Large-time behavior of solutions for unipolar euler-poisson equations with critical over-damping

This paper is concerned with the large-time behavior of solutions to the Cauchy problem for the one-dimensional unipolar Euler-Poisson equa-tions with critical time-dependent over-damping. We prove that the Cauchy problem admits a unique global smooth solution which time-asymptotically k converges to the stationary solution in the logarithmic form O(ln- 2 (1 + t)) for the integer k & ISIN; [1, +& INFIN;). In particular, the integer k can be large enough as the initial perturbation is small enough. This convergence rate is much better than the previous studies with critical over-damping. The proof is based on the technical time-weighted energy estimates and the mathematical induction.