Dynamics in a parabolic-elliptic chemotaxis system with growth source and nonlinear secretion

In this work, we are concerned with a class of parabolic-elliptic chemotaxis systems with the prototype given by

$\left\{ \begin{array}{lll}&u_t = \nabla\cdot(\nabla u-\chi u\nabla v)+au-bu^\theta, &x\in \Omega, t>0, \\&0 = \Delta v -v+u^\kappa, & x\in \Omega, t>0 \end{array}\right.$

with nonnegative initial condition for $u$ and homogeneous Neumann boundary conditions in a smooth bounded domain $Ω\subset \mathbb{R}^n(n≥ 2)$, where $χ, b, κ>0$, $a∈ \mathbb{R}$ and $θ>1$.

First, using different ideas from [9,11], we re-obtain the boundedness and global existence for the corresponding initial-boundary value problem under, either

$κ+1<\max\{θ, 1+\frac{2}{n}\}$

or

$θ = κ+1, \ \ b≥ \frac{(κ n-2)}{κ n}χ.$

Next, carrying out bifurcation from "old multiplicity", we show that the corresponding stationary system exhibits pattern formation for an unbounded range of chemosensitivity $χ$ and the emerging patterns converge weakly in $ L^θ(Ω)$ to some constants as $χ \to ∞$. This provides more details and also fills up a gap left in Kuto et al. [13] for the particular case that $θ = 2$ and $κ = 1$. Finally, for $θ = κ+1$, the global stabilities of the equilibria $((a/b)^{\frac{1}{κ}}, a/b)$ and $(0,0)$ are comprehensively studied and explicit convergence rates are computed out, which exhibits chemotaxis effects and logistic damping on long time dynamics of solutions. These stabilization results indicate that no pattern formation arises for small $χ$ or large damping rate $b$; on the other hand, they cover and extend He and Zheng's [6,Theorems 1 and 2] for logistic source and linear secretion ($θ = 2$ and $κ = 1$) (where convergence rate estimates were shown) to generalized logistic source and nonlinear secretion.