Blow-up prevention and rate of convergence of solutions for N-dimensional parabolic-parabolic systems with consumption of chemoattractant

This article studies the Neumann-boundary initial-value problem for a parabolic-parabolic chemotaxis-consumption system in a smooth bounded domain. For regular nonnegative initial data, we prove that the classical solution to the corresponding no-flux problem remains globally and uniformly bounded under structural assumptions. This is achieved through a novel trigonometric-type weight function rather than an exponential one; therefore we not only significantly improve previous results, but also providing a versatile context to resolve pertinent systems. More importantly, we confirm the convergence of the solution to an equilibrium constant. For more information an d the TEX file, see https://ejde.math.txstate.edu/Volumes/2025/98/abstr.html