A sixth-order parabolic system in semiconductors

Abstract

The authors investigate the global existence and semiclassical limit of weak solutions to a sixth-order parabolic system, which is a quantum-corrected macroscopic model derived recently to simulate the quantum effects in miniaturized semiconductor devices.

Keywords

Weak solution / Semiclassical limit / Sixth-order parabolic system

References

[1] Abdallah N. B., Unterreiter A.. On the stationary quantum drift diffusion model. Z. Angew. Math. Phys., 1998, 49(2): 251-275

[2] Bleher P. M., Lebowitz J. L., Speer E. R.. Existence and positivity of solutions of a fourth-order nonlinear PDE describing interface fluctuations. Comm. Pure Appl. Math., 1994, 47: 923-942

[3] Cáaceres M. J., Carrillo J. A., Toscani G.. Long-time behaviour for a nonlinear fourth-order parabolic equation. Trans. Amer. Math. Soc., 2004, 357(3): 1161-1175

[4] Chen L., Ju Q. C.. Existence of weak solution and semiclassical limit for quantum drift-diffusion model. Z. Angew. Math. Phys., 2007, 58(1): 1-15

[5] Chen L., Ju Q. C.. The semiclassical limit in the quantum drift-diffusion equations with isentropic pressure. Chin. Ann. Math., 2008, 29B(4): 369-384

[6] Chen X. Q.. The isentropic quantum drift-diffusion model in two or three space dimensions. Z. Angew. Math. Phys., 2009, 60(3): 416-437

[7] Chen X. Q., Chen L.. Initial time layer problem for quantum drift-diffusion model. J. Math. Anal. Appl., 2008, 343: 64-80

[8] Degond P., Méhats F., Ringhofer C.. Quantum energy-transport and drift-diffusion models. J. Stat. Phys., 2005, 118: 625-665

[9] Dolbeault J., Gentil I., Jüngel A.. A nonlinear fourth-order parabolic equation and related logarithmic Sobolev inequalities. Commun. Math. Sci., 2006, 4: 275-290

[10] Gasser I., Hsiao L., Li H. L.. Large time behavior of solutions of the bipolar hydrodynamical model for semiconductors. J. Diff. Eqs., 2003, 192(2): 326-359

[11] Gianazza U., Savarè G., Toscani G.. The wasserstein gradient flow of the fisher information and the quantum drift-diffusion equation. Archive for Rational Mechanics and Analysis, 2009, 194(1): 133-220

[12] Gualdani M., Jüngel A., Toscani G.. A nonlinear fourth-order parabolic equation with nonhomogeneous boundary conditions. SIAM J. Math. Anal., 2006, 37(6): 1761-1779

[13] Jia Y. L., Li H. L.. Large-time behavior of solutions of quantum hydrodynamic model for semiconductors. Acta Math. Sci., 2006, 26(1): 163-178

[14] Jüngel A.. Nonlinear problems on quantum semiconductor modeling. Nonlin. Anal., 2001, 47: 5873-5884

[15] Jüngel A.. Transport Equations for Semiconductors, 2009, Berlin: Springer-Verlag

[16] Jüngel A., Li H. L.. Quantum Euler-Poisson systems: global existence and exponential decay. Quart. Appl. Math., 2004, 62: 569-600

[17] Jüngel A., Li H. L., Matsumura A.. The relaxation-time limit in the quantum hydrodynamic equations for semicondcutors. J. Diff. Eqs., 2006, 225(2): 440-464

[18] Jüngel A., Pinnau R.. Global non-negative solutions of a nonlinear fourth order parabolic equation for quantum systems. SIAM J. Math. Anal., 2000, 32(4): 760-777

[19] Jüngel A., Pinnau R.. A positivity preserving numerical scheme for a nonlinear fourth-order parabolic system. SIAM J. Num. Anal., 2001, 39(2): 385-406

[20] Jüngel A., Pinnau R.. Convergent semidiscretization of a nonlinear fourth order parabolic system. Math. Mod. Num. Anal., 2003, 37(2): 277-289

[21] Jüngel A., Violet I.. The quasineutral limit in the quantum drift-diffusion equations. Asympt. Anal., 2007, 53: 139-157

[22] Jüngel A., Toscani G.. Exponential decay in time of solutions to a nonlinear fourth-order parabolic equation. Z. Angew. Math. Phys., 2003, 54: 377-386

[23] Jüngel A., Matthes D.. The Derrida-Lebowitz-Speer-Spohn equation: existence, non-uniqueness, and decay rates of the solutions. SIAM J. Math. Anal., 2008, 39: 1996-2015

[24] Jüngel A., Matthes D.. An algorithmic construction of entropies in higher-order nonlinear PDEs. Nonlinearity, 2007, 19: 633-659

[25] Jüngel A., Milišić J.. A sixth-order nonlinear parabolic equation for quantum systems. SIAM. J. Math. Anal., 2009, 41: 1472-1490

[26] Pinnau R.. A review on the quantum drift diffusion model. Transp. Theory Stat. Phys., 2002, 31(4–6): 367-395

[27] Simon J.. Compact sets in the space L ^p(0, T;B). Ann. Mat. Pura Appl., 1987, 146: 65-96

[28] Taylor M.. Partial Differential Equations III, Nonlinear Equations, 1997, New York: Springer-Verlag

[29] Zhang G. J., Li H. L., Zhang K. J.. Semiclassical and relaxation limits of bipolar quantum hydrodynamic model for semiconductors. J. Diff. Eqs., 2008, 245(6): 1433-1453