Existence of Solutions for a Class of Chemotaxis Models with Density-Dependent Sensitivity

DOI: https://doi.org/10.62517/jnse.202517509

Author(s)

Xueyong Chen

Affiliation(s)

School of Mathematics and Statistics, Taishan University, Taian, China

Abstract

Chemotaxis models of the Keller-Segel type are fundamental for describing the directed motion of biological entities in response to chemical gradients, and incorporating reaction terms enables them to capture more realistic biological processes. This paper carries out an investigation into a class of n-dimensional chemotaxis model with a reaction term, where a critical characteristic is the nonlinear kinetics term in the second equation. This nonlinearity, while reflecting practical biological scenarios, poses challenges to solution analysis—for instance, the risk of finite-time blow-up of solutions. To overcome this, we first impose appropriate growth conditions on the nonlinear kinetics function, and then derive essential priori estimates for the model’s solutions, including bounds on key Lp-norms and uniform boundedness of solution components. By leveraging these rigorous priori estimates, we further demonstrate that the proposed problem admits a bounded global solution, ensuring the solution remains well-defined and bounded for all positive time without developing finite-time singularities. This result provides insights into the long-term dynamical behavior of nonlinear chemotaxis systems in high-dimensional spaces.

Keywords

Chemotaxis; Global Existence; Blow-Up; Uniform Boundedness; Priori Estimates

References

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