Asymptotic stability in a fully parabolic quasilinear chemotaxis model with general logistic source and signal production

In this paper we study the asymptotic behaviors of global solutions to the fully parabolic chemotaxis system: u(t) = del . (D(u)del u - S(u)del v)+ ru - mu u( l+sigma),v(t) = Delta v - v u(gamma), subject to the homogeneous Neumann boundary conditions in a bounded and smooth domain Omega subset of R-n (n >= 2), where parameters mu, sigma, gamma > 0, r is an element of R, and the nonlinearity D, S is an element of C-2 ([0, infinity)) are supposed to generalize the prototypes D(u) >= a(0)(u+1)(-alpha), 0 <= S(u) <= b(0)u(u+1)(beta-1) with a(0), b(0) > 0 and alpha, beta is an element of R. We first consider the case of r > 0 and provide a boundedness result under alpha + beta + gamma < 2/n or beta + gamma < 1 + sigma, or beta + gamma = 1 + sigma with large mu > 0. The main result is concerned with the n asymptotic stability when damping effects of logistic source are strong enough. Specifically, there is mu(0) > 0 independent of initial data, such that the bounded classical solution (u, v) satisfies (u, v) -> ((r/mu)(1/sigma), (r/mu) (gamma/sigma)) in L-infinity(Omega) exponentially under conditions of mu > mu(0) and r > 0. For the case of r < 0, the trivial constant equilibria in the model is obtained in a priori way, that is, any bounded solution (u, v) satisfies (u,v) -> (0, 0) in L-infinity(Omega) exponentially, regardless of the size of mu > 0. (C) 2019 Elsevier Inc. All rights reserved.