Stabilization in a Keller-Segel system with gradient-dependent damping

This paper investigates the global dynamics of the fully parabolic Keller-Segel system equipped with a sub-quadratic logistic source and a nonlinear gradient-dependent damping: \ arrayll uₜ=Δ u-∇ · (u∇ v)+λ (u-u^κ )-b|∇ u|^γ , & x∈ Ω , \,t>0, \\ vₜ=Δ v-v+u, & x∈ Ω , \,t>0, array. in a bounded smooth domain Ω ⊂ Rⁿ . The parameters satisfy λ ≥ 0 , $$b>0$$ , κ ∈ (1,2) , and γ ∈ [1,2] . Within this framework, uniform asymptotic stabilization toward spatially homogeneous equilibria is established in two distinct regimes. First, in the presence of the logistic source ( λ >0 ), we exploit the dissipation structure of a Lyapunov-type functional to prove that for sufficiently large λ and appropriate conditions on κ and γ , global solutions converge exponentially to the steady state (1, 1) in L^∞ -norm. Second, in the absence of the logistic source ( λ =0 ), we adopt the framework of LaSalle’s invariance principle. By establishing the relative compactness of trajectories and characterizing the ω -limit set via the mass dissipation induced by the gradient damping, we prove that the solution stabilizes in the C¹ -topology to a constant state uniquely determined by the asymptotic mass.