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A sixth-order nonlinear parabolic equation for quantum systems. (English) Zbl 1197.35151
Summary: The global-in-time existence of weak nonnegative solutions to a sixth-order nonlinear parabolic equation in one space dimension with periodic boundary conditions is proved. The equation arises from an approximation of the quantum drift-diffusion model for semiconductors and describes the evolution of the electron density in the semiconductor crystal. The existence result is based on two techniques. First, the equation is reformulated in terms of exponential and power variables, which allows for the proof of nonnegativity of solutions. The existence of solutions to an approximate equation is shown by fixed point arguments. Second, a priori bounds uniformly in the approximation parameters are derived from the algorithmic entropy construction method which translates systematic integration by parts into polynomial decision problems. The a priori estimates are employed to show the exponential time decay of the solution to the constant steady state in the $L^1$ norm with an explicit decay rate. Furthermore, some numerical examples are presented.

##### MSC:
 35K59 Quasilinear parabolic equations 35Q40 PDEs in connection with quantum mechanics 35B40 Asymptotic behavior of solutions of PDE 35B40 Asymptotic behavior of solutions of PDE 35K35 Higher order parabolic equations, boundary value problems
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