Asymptotic Behavior of Solutions of the Bipolar Quantum Drift-Diffusion Model in the Quarter Plane

Time:2019-11-15

LIU fang, LI Yeping Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China

Abstract:

In this study, we consider the one-dimensional bipolar quantum drift-diffusion model, which consists of the coupled nonlinear fourth-order parabolic equation and the electric field equation. We first show the global existence of the strong solution of the initial boundary value problem in the quarter plane. Moreover, we show the self-similarity property of the strong solution of the bipolar quantum drift-diffusion model in the large time. Namely, we show the unique global strong solution with strictly positive density to the initial boundary value problem of the quantum drift-diffusion model, which in large time, tends to have a self-similar wave at an algebraic time-decay rate. We prove them in an energy method.

Key words:asymptotic behavior; quantum drift-diffusion model; self-similar wave; energy estimate

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