Optimal convergence rate to nonlinear diffusion waves for Euler equations with critical overdamping

This paper is concerned with the Cauchy problem for the one-dimensional compressible Euler equations with critical time-dependent overdamping. In this critical case, the damped Euler system is proved to admit a unique global solution that converges time-asymptotically to the corresponding nonlinear diffusion wave. Particularly, the optimal convergence rate in logarithmic form is first derived by the technical time-weighted method.