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This work deals with a fully parabolic chemotaxis model with nonlinear production and chemoattractant. The problem is formulated on a bounded domain and, depending on a specific interplay between the coefficients associated to such production and chemoattractant, we establish that the related initial-boundary value problem has a unique classical solution which is uniformly bounded in time. To be precise, we study this zero-flux problem $$ \textstyle\begin{cases} u_{t}= \Delta u - \nabla \cdot (f(u) \nabla v) & \text{ in } \Omega \times (0,T_{max}), \\ v_{t}=\Delta v-v+g(u) & \text{ in } \Omega \times (0,T_{max}), \end{cases} $$ { u t = Δ u − ∇ ⋅ ( f ( u ) ∇ v )  in  Ω × ( 0 , T m a x ) , v t = Δ v − v + g ( u )  in  Ω × ( 0 , T m a x ) , where $\Omega $ Ω is a bounded and smooth domain of $\mathbb{R}^{n}$ R n , for $n\geq 2$ n ≥ 2 , and $f(u)$ f ( u ) and $g(u)$ g ( u ) are reasonably regular functions generalizing, respectively, the prototypes $f(u)=u^{\alpha }$ f ( u ) = u α and $g(u)=u^{l}$ g ( u ) = u l , with proper $\alpha , l>0$ α , l > 0 . After having shown that any sufficiently smooth $u(x,0)=u_{0}(x)\geq 0$ u ( x , 0 ) = u 0 ( x ) ≥ 0 and $v(x,0)=v_{0}(x)\geq 0$ v ( x , 0 ) = v 0 ( x ) ≥ 0 produce a unique classical and nonnegative solution $(u,v)$ ( u , v ) to problem (◊), which is defined on $\Omega \times (0,T_{max})$ Ω × ( 0 , T m a x ) with $T_{max}$ T m a x denoting the maximum time of existence, we establish that for any $l\in (0,\frac{2}{n})$ l ∈ ( 0 , 2 n ) and $\frac{2}{n}\leq \alpha <1+\frac{1}{n}-\frac{l}{2}$ 2 n ≤ α < 1 + 1 n − l 2 , $T_{max}=\infty $ T m a x = ∞ and $u$ u and $v$ v are actually uniformly bounded in time.The paper is in line with the contribution by Horstmann and Winkler (J. Differ. Equ. 215(1):52–107, 2005) and, moreover, extends the result by Liu and Tao (Appl. Math. J. Chin. Univ. Ser. B 31(4):379–388, 2016). Indeed, in the first work it is proved that for $g(u)=u$ g ( u ) = u the value $\alpha =\frac{2}{n}$ α = 2 n represents the critical blow-up exponent to the model, whereas in the second, for $f(u)=u$ f ( u ) = u , corresponding to $\alpha =1$ α = 1 , boundedness of solutions is shown under the assumption $0< l<\frac{2}{n}$ 0 < l < 2 n .

We enter the details of two recent articles concerning as many chemotaxis models, one nonlinear and the other linear, and both with produced chemoattractant and saturated chemorepellent. More precisely, we are referring respectively to the papers "Boundedness in a nonlinear attraction-repulsion Keller-Segel system with production and consumption," by S. Frassu, C. van der Mee and G.Viglialoro [J. Math. Anal. Appl. 504(2):125428, 2021] and "Boundedness in a chemotaxis system with consumed chemoattractant and produced chemorepellent," by S. Frassu and G. Viglialoro [Nonlinear Anal. 213:112505, 2021]. These works, when properly analyzed, leave open room for some improvement of their results. We generalize the outcomes of the mentioned articles, establish other statements, and put all the claims together; in particular, we select the sharpest ones and schematize them. Moreover, we complement our research also when logistic sources are considered in the overall study.