engleski

Semiconductor modeling and nonlinear higher-order PDEs.

First we consider the sixth-order nonlinear parabolic equation which arise from an approximation of the quantum drift-di usion model and describe the evolution of the electron density in the semiconductor crystal. We prove the global{;in{;time existence of weak nonnegative solutions in one space dimension with periodic boundary conditions. The apriori estimates are obtained using the entropy{;construction method developed by Juengel and Matthes which translates systematic integration by parts into polynomial decision problems. Furthermore, the apriori 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. In the second part of the talk we consider quantum systems. The first quantum system of the interest will be the simpli ed quantum energy-transport (SQET) model obtained from the quantum hydrodynamic model in a large-time and small-velocity regime. This model consists of a nonlinear parabolic fourth-order equation for the electron density, including temperature gradients ; an elliptic nonlinear heat equation for the electron temperature and the Poisson equation for the electric potential. We prove the existence of global-in-time weak solutions with periodic boundary conditions. Finally, the full compressible quantum Navier Stokes (QNS) model will be presented. It consists of the continuity equation for the electron density following by nonlinear momentum and energy equations containing terms of the third-order with viscosity corrections. Up to our knowledge, no analytical or numerical results are available for this model.

sixth-order parabolic equation; quantum semiconductors; global existence of solutions; long-time behaviour; quantum systems

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