Stability of planar diffusion wave for three‐dimensional bipolar Euler–Poisson system with time‐dependent over‐damping

This paper is concerned with the asymptotic behavior of the solution to the Cauchy problem for the three‐dimensional bipolar Euler–Poisson system with time‐dependent over‐damping. We prove that the Cauchy problem admits a unique global‐in‐time smooth solution that tends to the planar diffusion wave as time goes to infinity provided that the initial perturbation is sufficiently small. Moreover, we also obtain the convergence rate of the solution toward the planar diffusion wave. The proof is completed by the time‐weighted energy method.