Refined criteria toward boundedness in an attraction–repulsion chemotaxis system with nonlinear productions.

We study some zero-flux attraction-repulsion chemotaxis models, with nonlinear production rates for the chemorepellent and the chemoattractant, whose formulation can be schematized as (⋄) $$\begin{equation} \begin{cases} u_t= \Delta u - \chi \nabla \cdot (u \nabla v)+\xi \nabla \cdot (u \nabla w) & {\rm in}\ \Omega \times (0,T_{\max}),\\ \tau v_t=\Delta v-\varphi(t,v)+f(u) & {\rm in}\ \Omega \times (0,T_{\max}),\\ \tau w= \Delta w - \psi(t,w) + g(u) & {\rm in}\ \Omega \times (0,T_{\max}). \end{cases} \end{equation}$$ { u t = Δ u − χ ∇ ⋅ (u ∇ v) + ξ ∇ ⋅ (u ∇ w) in Ω × (0 , T max) , τ v t = Δ v − φ (t , v) + f (u) in Ω × (0 , T max) , τ w = Δ w − ψ (t , w) + g (u) in Ω × (0 , T max). In this problem, Ω is a bounded and smooth domain of $ \mathbb R^n $ R n , for $ n\geq 2 $ n ≥ 2 , $ \chi,\xi \gt 0 $ χ , ξ > 0 , $ f(u) $ f (u) , $ g(u) $ g (u) reasonably regular functions generalizing, respectively, the prototypes $ f(u)=\alpha u^k $ f (u) = α u k and $ g(u)= \gamma u^l $ g (u) = γ u l , for some $ k,l,\alpha,\gamma \gt 0 $ k , l , α , γ > 0 and all $ u\geq 0 $ u ≥ 0. Moreover, $ \varphi (t,v) $ φ (t , v) and $ \psi (t,w) $ ψ (t , w) have specific expressions, $ \tau \in \{0,1\} $ τ ∈ { 0 , 1 } and $ \Theta :=\chi \alpha -\xi \gamma $ Θ := χ α − ξ γ. Once for any sufficiently smooth $ u(x,0)=u_0(x)\geq 0 $ u (x , 0) = u 0 (x) ≥ 0 , $ \tau v(x,0)=\tau v_0(x)\geq 0 $ τ v (x , 0) = τ v 0 (x) ≥ 0 and $ \tau w(x,0)=\tau w_0(x)\geq 0, $ τ w (x , 0) = τ w 0 (x) ≥ 0 , the local well-posedness of problem ( $ \Diamond $ ◊) is ensured, and we establish for the classical solution $ (u,v,w) $ (u , v , w) defined in $ \Omega \times (0,T_{\max }) $ Ω × (0 , T max) that the life span is indeed $ T_{\max }=\infty $ T max = ∞ and u, v and w are uniformly bounded in $ \Omega \times (0,\infty) $ Ω × (0 , ∞) in the following cases: For $ \varphi (t,v)=\beta v $ φ (t , v) = β v , $ \beta \gt 0 $ β > 0 , $ \psi (t,w)=\delta w $ ψ (t , w) = δ w , $ \delta \gt 0 $ δ > 0 and $ \tau =0 $ τ = 0 , provided (I.1) k<l; (I.2) $ k,l\in \left (0,\frac {2}{n}\right) $ k , l ∈ (0 , 2 n) ; (I.3) k = l and $ \Theta \lt 0 $ Θ < 0 , or $ l=k\in \left (0,\frac {2}{n}\right) $ l = k ∈ (0 , 2 n) and $ \Theta \geq 0 $ Θ ≥ 0. For $ \varphi (t,v)=\beta v $ φ (t , v) = β v , $ \beta \gt 0 $ β > 0 , $ \psi (t,w)=\delta w $ ψ (t , w) = δ w , $ \delta \gt 0 $ δ > 0 and $ \tau =1 $ τ = 1 , whenever (II.1) $ l,k\in \left (0,\frac {1}{n}\right ] $ l , k ∈ (0 , 1 n ] ; (II.2) $ l\in \left (\frac {1}{n},\frac {1}{n}+\frac {2}{n^2+4}\right) $ l ∈ (1 n , 1 n + 2 n 2 + 4) and $ k\in \left (0,\frac {1}{n}\right ] $ k ∈ (0 , 1 n ] , or $ k\in \left (\frac {1}{n},\frac {1}{n}+\frac {2}{n^2+4}\right) $ k ∈ (1 n , 1 n + 2 n 2 + 4) and $ l\in \left (0,\frac {1}{n}\right ] $ l ∈ (0 , 1 n ] ; (II.3) $ l,k\in \left (\frac {1}{n},\frac {1}{n}+\frac {2}{n^2+4}\right) $ l , k ∈ (1 n , 1 n + 2 n 2 + 4) . For $ \varphi (t,v)=\frac {1}{|\Omega |}\int _\Omega f(u) $ φ (t , v) = 1 | Ω | ∫ Ω f (u) and $ \psi (t,w)=\frac {1}{|\Omega |}\int _\Omega g(u) $ ψ (t , w) = 1 | Ω | ∫ Ω g (u) and $ \tau =0 $ τ = 0 , under the assumptions k<l or (I.3)). In particular, in this paper we partially improve what derived in Viglialoro [Influence of nonlinear production on the global solvability of an attraction-repulsion chemotaxis system. Math Nachr. 2021;294(12):2441–2454] and solve an open question given in Liu and Li [Finite-time blowup in attraction-repulsion systems with nonlinear signal production. Nonlinear Anal Real World Appl. 2021;61:Paper No. 103305, 21]. Finally, the research is complemented with numerical simulations in bi-dimensional domains. [ABSTRACT FROM AUTHOR]