Two-species chemotaxis-competition system with singular sensitivity: Global existence, boundedness, and persistence

This paper is concerned with the following parabolic-parabolic-elliptic chemotaxis system with singular sensitivity and Lotka-Volterra competitive kinetics, {u(t) = Delta u - chi(1)del.(u/w del w) + u(a(1) - b(1)u - c(1)v), x is an element of Omega v(t) = Delta v - chi(2)del.(v/w del w) + v(a(2) - b(2)v - c(2)u), x is an element of Omega (0.1) 0 = Delta w - mu w + nu u + lambda v, x is an element of Omega partial derivative u/partial derivative n = partial derivative v/partial derivative n = partial derivative w/partial derivative n = 0, x is an element of partial derivative Omega, where Omega subset of R-N is a bounded smooth domain, and chi(i), a(i), b(i), c(i) (i = 1, 2) and mu, nu, lambda are positive constants. This is the first work on two-species chemotaxis-competition system with singular sensitivity and Lotka-Volterra competitive kinetics. Among others, we prove that for any given nonnegative initial data u(0), v(0) is an element of C-0((Omega) over bar) with u(0) + v(0) not equivalent to 0, (0.1) has a unique globally defined classical solution (u(t, x; u(0), v(0)), v(t, x; u(0), v(0)), w(t, x; u(0), v(0))) with u(0, x; u(0), v(0)) = u(0)(x) and v(0, x; u(0), v(0)) = v(0)(x) in any space dimensional setting with any positive constants chi(i), a(i), b(i), c(i) (i = 1, 2) and mu, nu, lambda. Moreover, we prove that there is chi*(mu, chi(1), chi(2)) > 0 satisfying chi*(mu, chi(1), chi(2)) = {mu chi(2)/4 if 0 < chi < 2 when chi(1) = chi(2) := chi mu(chi - 1) if chi >= 2, and chi*(mu, chi(1), chi(2)) <= min {mu chi(2) + mu(chi(1) - chi(2))(2)/4, mu chi(1) + mu(chi(2) - chi(1))(2)/4}, when chi(1) not equal chi(2) such that the condition min{a(1), a(2)} > chi* (mu, chi(1), chi(2)) implies lim sup(t -> infinity) parallel to u(t, .; u(0), v(0)) + v(t, .; u(0), v(0))parallel to(infinity) <= M* and lim inf(t ->infinity) inf(x is an element of Omega) (u(t, x, u(0), v(0)) + v(t, x; u(0), v(0))) >= m* for some positive constants M*, m* independent of u(0), v(0), the latter is referred to as combined pointwise persistence. (c) 2023 Elsevier Inc. All rights reserved.