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A. Jüngel, J. Milisic:
"A sixth-order nonlinear parabolic equation for quantum sytems";
in: "ASC Report 34/2008", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2008, ISBN: 978-3-902627-01-8.

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 L1 norm with an
explicit decay rate. Furthermore, some numerical examples are presented.

Sixth-order diffusive equation, global existence of periodic solutions, algorithmic

http://www.asc.tuwien.ac.at/preprint/2008/asc34x2008.pdf