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

Abstract

AbstractThis 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 $$\begin{aligned} {\left\{ \begin{array}{ll} u_t=\Delta u-\chi \nabla \cdot (u\nabla v)+au^\alpha -bu^\alpha \int _\Omega u^\beta &{}\textrm{in}\ \Omega \times (0, T_{max}),\\ \tau v_t=\Delta v-v+u &{}\textrm{in}\ \Omega \times (0, T_{max}),\\ u_\nu =v_\nu =0 &{}\textrm{on}\ \partial \Omega \times (0, T_{max}),\\ u(x, 0)=u_0(x)\ge 0, v(x,0)=v_0(x)\ge 0, &{}x \in {\bar{\Omega }}, \end{array}\right. } \quad {\Diamond } \end{aligned}$$ u t = Δ u - χ ∇ · ( u ∇ v ) + a u α - b u α ∫ Ω u β in Ω × ( 0 , T max ) , τ v t = Δ v - v + u in Ω × ( 0 , T max ) , u ν = v ν = 0 on ∂ Ω × ( 0 , T max ) , u ( x , 0 ) = u 0 ( x ) ≥ 0 , v ( x , 0 ) = v 0 ( x ) ≥ 0 , x ∈ Ω ¯ , ◊ for $$\tau =1$$ τ = 1 , $$n\in {\mathbb {N}}$$ n ∈ N , $$\chi ,a,b>0$$ χ , a , b > 0 and $$\alpha , \beta \ge 1$$ α , β ≥ 1 . Herein u stands for the population density, v for the chemical signal and $$T_{max}$$ T max for the maximal time of existence of any nonnegative classical solution (u, v) to system ($$\Diamond $$ ◊ ). We prove that despite any large-mass initial data $$u_0$$ u 0 , whenever (The subquadratic case) $$1\le \alpha <2 \quad \text {and} \quad \beta >\frac{n+4}{2}-\alpha ,$$ 1 ≤ α < 2 and β > n + 4 2 - α , (The superquadratic case) $$\beta >\frac{n}{2} \quad \text {and} \quad 2\le \alpha < 1+ \frac{2\beta }{n},$$ β > n 2 and 2 ≤ α < 1 + 2 β n , actually $$T_{max}=\infty $$ T max = ∞ 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 ($$\Diamond $$ ◊ ), corresponding to $$\tau =0$$ τ = 0 ; more exactly, this work extends the study to the fully parabolic case Bian et al. (Nonlinear Anal 176:178–191, 2018).