Finite-time blow-up and global boundedness for chemotaxis system with strong logistic dampening

In the present study, we consider the chemotaxis system with logistic-type superlinear degradation {partial derivative(t)u(1)= tau(1)Delta u(1) - chi(1)del . (u(1)del(v)) + lambda(1)u(1) - mu(1)u(1)(k1), x is an element of Omega, t > 0, partial derivative(tau)u(2) = tau(2)Delta u(2) - chi(2)del . (u(2)del v) + lambda(2)u(2) - mu(2)u(2)(k2), x is an element of Omega, t > 0, 0 = Delta v - gamma v + alpha(1)u(1) + alpha(2)u(2), x is an element of Omega, t > 0, under the homogeneous Neumann boundary condition, where gamma > 0, tau(i) > 0, chi(i) > 0, lambda(i) is an element of R, mu(i) > 0, alpha(i) > 0 (i = 1, 2). Consider an arbitrary ball Omega = B-R(0) subset of R-n, n >= 3, R > 0, when k(i) > 1(i = 1,2), it is shown that for any parameter (k) over cap = max{k(1), k(2)} satisfies (k) over cap < {7/6 if n is an element of {3,4}, 1 + 1/2(n-1) if n >= 5, there exist nonnegative radially symmetric initial data under suitable conditions such that the corresponding solutions blow up in finite time in the sense that lim sup(t NE arrow Tmax) (parallel to u(1)(.t)parallel to(L infinity(Omega)) + parallel to u(2)(.,t)parallel to(L infinity(Omega))) = infinity for some 0 < T-max < infinity. Furthermore, for any smooth bounded domain Omega C R-n(n >= 1), when k(i )>= 2(i = 1, 2), we prove that the system admits a unique global bounded solution. (C) 2020 Elsevier Inc. All rights reserved.