Boundedness in a two-species chemotaxis system with nonlinear sensitivity and signal secretion

The fully parabolic two-species chemotaxis system with nonlinear sensitivity, Lotka-Volterra competitive kinetics, and signal secretion {u(t) = Delta u - del . (S-1(u)del v) + mu(1)u(1 - u(alpha 1) - a(1)w), x is an element of Omega, t > 0, v(t) = Delta v - v + g(1)(w), x is an element of Omega, t > 0, w(t) = Delta w - del . (S-2(w)del z) + mu(2)w(1 - w(alpha 2) - a(2)u), x is an element of Omega, t > 0, z(t) = Delta z - z + g(2)(u), x is an element of Omega, t > 0 is considered in a bounded domain Omega subset of R-3 with homogeneous Neumann boundary conditions and the parameters mu(i), a(i), a(i) > 0 (i = 1, 2). The chemotactic sensitivity function vertical bar S-i(s)vertical bar <= vertical bar chi(i)vertical bar s(qi) for all s > 0 with chi(i) is an element of R and q(i) > 0 (i = 1, 2). The nonlinear signal secretion function g(i)(s) <= k(i)s(gamma i) for all s > 0 with k(i) > 0 and gamma(i) > 0 (i = 1, 2). When the initial data satisfy the appropriate regularization assumptions, we prove that the system has a global bounded classical solution under different parameter conditions. Our results partially improve on results reported by Ren and Liu (2019) [22]. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

This study investigates a fully parabolic two-species chemotaxis system with nonlinear sensitivity and signal secretion. It is proven that the system has a global bounded classical solution under different parameter conditions when the initial data satisfy appropriate regularization assumptions. This work partially builds on results reported by Ren and Liu (2019) [22].