Long time asymptotics of small mass solutions for a chemotaxis-consumption system involving prescribed signal concentrations on the boundary

Abstract

This paper investigates a parabolic–elliptic chemotaxis-consumption system with signal dependent sensitivity $\chi =\chi \left(c\right)$ under no-flux/Dirichlet boundary conditions. For general $\chi$ which may allow singularities at $c=0$, the global existence and boundedness of radial large data solutions are established in dimensions $d\ge 2$. In particular, when $\chi \left(c\right)=1$, we also find that the constructed solution converges asymptotically to a nonhomogeneous steady state if the initial mass is small. On the other hand, for the system with $\chi \left(c\right)=c$, a Lyapunov-type inequality is derived. This inequality not only leads to a result on global existence of smooth solutions with non-radial large data in two dimensions but moreover, provides long-time asymptotics of non-radial $(d=2)$ and radial $(d\ge 2)$ solutions at suitably small mass levels.