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: $$ {\left\{ \begin{array}{ll} u_t=\Delta u-\nabla \cdot (u\nabla v)+\lambda (u-u^\kappa )-b|\nabla u|^\gamma , & x\in \Omega , \,t>0, \\ v_t=\Delta v-v+u, & x\in \Omega , \,t>0, \end{array}\right. } $$ in a bounded smooth domain $$\Omega \subset \mathbb {R}^n$$ . The parameters satisfy $$\lambda \ge 0$$ , $$b>0$$ , $$\kappa \in (1,2)$$ , and $$\gamma \in [1,2]$$ . Within this framework, uniform asymptotic stabilization toward spatially homogeneous equilibria is established in two distinct regimes. First, in the presence of the logistic source ( $$\lambda >0$$ ), we exploit the dissipation structure of a Lyapunov-type functional to prove that for sufficiently large $$\lambda $$ and appropriate conditions on $$\kappa $$ and $$\gamma $$ , global solutions converge exponentially to the steady state (1, 1) in $$L^\infty $$ -norm. Second, in the absence of the logistic source ( $$\lambda =0$$ ), we adopt the framework of LaSalle’s invariance principle. By establishing the relative compactness of trajectories and characterizing the $$\omega $$ -limit set via the mass dissipation induced by the gradient damping, we prove that the solution stabilizes in the $$C^1$$ -topology to a constant state uniquely determined by the asymptotic mass.