A quasilinear attraction-repulsion chemotaxis system with logistic source

This paper is concerned with the quasilinear attraction-repulsion chemotaxis sys-tem with logistic source under Neumann boundary conditions in a bounded domain ohm subset of RN(N >= 1), where D, Phi, Psi is an element of C2[0, infinity) are nonnegative functions with D(s) >= (s + 1)p for s >= 0, Phi(s) <= chi sq, xi sr <= Psi (s) <= zeta sr for s > 1, and the smooth function f satisfies f (s) <= mu s(1 - sk) for s > 0, f (0) >= 0. Tian et al. (2016) proved that when q = max{r, k} and q -p >= 2N, if one of the following assumptions holds: (i) q = r = k, mu > (alpha chi-gamma xi)(1-N(q2-p))/(1+2(q-1) N(q-p) ); (ii) q = r > k, alpha chi-gamma xi< 0; (iii) q = k > r, mu > alpha chi(1- N2(p-q) )/(1 + 2(q-1) N(p-q) ), then the solution of the equations is globally bounded. The present work further shows that the same conclusion still holds for the critical cases: (a) q = r = k, mu = (alpha chi-gamma xi)(1-N(q2-p) )/(1 + 2(q-1) N(q-p) ); (b) q = r > k, alpha chi -gamma xi = 0 with q -p < 1Nmin{ 4(N+1) N+2 , N + 2}; (c) q = k > r, mu = alpha chi(1-N2(p-q))/(1 + 2(q-1) N(p-q) ).(c) 2022 Elsevier Ltd. All rights reserved.