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The existence and long-time behavior of weak solution to bipolar quantum drift-diffusion model. (English) Zbl 1145.35384

Summary: The authors study the existence and long-time behavior of weak solutions to the bipolar transient quantum drift-diffusion model, a fourth order parabolic system. Using semi-discretization in time and entropy estimate, the authors get the global existence of nonnegative weak solutions to the one-dimensional model with nonnegative initial and homogeneous Neumann (or periodic) boundary conditions. Furthermore, by a logarithmic Sobolev inequality, it is proved that the periodic weak solution exponentially approaches its mean value as time increases to infinity.

MSC:

35K35 Initial-boundary value problems for higher-order parabolic equations
35J60 Nonlinear elliptic equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
35K50 Systems of parabolic equations, boundary value problems (MSC2000)
35D05 Existence of generalized solutions of PDE (MSC2000)
35B40 Asymptotic behavior of solutions to PDEs
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