu$ are positive constants, $c\in R$, the resource function $r(x)$ is globally H\"older continuous, bounded, $r^*=\sup_{x\in R}r(x)>0$, $r(\pm \infty):=\lim_{x\to \pm\infty}r(x)$ exist, and either $r(-\infty)<0<r(\infty)$, or $r(\pm\infty)<0$. Assume that $b>2\chi\mu$. In the case that $r(-\infty)<0<r(\infty)$, it is shown that (1) has a forced wave solution connecting $(\frac{r^*}{b},\frac{\mu}{\nu}\frac{r^*}{b})$ and $(0,0)$ with speed $c$ provided that $c>\frac{\chi\mu r^*}{2\sqrt \nu (b-\chi\mu)}- 2\sqrt{\frac{r^*(b-2\chi\mu)}{b-\chi\mu}}$. In the case that $r(\pm\infty)<0$, it is shown that (1) has a forced wave solution connecting $(0,0)$ and $(0,0)$ with speed $c$ provided that $\chi$ is sufficiently small and $\lambda_\infty>0$, where $\lambda_\infty$ is the generalized principal eigenvalue of the operator $u(\cdot)\mapsto u_{xx}(\cdot)+cu_{x}(\cdot)+r(\cdot)u(\cdot)$ on $R$ in certain sense. Some numerical simulations are also carried out. The simulations indicate the existence of forced wave solutions in some parameter regions which are not covered in the theoretical results, induce several problems to be further studied, and also provide some illustration of the theoretical results.