Global stability of a logarithmically sensitive chemotaxis model under time-dependent boundary conditions

TL;DR

This paper proves the global stability of solutions to a chemotaxis model with logarithmic sensitivity under time-dependent boundary conditions, showing convergence to boundary data without small initial perturbation restrictions.

Contribution

It establishes the global existence and stability of classical solutions for a chemotaxis system with time-dependent boundaries, without requiring small initial data.

Findings

Solutions exist globally in time under suitable boundary conditions.

Solutions converge to boundary data as time approaches infinity.

Numerical simulations confirm the necessity of boundary assumptions.

Abstract

This paper studies the dynamical behavior of classical solutions to a hyperbolic system of balance laws, derived from a chemotaxis model with logarithmic sensitivity, subject to time-dependent boundary conditions. It is shown that under suitable assumptions on the boundary data, solutions starting in $H^2$-space exist globally in time and the differences between the solutions and their corresponding boundary data converge to zero, as time goes to infinity. There is no smallness restriction on the magnitude of initial perturbations. Moreover, numerical simulations show that the assumptions on the boundary data are necessary for the above mentioned results.