THE KELLER-SEGEL SYSTEM WITH LOGISTIC GROWTH AND SIGNAL-DEPENDENT MOTILITY

作者：Jin, Hai-Yang; Wang, Zhi-An^*

来源：Discrete and Continuous Dynamical Systems - Series B, 2021, 26(6): 3023-3041.

DOI：10.3934/dcdsb.2020218

• Chemotaxis
• density-dependent motility
• global boundedness
• exponential decay

The paper is concerned with the following chemotaxis system with nonlinear motility functions @@@ {u(t) = del . (gamma(v)del u - u chi(v)del v) + mu u(1 - u), x is an element of Omega, t > 0, 0 = Delta v + u - v, x is an element of Omega, t > 0, (*) u(x, 0) = u(0)(x), x is an element of Omega, @@@ subject to homogeneous Neumann boundary conditions in a bounded domain Omega subset of R-2 with smooth boundary, where the motility functions gamma(v) and chi(v) satisfy the following conditions @@@ (gamma, chi) is an element of [C-2 [0, infinity)](2) with gamma(v) > 0 and vertical bar chi(v)vertical bar(2)/gamma(v) is bounded for all v >= 0. @@@ By employing the method of energy estimates, we establish the existence of globally bounded solutions of (*) with mu > 0 for any u(0) is an element of W-1,W-infinity (Omega) with u(0) >= (not equivalent to)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 mu > K-0/16 with K-0 = max(0 <= v <=infinity) vertical bar chi(v)vertical bar(2)/gamma(v).