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

This is the authors' abstract. We don't add key points for this paper.

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 {equation}{problem_abstract} {} {cases} ρ_t=Δ ρ^m ± ∇ · (ρ ∇ ({|x|²⁻ⁿ}{2-n}*ρ)) \\[10pt] 50pt +aρ^η-bρ^α ∫R^n ρ^β dx & x∈ R^n, t∈ (0,Tₘₐₓ), \\ ρ(x,0)=ρ₀(x) & x∈ R^n, {cases} {equation} for n≥ 3, m,a,b,α,η > 0 and β≥ 1. By denoting with Tₘₐₓ the maximum time of existence of any nonnegative weak solution ρ to problems {problem_abstract}, we prove that despite any large-mass initial data ρ₀, for any η > 0 and arbitrarily small diffusive parameter $m > 0$, whenever α+β surpasses some computable expression depending on m, η and n, Tₘₐₓ=∞ and ρ 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: {align*} • \ & the repulsive scenario, under the assumption $m > 0$ 300pt \\ & (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]). {align*} On the other hand, for the attractive case with $a=b=m=1$ and η=α, this investigation also extends a result derived by Bian, Chen and Latos in [3].