Stabilization and pattern formation in chemotaxis models with acceleration and logistic source.

We consider the following chemotaxis-growth system with an acceleration assumption, \begin{align*} \begin{cases} u_t= \Delta u -\nabla \cdot\left(u \bw \right)+\gamma\xkh{u-u^\alpha}, & x\in\Omega,\ t>0,\\ v_t=\Delta v- v+u, & x\in\Omega,\ t>0,\\ \bw_t= \Delta \bw -\bw +\chi\nabla v, & x\in\Omega,\ t>0, \end{cases} \end{align*} under the homogeneous Neumann boundary condition for $u,v$ and the homogeneous Dirichlet boundary condition for $\bw$ in a smooth bounded domain $\Omega\subset\R^{n}$ ($n\geq1$) with given parameters $\chi>0$, $\gamma\geq0$ and $\alpha>1$. It is proved that for reasonable initial data with either $n\leq3$, $\gamma\geq0$, $\alpha>1$ or $n\geq4,\ \gamma>0,\ \alpha>\frac12+\frac n4$, the system admits global bounded solutions, which significantly differs from the classical chemotaxis model that may have blow-up solutions in two and three dimensions. For given $\gamma$ and $\alpha$, the obtained global bounded solutions are shown to convergence exponentially to the spatially homogeneous steady state $(m,m,\mathbf 0$) in the large time limit for appropriately small $\chi$, where $m=\frac1{|\Omega|}\jfo u_0(x)$ if $\gamma=0$ and $m=1$ if $\gamma>0$.