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 equationproblemₐbstract cases ₜ= ᵐ ( (|x|^2-n2-n*) ) \\10pt 50pt +a^-b^ ₑ䂞 ^ dx } \\ \ & the attraction scenario, under the assumption 2nn+2 < m < 2-2{n} \\ & and for small initial data (Chen and Wang in 18). align* 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.