Global boundedness of weak solutions to a flux-limited Keller--Segel system with superlinear production

The flux-limited Keller–Segel system \begin{align*} \begin{cases} u_t = Δu - χ\nabla \cdot (u|\nabla v|^{p-2}\nabla v), \[] v_t = Δv - v + u^θ \end{cases} \end{align*} is considered under homogeneous Neumann boundary conditions in a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ $(n\in \mathbb{N})$. In the case that $\theta \le 1$, existence of global bounded weak solutions was established in the previous work (arXiv:2501.04370 ; to be appear in Proceedings of the conference “Critical Phenomena in Nonlinear Partial Differential Equations, Harmonic Analysis, and Functional Inequalities.”). The purpose of this paper is to prove that global bounded weak solutions can also be constructed in the case $\theta >1$ with a smallness condition on $p$.