Boundedness for a nonlocal reaction chemotaxis model even in the attraction-dominated regime

This work deals with a parabolic chemotaxis model with nonlinear diffusion and nonlocal reaction source. The problem is formulated on the whole space and, depending on a specific interplay between the coefficients associated to such diffusion and reaction, we establish that all given solutions are uniformly bounded in time. To be precise, we study these attractive (sign "+") and repulsive (sign "") following models, formally described by the Cauchy problems (For Example for n 3, m; a; b;; > 0 and 1. By denoting with Tmax the maximum time of existence of any nonnegative weak solution to problems (), we prove that despite any large-mass initial data 0, for any > 0 and arbitrarily small diffusive parameter m > 0, whenever + surpasses some computable expression depending on m; and n, Tmax = 1 and is uniformly bounded. On the one hand, this paper is in line with claims established for a = b = 0, where the same conclusion holds true in, respectively: The repulsive scenario, under the assumption m > 0 (adaptation of the case m > 1 2 n , in Carrillo and Wang [17]); the attraction scenario, under the assumption 2n n+2 < m < 2 2 n and for small initial data (Chen and Wang in [18]). On the other hand, for the attractive case with a = b = m = 1 and =, this investigation also extends a result derived by Bian, Chen and Latos in [3].