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

where \(\Omega \) is a bounded and smooth domain of \(\mathbb{R}^{n}\), for \(n\geq 2\), and \(f(u)\) and \(g(u)\) are reasonably regular functions generalizing, respectively, the prototypes \(f(u)=u^{\alpha }\) and \(g(u)=u^{l}\), with proper \(\alpha , l>0\). After having shown that any sufficiently smooth \(u(x,0)=u_{0}(x)\geq 0\) and \(v(x,0)=v_{0}(x)\geq 0\) produce a unique classical and nonnegative solution \((u,v)\) to problem (◊), which is defined on \(\Omega \times (0,T_{max})\) with \(T_{max}\) denoting the maximum 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.

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 \(g(u)=u\) the value \(\alpha =\frac{2}{n}\) represents the critical blow-up exponent to the model, whereas in the second, for \(f(u)=u\), corresponding to \(\alpha =1\), boundedness of solutions is shown under the assumption \(0< l<\frac{2}{n}\).

这项工作涉及具有非线性产生和化学吸引剂的完全抛物线趋化模型。该问题是在有界域上提出的，并且取决于与此类生产和化学引诱剂相关的系数之间的特定相互作用，我们确定相关的初始边界值问题具有唯一的经典解决方案，该解决方案在时间上是一致有界的。确切地说，我们研究这个零通量问题

其中\（\ Omega \）是\（\ mathbb {R} ^ {n} \）的有界且平滑域，对于\（n \ geq 2 \）以及\（f（u）\）和\（ g（u）\）是合理的常规函数​​，分别推广原型\（f（u）= u ^ {\ alpha} \）和\（g（u）= u ^ {l} \），并带有适当的\ （\ alpha，l> 0 \）。在证明任何足够平滑的\（u（x，0）= u_ {0}（x）\ geq 0 \）和\（v（x，0）= v_ {0}（x）\ geq 0 \）产生问题（◊）的唯一经典非负解\（（u，v）\），在\（\ Omega \ times（0，T_ {max}）\）上用\（T_ {max} \ ）表示存在的最长时间，我们确定对于\（l \ in（0，\ frac {2} {n}）\）和\（\ frac {2} {n} \ leq \ alpha <1+ \ frac {1} {n}-\ frac {l} {2} \），\（T_ {max} = \ infty \）以及\（u \）和\（v \）实际上是时间均匀的。

该论文与Horstmann和Winkler（J. Differ。Equ。215（1）：52-107，2005）的贡献是一致的，此外，还扩展了Liu和Tao（Appl。Math。J. Chin。 B系列31（4）：379–388，2016）。确实，在第一篇论文中，证明了对于\（g（u）= u \），值\（\ alpha = \ frac {2} {n} \）表示模型的临界爆炸指数，而在第二个中，对于\（f（u）= u \），它对应于\（\ alpha = 1 \），在假设\（0 <l <\ frac {2} {n} \的情况下显示了解的有界性）。