Publication: On a quasilinear degenerate system arising in semiconductor theory. Part II: Localization of vacuum solutions
Full text at PDC
Advisors (or tutors)
Pergamon Elsevier Science Ltd
The temporal and spatial localization of vacuum sets of the solutions to the drift-diffusion equations for semiconductors is studied in this paper. It is shown that if there are vacuum sets initially then there are vacuum sets for a small time, which shows the finite propagation speed of the support of the densities. It is also shown that for a certain recombination-generation rate there is no dilation of the initial support, and under some condition on the recombination-generation rate the vacuum will develop after a certain time even if there is no vacuum initially. These results are proved based on a local energy method for free boundary problems.
S. Antontsev, On the localization of solutions of nonlinear degenerate elliptic and parabolic equations Sov. Math. Dokl., 24 (1981), pp. 420–424 S. Antontsev, J.I. Díaz, New results on localization of solutions of nonlinear elliptic and parabolic equations obtained by the energy method, Sov. Math. Dokl. 38 (1989) 535–539. S. Antontsev, J.I. Díaz, On space an time localization of solutions of nonlinear elliptic and parabolic equations via energy methods, in: P. Bénilan et al. (Ed.), Recent Advances in Nonlinear Elliptic and Parabolic Problems, Longman, London, 1989, pp. 3–14. S. Antontsev, J.I. Díaz, Space and time localization in the flow of two immiscible fluids through a porous medium: energy methods applied to systems, Nonlinear Anal. 16(1991) 299–313. S. Antontsev, J.I. Díaz, Energy Methods for Free Boundary Problems in Continuum Mechanics, to appear. S. Antontsev, J.I. Díaz, S. Shmarev, The support shrinking in solutions of parabolic equations with nonhomogeneous absorption terms, Ann. Fac. Sci. Toulouse VI. Ser., Math. 4 (1995) 5–30. F. Bernis, Compactness of the support for some nonlinear elliptic problems of arbitrary order in dimension n Comm. PDE, 9 (1984), pp. 271–312 F. Bernis, Finite speed of propagation and continuity of the interface of thin viscous films Adv. Diff. Eqs., 1 (1996), pp. 337–368 R. Courant, K.O. Friedrichs, Supersonic Flow and Shock WavesInterscience, New York (1967) J.I. Díaz G. Galiano, A. Jüngel, On a quasilinear degenerate system arising in semiconductor theory, Part I: existence and uniqueness of solutions, submitted. J.I. Díaz, G. Galiano, A. Jüngel, Space localization and uniqueness of vacuum solutions to a degenerate parabolic problem in semiconductor theory, C. R. Acad. Sci. Paris 325 (1997) 267–272. J.I. Díaz, L. Véron, Local vanishing properties of solutions of elliptic and parabolic quasilinear equations, Trans. AMS 290 (1985) 787–814. G. Galiano, Sobre algunos problemas de la Mecánica de Medios Continuos en los que se originan Fronteras Libres, Ph.D. Thesis, Universidad Complutense de Madrid, Departamento de Matemática Aplicada, Spain, 1997. G. Galiano, M.A. Peletier, Spatial localization for a general reaction-diffusion system, Ann. Fac. Sci. Toulouse, to appear. D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer, Berlin, 1983. A. Jüngel, On the existence and uniqueness of transient solutions of a degenerate nonlinear drift-diffusion model for semiconductors, Math. Models Meth. Appl. Sci., 4 (1994), pp. 677–703 A. Jüngel, Numerical approximation of a drift-diffusion model for semiconductors with nonlinear diffusion, ZAMM, 75 (1995), pp. 783–799 A. Jüngel, Qualitative behavior of solutions of a degenerate nonlinear drift-diffusion model for semiconductors, Math. Models Meth. Appl. Sci., 5 (1995), pp. 497–518 A. Jüngel, Asymptotic analysis of a semiconductor model based on Fermi–Dirac statistics Math. Meth. Appl. Sci., 19 (1996), pp. 401–424 A. Jüngel, A nonlinear drift-diffusion system with electric convection arising in semiconductor and electrophoretic modelling, Math. Nachr., 185 (1997), pp. 85–110 A. Jüngel and P. Pietra, A discretization scheme of a quasi-hydrodynamic semiconductor model Math. Models Meth. Appl. Sci., 7 (1997), pp. 935–955 R. Kersner, A. Shishkov, Instantaneous shrinking of the support of energy solutions J. Math. Anal. Appl., 198 (1996), pp. 729–750 P.A. Markowich, C.A. Ringhofer, C. Schmeiser, Semiconductor EquationsSpringer, Berlin (1990) M.S. Mock, On equations describing steady state carrier distributions in a semiconductor device Comm. Pure Appl. Math., 25 (1972), pp. 781–792 R. Natalini, The bipolar hydrodynamic model for semiconductors and the drift-diffusion equations J. Math. Anal. Appl., 198 (1996), pp. 262–281