Boundedness in a nonlinear chemotaxis-consumption model with gradient terms
Daniel Acosta Soba, Alessandro Columbu, Giuseppe Viglialoro
TL;DR
The paper analyzes a nonlinear chemotaxis–consumption system with gradient nonlinearities on a bounded domain, incorporating nonlinear diffusion, gradient-driven chemotaxis, logistic degradation, and a gradient damping term $-c|\nabla u|^\gamma$. The main result shows that, under either $\gamma > \max\{ \frac{2n}{n+1}, \frac{n}{n+1}(2m_2 - m_1 + 1)\}$ with $\gamma \le 2$, or $\frac{n}{n+1}(2m_2 - m_1 + 1) < \gamma \le 2$ with a sufficiently large $\mu$, the system admits a unique globally bounded classical solution; the gradient term can also allow boundedness when $\gamma$ is favorable even for strong attraction. The approach hinges on an energy method with $\varphi(t)=\int (u+1)^p + \chi^{2p}\int |\nabla v|^{2p}$, deriving a differential inequality $\varphi'(t)+\varphi(t)\le C$, and employing a Lebesgue-space boundedness criterion plus an extensibility lemma to promote local solutions to global ones. This work extends gradient-based damping results to a broader chemotaxis–consumption framework and provides rigorous conditions guaranteeing global regularity for models with gradient nonlinearities.
Abstract
We study a chemotaxis-consumption mechanism, in which some chemical signal and cells density interact each other. In order to control the concentration of such a population, sources involving gradient nonlinearities, which introduce a dampening effect on the model, are considered. Moreover, the system is characterized by nonlinear diffusion and sensitivity terms. We derive conditions on some data of the problem so to ensure the boundedness of related solutions.