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

Abstract

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 $\gamma>0$. Outside the stable parameter regime, we conduct linear analysis to specify possible patterning regimes. In weakly nonlinear parameter regimes, with a standard perturbation expansion approach, we show that the above asymmetric model can generate pitchfork bifurcations which occur generically in symmetric systems. Moreover, our numerical simulations demonstrate that the model can generate rich aggregation patterns, including stationary, single merging aggregation, merging and emerging chaotic, and spatially inhomogeneous time-periodic. Some open questions for further research are discussed.