Large-time dynamics of solutions in a logistic chemotaxis system with weak singular sensitivity: Uniform boundedness, pointwise persistence and stability

This paper investigates the long-term dynamics of solutions to a parabolic-elliptic chemotaxis system with weakly singular sensitivity and a logistic source under the Neumann boundary conditions in a smooth bounded domain $ \Omega \subset \mathbb{R}^N $ with $ N \geq 2 $

$ \begin{equation*} \begin{cases} u_t = \Delta u-\chi \nabla\cdot \Big(\frac{u}{v^{\lambda}} \nabla v \Big) +ru- \mu u^2, \quad &x\in \Omega, \\ 0 = \Delta v- \alpha v +\beta u, \quad &x\in \Omega, \end{cases} \end{equation*} $

where the parameters $ \chi, \, r, \, \mu, \, \alpha, \, \beta $ are positive constants and $ \lambda \in (0, 1). $ The present study improves previously established results on the global existence and boundedness of classical solutions; and constitutes the first investigation of the large-time dynamics of globally bounded solutions. In particular, it has provide a detailed analysis of their uniform boundedness, pointwise persistence, and asymptotic stability.

For all suitably smooth initial data $ u_0\in C^0(\bar\Omega) $ with $ u_0 \not \equiv 0, $ the following results have been established. First, there exists $ \mu > \mu_1^*(N, \lambda, \chi, \beta) $ such that all classical solutions exist globally and remain bounded. Next, there exists $ \mu > \mu_2^*(N, \lambda, \chi, \beta) $ such that every global positive solution is uniformly bounded from above and below by positive constants independent of its initial function $ u_0. $ Last, there exists $ \mu > \mu_3^*(N, \lambda, \chi, \alpha, \beta, r, \Omega) $ such that any globally bounded classical solution exponentially converges to the constant steady state $ (\frac{r}{\mu}, \frac{\beta}{\alpha}\frac{r}{\mu}). $