RT Journal Article
T1 On a quasilinear degenerate system arising in semiconductors theory. Part 1: Existence and uniqueness of solutions
A1 Díaz Díaz, Jesús Ildefonso
A1 Galiano, Gonzalo
A1 Jungel, Ansgar
AB This paper is about the drift-diffusion equations for semiconductors. Existence and uniqueness of weak solutions are obtained. The existence is proved by using the regularization technique. The proof of the uniqueness is interesting.
PB Pergamon Elsevier Science Ltd.
SN 1468-1218
YR 2001
FD 2001-09
LK https://hdl.handle.net/20.500.14352/57337
UL https://hdl.handle.net/20.500.14352/57337
LA eng
NO H.W. Alt, S. Luckhaus, Quasilinear elliptic–parabolic differential equations, Math. Z. 183 (1983)311–341.S.N. Antontsev, J.I. Díaz, A.V. Domansky, Continuous dependence and stabilization of solutions of the degenerate system in two-phase filtration, Din. Sploshnoi Sredy (107) (1993) 11–25.S.N. Antontsev, A.V. Kazhikov, V.N. Monkhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, North-Holland, Amsterdam, 1990.O. Arino, S. Gauthier, J.P. Penot, Afixed point theorem for sequentially continuous mappings with application to ordinary Differential equations, Funkcial Ekvac. 27 (1987) 273–279.P. Benilan, R. Gariepy, Strong solutions in L1 of degenerate parabolic equations, J. Differential Equation 119 (1995) 473–502.H. Brézis, OpérateursMaximaux Monotones et Semigroupes de Contractions dans les Espaces de Hilbert, North-Holland, Amsterdam, 1973.J. Carrillo, On the uniqueness of the solution of the evolution dam problem, Nonlinear Anal. Theory Methods Appl. 22 (5) (1994) 573–607.R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience, New York, 1967.J.I. Díaz, G. Galiano, A. Jüngel, On a quasilinear degenerate system arising in semiconductors theory.Part II: Localization of vacuum solutions, Nonlinear Anal. 36 (5) (1998) 569–594.J.I. Díaz, R. Kersner, On a nonlinear degenerate parabolic equation in infiltration or evaporation through a porous medium, J. Differential Equations 69 (1987) 368–403.J.I. Díaz, I.I. Vrabie, Existence for reaction diffusion systems: a compactness approach, J. Math. Anal. Appl. 188 (1994) 521–540.J.I. Díaz, I.I. Vrabie, Compactness of the Green operator of nonlinear diffusion equations: applications to Boussinesq type systems in ffuid dynamics, Topol. Methods Nonlinear Anal. 4 (1994) 399–416.J.A. Dubinskii, Weak convergence in nonlinear elliptic and parabolic equations, AMS. Transl. 67 (1968)226–258.G. Gagneux, M. Madaune-Tort, Unicitédes solutions faibles d’équations de diffusion-convection, C. R. Acad. Sci. t. 318, Ser. I (1994) 919–924.G. Gagneux, M. Madaune-Tort, Analyse mathématique de modèles non linéaires de l’ingeniérie pétrolière, Springer, Paris, 1996.G. Galiano, Sobre algunos problemas de la Mecáanica de Medios Continuos en los que se originan Fronteras Libres, Thesis, Universidad Complutense de Madrid, 1997.P. Grisvard, Elliptic Problems in Nonsmooth Domains,Pitman, Boston, 1985.A. Jüngel, On the existence and uniqueness of transient solutions of a degenerate nonlinear drift-diffusion model for semiconductors, Math. Models Methods Appl. Sci. 4 (1994) 677–703.A. Jüngel, Qualitative behavior of solutions of a degenerate nonlinear drift-diffusion model for semiconductors, Math. Models Methods Appl. Sci. 5 (1995) 497–518.A. Jüngel, Asymptotic analysis of a semiconductor model based on Fermi–Dirac statistics, Math.Methods Appl. Sci. 19 (1996) 401–424.A. Jüngel, A nonlinear drift-diffusion system with electric convection arising in semiconductor and electrophoretic modeling, Math. Nachr. 185 (1997) 85–110.A.S. Kalashnikov, The Cauchy problem in a class of growing functions for equations of unsteady filtration type, Vestnik Moskov Univ. Ser. VI Mat. Mech. 6 (1963) 17–27.S.N. Kruzhkov, First order quasilinear equations in several independent variables, Math. USSR Sb. 10 (2) (1970) 217–243.S.N. Kruzhkov, S.M. Sukorjanski, Boundary value problems for systems of equations of two phase porous ffow type: statement of the problems, questions of solvability, justification of approximate methods, Math USSR Sb. 33 (1) (1977) 62–80.N.V. Krylov, Nonlinear elliptic and parabolic equations of the second order, D. Reidel Publishing Company, Dordrecht, 1987.O.A. Ladyzenskaya, V.A. Solonnikov, N.N. Ural’ceva, in: Quasilinear Equations of Parabolic Type,Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, RI, 1968.J.L. Lions, Quelques méthodes de résolution des probl_emes aux limites non linéaires, Dunod, Gauthiers-Villars, Paris, 1969.P.A. Markowich, The Stationary Semiconductor Device Equations, Springer, Wien, 1986.P.A. Markowich, C.A. Ringhofer, C. Schmeiser, Semiconductor Equations, Springer, Wien, 1990.M.S. Mock, On equations describing steady state carrier distributions in a semiconductor device, Comm.Pure Appl. Math. 25 (1972) 781–792.F. Otto, L1-contraction and uniqueness for quasilinear elliptic-parabolic equations, J. Differential Equations 131 (1996) 20–38.J.F. Padial, A comparison principle for weak solutions of some nonlinear parabolic systems, preprint, 1997.W. Rudin, Real and Complex Analysis, 2nd Edition, McGraw-Hill, New York, 1974.J. Rulla, Weak solutions to Stefan Problems with prescribed convection, SIAM J. Math. Anal. 18 (6)(1987) 1784–1800.J. Simon, Compact sets in the space Lp(0ff Tff B), Ann. Math. Pura Appl. 146 (1987) 65–96.G. Stampacchia, Equations elliptiques a donnees discontinues, Seminaire Schwartz, 1960–61: Equations aux d-eriv-ees partialles et interpolation, pp. 4.01–4.16.Further ReadingP. Benilan, Equations d’evolution dans un espace de Banach quelconque et applications, Thèse, Orsay, 1972.M. Bertsch, D. Hilhorst, A density dependent diffusion equation in population dynamics: stabilization to equilibrium, SIAM J. Math. Anal. 17 (4) (1986) 863–883.J.I. Díaz, G. Galiano, A. Jüngel, Space localization and uniqueness of solutions of a quasilinear parabolic system arising in semiconductor theory, C.R. Acad. Sci. t. 325, Ser. I (1997) 267–272.J.I. Díaz, J.F. Padial, Uniqueness and existence of solutions in BVt (Q) space to a doubly nonlinear parabolic problem, Publ. Mat. 40 (1996) 527–560.B.H. Gilding, Improved theory for a nonlinear degenerate parabolic equation, Ann. Sci. Norm. Sup.Pisa, Cl. Sci. IV 16 (1989) 165–224.
NO DGICYT (Spain)
DS Docta Complutense
RD 28 abr 2024