Uniform-in-time boundedness in a class of local and nonlocal nonlinear attraction–repulsion chemotaxis models with logistics

The following fully nonlinear attraction-repulsion and zero-flux chemotaxis model is studied:{u(t)= del center dot((u + 1)(m1-1)del u - chi u(u+1)(m2-1)del v+xi u(u+ 1)(m3-1)del w) + lambda u - mu u in Omega x (0, T-max),(lozenge)tau v(t) = Delta v-phi(t, v) +f(u) in Omega x (0, T-max),tau w(t) = Delta w - psi(t, w) +g(u) in Omega x (0, T-max).Herein, Omega is a bounded and smooth domain of R-n, for n is an element of N, chi, xi, lambda, mu, r proper positive numbers, m(1), m(2), m(3) is an element of R, and f(u) and g(u) regular functions that generalize the prototypes f(u) similar or equal to u(k) and g(u) similar or equal to ul, for some k, l > 0 and all u >= 0. Moreover, tau is an element of {0, 1}, and T-max is an element of (0, infinity] is the maximalinterval of existence of solutions to the model. Once suitable initial data u(0)(x), tau v(0)(x), tau w(0)(x) are fixed, we are interested in deriving sufficient conditions implying globality (i.e., T-max= infinity) and boundedness (i.e., parallel to u(center dot, t)parallel to(L infinity(Omega)) + parallel to v(center dot, t)parallel to(L infinity(Omega)) + parallel to w(center dot, t)parallel to(L infinity(Omega)) <= C for all t is an element of (0, infinity)) of solutions to problem (lozenge). This is achieved in the following scenarios:For phi(t, v) proportional to v and psi(t, w) to w, whenever tau = 0 and provided one of thefollowing conditions(I) m(2) + k < m(3) + l, (II) m(2) + k