On the asymptotic behavior of a run and tumble equation for bacterial chemotaxis

We prove that linear and weakly nonlinear run and tumble equations converge to a unique steady state solution with an exponential rate in a weighted total variation distance. In the linear setting, our result extends the previous results to an arbitary dimension while relaxing the assumptions. The main challenge is that even though the equation is a mass-preserving, Boltzmann-type kinetic-transport equation, the classical hypocoercivity methods, e.g., by J. Dolbeault, C. Mouhot, and C. Schmeiser [Trans. Amer. Math. Soc., 367 (2015), pp. 3807–3828], are not applicable for dimension . We overcome this difficulty by using a probabilistic technique, known as Harris’s theorem. We also introduce a weakly nonlinear model via a nonlocal coupling on the chemoattractant concentration. This toy model serves as an intermediate step between the linear model and the physically more relevant nonlinear models. We build a stationary solution for this equation and provide a hypocoercivity result