Nonlinear stability of strong traveling waves for a chemotaxis model with logarithmic sensitivity and periodic perturbations

In this paper, we are concerned with the nonlinear stability of traveling wave solutions for a conservation laws arising from the well‐known chemotaxis model with logarithmic sensitivity. By constructing appropriate “ansatz,” we shall prove the nonlinear stability of traveling waves even though the perturbations oscillate at the far fields. More precisely, it is shown that as time tends to infinite, the solutions converge to arbitrarily large amplitude viscous shock wave solutions with a shift, which are partially determined by the periodic oscillations.