Boundedness Through Nonlocal Dampening Effects in a Fully Parabolic Chemotaxis Model with Sub and Superquadratic Growth

This work deals with a chemotaxis model where an external source involving a sub and superquadratic growth effect contrasted by nonlocal dampening reaction influences the motion of a cell density attracted by a chemical signal. We study the mechanism of the two densities once their initial configurations are fixed in bounded impenetrable regions; in the specific, we establish that no gathering effect for the cells can appear in time provided that the dampening effect is strong enough. Mathematically, we are concerned with this problem {ut=Δu-χ∇·(u∇v)+auα-buα∫ΩuβinΩ×(0,Tmax),τvt=Δv-v+uinΩ×(0,Tmax),uν=vν=0on∂Ω×(0,Tmax),u(x,0)=u0(x)≥0,v(x,0)=v0(x)≥0,x∈Ω¯,◊ for τ= 1 , n∈ N , χ, a, b> 0 and α, β≥ 1 . Herein u stands for the population density, v for the chemical signal and Tmax for the maximal time of existence of any nonnegative classical solution (u, v) to system (◊). We prove that despite any large-mass initial data u , whenever (The subquadratic case) 1≤α[removed]n+42-α,(The superquadratic case) β>n2and2≤α<1+2βn, actually Tmax= ∞ and u and v are uniformly bounded. This paper is in line with the result in Bian et al. (Nonlinear Anal 176:178–191, 2018), where the same conclusion is established for the simplified parabolic-elliptic version of model (◊), corresponding to τ= 0 ; more exactly, this work extends the study to the fully parabolic case Bian et al. (Nonlinear Anal 176:178–191, 2018)