The primary intention of this article centers on the global existence of uniform-in-time boundedness for the chemotaxis system capturing T-cell dynamics, posed in a general bounded domain \(\Omega \subset \mathbb {R}^{N}(N\ge 2)\) with smooth boundary \(\partial \Omega \) , where the parameter \(\alpha \) is anticipated to be positive. It is clearly confirmed that for any choice of the nonnegative initial data \(u_{0}\in C^{0}(\bar{\Omega })\) and \(v_{0}\in W^{1,\infty }(\Omega )\) , then the corresponding initial-boundary value problem ( \(*\) ) possesses a unique globally bounded classical solution in the structurally concrete sense of \( 0<\max \left\{ \Vert v_{0}\Vert _{L^{\infty }(\Omega )},\frac{1}{\alpha }\right\} <\pi \sqrt{\frac{2}{N}}, \) which evidently goes beyond the existing discovery of Tao-Winkler (European J. Appl. Math., 36(2025), 570–583), who asserted the same results under quite ambiguous conditions. Particularly, one of the most innovative aspects here is the construction of the weight function in trigonometric form, as opposed to the more conventionally exponential one or other types, showcasing remarkable versatility that can be effectively implemented to resolve a broad class of relevant chemotaxis-consumption systems.
