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Asymptotic convergence to stationary waves for unipolar hydrodynamic model of semiconductors. (English) Zbl 1227.35063

Summary: We study the one-dimensional unipolar hydrodynamic model for semiconductors in the form of Euler-Poisson equations. In the case when the state constants on the current density and the electric field are nonzero (switch-on case), the stability of stationary waves of one-dimensional isentropic Euler-Poisson equations for the unipolar hydrodynamic model has been open. In order to overcome this difficulty, we first analyze the behaviors of the solutions at \(x=\pm\infty\), and observe what are the exact gaps between the original solutions and the stationary solutions in \(L^2\)-space; then we technically construct some new correction functions to delete these gaps. Finally, based on the energy methods, we prove that the solutions of one-dimensional isentropic Euler-Poisson equations for the unipolar hydrodynamic model decay exponentially fast to the stationary solutions.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35L50 Initial-boundary value problems for first-order hyperbolic systems
35L60 First-order nonlinear hyperbolic equations
35L65 Hyperbolic conservation laws
76R50 Diffusion
82D37 Statistical mechanics of semiconductors
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