Boundedness and exponential stabilization in a degenerate migration-consumption system with logistic damping

We study the no-flux problem for a taxis-type migration-consumption system with logistic damping in a bounded planar domain, where the motility function $\phi$ vanishes at zero and degenerates algebraically of order $\alpha \ge 1$. We prove uniform boundedness in $L^{\mathrm{\infty }}(\mathrm{\Omega })\times W^{1,\mathrm{\infty }}(\mathrm{\Omega })$ for every fixed damping coefficient $b>0$, without any smallness assumption on the initial data, thus improving the previous $L^2$-boundedness results. For $\alpha >1$, we further prove that the solution converges exponentially to the spatially homogeneous state $(a\mathrm{/}b,0)$, with an explicit admissible rate. The proof combines a weighted Gagliardo–Nirenberg type inequality adapted to the degeneracy with the dissipative structure of the signal equation. Therefore logistic damping drives the system toward homogeneity, despite the loss of uniform elliptic regularization.