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

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).

Automated summary

This paper investigates the chemotaxis system with nonlinear motility functions and establishes the existence of globally bounded solutions under specific conditions, showing that all solutions will exponentially converge to the unique constant steady state (1, 1) when mu > K-0/16.