Large time behavior of solutions for density-suppressed motility system in higher dimensions

In this paper, we study the asymptotic stability of the following density-suppressed motility system {u(t) = Delta(gamma(v)u) + mu u(1 - u), x is an element of Omega, t > 0, v(t) = Delta v + u - v, x is an element of Omega, t > 0, in a bounded domain with smooth boundary, where the motility function gamma(v) and mu are positive. Under the conditions that the motility function-gamma(v) has the lower upper bounded and that mu, is large enough, we derive that system (*) possesses a unique global solution in three-dimensional space. Moreover, we show that if the global classical solution exists of system (*) in any dimensional space, then the solution will converge to the equilibrium (1, 1) exponentially as t -> +infinity when mu > K-0/16 with K-0 = max(0 <= v <=infinity) vertical bar gamma'(v)vertical bar(2)/gamma(v). (C) 2019 Elsevier Inc. All rights reserved.