The keller-segel system with logistic growth and signal-dependent motility

The paper is concerned with the following chemotaxis system with nonlinear motility functions (equation presented) subject to homogeneous Neumann boundary conditions in a bounded domain Ω⊂ R2 with smooth boundary, where the motility functions γ(v) and χ(v) satisfy the following conditions • (γ; χ) 2 [C2[0;1)]2 with (v) >0 and jχ(v)j2 γ(v) is bounded for all v ≥ 0. By employing the method of energy estimates, we establish the existence of globally bounded solutions of (*) with μ >0 for any u0 ∈ W1;1() with u0 ≥ (≢)0. Then based on a Lyapunov function, we show that all solutions (u; v) of (*) will exponentially converge to the unique constant steady state (1; 1) provided μ>K0 16 with K0 = max 0≤v≤1 Jχ(v)j2 γ(v) .