Asymptotic stability in a quasilinear single-species chemotaxis system with one stimulus

This work focuses on a Neumann initial boundary value problem in smoothly bounded domain Ω⊂Rn(n≥2). The first part of the paper shows that the system possesses globally bounded classical solution whether for the parabolic-elliptic case or fully parabolic case. The second part proves that for parabolic-elliptic case, the solutions converge to (0, 0) and another constant steady state solutions exponentially or algebraically in three cases; whereas, for fully parabolic case, the solution decays to (0, 0) algebraically in a certain case. We extend the results obtained recently by Tao et al. [J. Math. Anal. Appl. 474, 733–747 (2019)] and Ding et al. [J. Differ. Equations 268, 6729–6777 (2020)].