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This work deals with a fully parabolic chemotaxis model with nonlinearproduction and chemoattractant. The problem is formulated on a bounded domainand, depending on a specific interplay between the coefficients associated tosuch production and chemoattractant, we establish that the relatedinitial-boundary value problem has a unique classical solution which isuniformly bounded in time. To be precise, we study this zero-flux problem\begin{equation}\label{problem_abstract} \tag{$\Diamond$} \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} \end{equation} where $\Omega$ is a bounded and smooth domain of$\mathbb{R}^n$, for $n\geq 2$, and $f(u)$ and $g(u)$ are reasonably regularfunctions generalizing, respectively, the prototypes $f(u)=u^\alpha$ and$g(u)=u^l$, with proper $\alpha, l>0$. After having shown that any sufficientlysmooth $ u(x,0)=u_0(x)\geq 0, \, v(x,0)=v_0(x)\geq 0$ emanate a uniqueclassical and nonnegative solution $(u,v)$ to problem \eqref{problem_abstract},which is defined on $\Omega \times (0,T_{max})$ with $T_{max}$ denoting themaximum time of existence, we establish that for any $l\in (0,\frac{2}{n})$ and$\frac{2}{n}\leq \alpha<1+\frac{1}{n}-\frac{l}{2}$, $T_{max}=\infty$ and $u$and $v$ are actually uniformly bounded in time.  This paper is in line with the contribution by Horstmann and Winkler,moreover, extends the result by Liu and Tao. Indeed, in the first work it isproved that for $g(u)=u$ the value $\alpha=\frac{2}{n}$ represents the criticalblow-up exponent to the model, whereas in the second, for $f(u)=u$,corresponding to $\alpha=1$, boundedness of solutions is shown under theassumption $0<l<\frac{2}{n}.$

Boundedness for a fully parabolic Keller-Segel model with sublinear segregation and superlinear aggregation