Naslov
A sixth-order nonlinear parabolic equation for quantum systems

Autori
Juengel, Ansgar ; Milišić, Josipa Pina

Izvornik
Siam Journal of Mathematical Analysis (0036-1410) 41 (2009), 4; 1472-1490

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
sixth-order parabolic equation; quantum semiconductors; global existence of solutions; algorithmic entropy construction; convergence of transport plans; long-time behavior of the solutions

Sažetak
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.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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