$L^\infty$ bounds and asymptotic behavior in a doubly degenerate chemotaxis system below six dimensions

Abstract

We investigate a doubly degenerate nutrient-taxis system of the form \begin{equation*} \begin{cases} u_t = \nabla \cdot (u v \nabla u) - χ\nabla \cdot (u^αv \nabla v) + \ell u v, \qquad &x \in Ω, \ t > 0, v_t = Δv - u v, \qquad &x \in Ω, \ t > 0, \end{cases} \end{equation*} subject to the homogeneous Neumann boundary conditions in a smoothly bounded convex domain $Ω\subset \mathbb{R}^n$ with $n\in \left \{ 3,4,5 \right \}$, where $α\geq 1$, $χ>0$ and $\ell \geq 0$. For any suitably regular initial data, we establish the global existence of a weak solution that remains uniformly bounded in time, provided that $α$ lies in the range $\left[1, \frac{5}{2} - \frac{n}{4}\right)$, and we also determine the large-time behavior of these solutions. Our proof relies on several novel functional inequalities, a bootstrap argument, and a Moser iteration method.

Disclosure

“1.6) is a consequence of (2.3) and (1.7) is a result of (6.4) and
(6.14). The proof is now complete. Declarations Conflict of Interest The authors declare that they have no conflict of interest. Declaration of Generative AI and AI-Assisted Technologies in the Manuscript Preparation Process During the preparation of this work, the author(s) used DeepSeek (deepseek.com) for language polishing, grammar correction, and overall readability improvement. The author(”

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2026-08-23T19:14:44.203658Z

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