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Dynamical system modeling fermionic limit
NLS-like equations in bounded domains: Parabolic approximation procedure
1. | Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo-Campus de São Carlos, Caixa Postal 668, 13560-970 São Carlos SP, Brazil |
2. | Institute of Mathematics, University of Silesia in Katowice, 40-007 Katowice, Poland |
The article is devoted to semilinear Schrödinger equations in bounded domains. A unified semigroup approach is applied following a concept of Trotter-Kato approximations.Critical exponents are exhibited and global solutions are constructed for nonlinearities satisfying even a certain critical growth condition in $ H^1_0(Ω)$.
References:
[1] | |
[2] |
H. Amann,
Linear and Quasilinear Parabolic Problems, Volume Ⅰ, Abstract Linear Theory Birkhäuser Verlag, Basel, 1995.
doi: 10.1007/978-3-0348-9221-6. |
[3] |
J. Arrieta and A. N. Carvalho,
Abstract parabolic problems with critical nonlinearities and applications to Navier-Stokes and heat equations, Trans. Amer. Math. Soc., 352 (2000), 285-310.
doi: 10.1090/S0002-9947-99-02528-3. |
[4] |
A. N. Carvalho and J. W. Cholewa,
Continuation and asymptotics of solutions to semilinear parabolic equations with critical nonlinearities, J. Math. Anal. Appl., 310 (2005), 557-578.
doi: 10.1016/j.jmaa.2005.02.024. |
[5] |
A. N. Carvalho, J. W. Cholewa and T. Dlotko,
Damped wave equations with fast growing dissipative nonlinearities, Discrete Contin. Dyn. Syst., 24 (2009), 1147-1165.
doi: 10.3934/dcds.2009.24.1147. |
[6] |
T. Cazenave,
Semilinear Schrödinger Equations, Courant lecture notes, 10, American Mathematical Society, Providence, RI, 2003.
doi: 10.1090/cln/010. |
[7] |
T. Cazenave and A. Haraux,
An Introduction to Semilinear Evolution Equations, The Clarendon Press, Oxford University Press, New York 1998. |
[8] |
J. Chabrowski and A. Szulkin,
On a semilinear Schrödinger equation with critical Sobolev exponent, Proc. Amer. Math. Soc., 130 (2002), 85-93.
doi: 10.1090/S0002-9939-01-06143-3. |
[9] |
J. W. Cholewa and A. Rodriguez-Bernal,
Linear and semilinear higher order parabolic equations in $ \mathbb{R}^N$, Nonlinear Analysis TMA, 75 (2012), 194-210.
doi: 10.1016/j.na.2011.08.022. |
[10] |
J. W. Cholewa and A. Rodriguez-Bernal,
Dissipative mechanism of a semilinear higher order parabolic equation in $ \mathbb{R}^N$, Nonlinear Analysis TMA, 75 (2012), 3510-3530.
doi: 10.1016/j.na.2012.01.011. |
[11] |
R. Denk, G. Dore, M. Hieber, J. Prüss and A. Venni,
New thoughts on old results of R. T. Seeley, Math. Ann., 328 (2004), 545-583.
doi: 10.1007/s00208-003-0493-y. |
[12] |
J. Dieudonné,
Éléments D'analyse. Tome Ⅰ: Fondements de L'analyse Moderne Gauthier-Villars, Paris 1968. |
[13] |
L. Fanelli,
Semilinear Schrödinger equation with time dependent coefficients, Math. Nachr., 282 (2009), 976-994.
doi: 10.1002/mana.200610784. |
[14] |
D. Henry,
Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin, 1981. |
[15] |
T. Kato,
Fractional powers of dissipative operators Ⅱ, J. Math. Soc. Japan, 14 (1962), 242-248.
doi: 10.2969/jmsj/01420242. |
[16] |
A. Pazy,
Semigroups of Linear Operators and Applications to Partial Differential Equations Springer, Berlin, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[17] |
T. Saanouni,
Remarks on the semilinear Schrödinger equation, J. Math. Anal. Appl., 400 (2013), 331-344.
doi: 10.1016/j.jmaa.2012.11.037. |
[18] |
K. Taira,
Analytic Semigroups and Semilinear Initial Boundary Value Problems Cambridge University Press, Cambridge, 1995.
doi: 10.1017/CBO9780511662362. |
[19] |
R. Temam,
Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[20] |
H. Triebel,
Interpolation Theory, Function Spaces, Differential Operators, Veb Deutscher, Berlin, 1978. |
[21] |
W. von Wahl,
Global solutions to evolution equations of parabolic type,
in: Differential Equations in Banach Spaces, Proceedings, 1985 (Eds. A. Favini, E. Obrecht), Springer-Verlag, Berlin, 1223 (1986), 254-266.
doi: 10.1007/BFb0099198. |
show all references
References:
[1] | |
[2] |
H. Amann,
Linear and Quasilinear Parabolic Problems, Volume Ⅰ, Abstract Linear Theory Birkhäuser Verlag, Basel, 1995.
doi: 10.1007/978-3-0348-9221-6. |
[3] |
J. Arrieta and A. N. Carvalho,
Abstract parabolic problems with critical nonlinearities and applications to Navier-Stokes and heat equations, Trans. Amer. Math. Soc., 352 (2000), 285-310.
doi: 10.1090/S0002-9947-99-02528-3. |
[4] |
A. N. Carvalho and J. W. Cholewa,
Continuation and asymptotics of solutions to semilinear parabolic equations with critical nonlinearities, J. Math. Anal. Appl., 310 (2005), 557-578.
doi: 10.1016/j.jmaa.2005.02.024. |
[5] |
A. N. Carvalho, J. W. Cholewa and T. Dlotko,
Damped wave equations with fast growing dissipative nonlinearities, Discrete Contin. Dyn. Syst., 24 (2009), 1147-1165.
doi: 10.3934/dcds.2009.24.1147. |
[6] |
T. Cazenave,
Semilinear Schrödinger Equations, Courant lecture notes, 10, American Mathematical Society, Providence, RI, 2003.
doi: 10.1090/cln/010. |
[7] |
T. Cazenave and A. Haraux,
An Introduction to Semilinear Evolution Equations, The Clarendon Press, Oxford University Press, New York 1998. |
[8] |
J. Chabrowski and A. Szulkin,
On a semilinear Schrödinger equation with critical Sobolev exponent, Proc. Amer. Math. Soc., 130 (2002), 85-93.
doi: 10.1090/S0002-9939-01-06143-3. |
[9] |
J. W. Cholewa and A. Rodriguez-Bernal,
Linear and semilinear higher order parabolic equations in $ \mathbb{R}^N$, Nonlinear Analysis TMA, 75 (2012), 194-210.
doi: 10.1016/j.na.2011.08.022. |
[10] |
J. W. Cholewa and A. Rodriguez-Bernal,
Dissipative mechanism of a semilinear higher order parabolic equation in $ \mathbb{R}^N$, Nonlinear Analysis TMA, 75 (2012), 3510-3530.
doi: 10.1016/j.na.2012.01.011. |
[11] |
R. Denk, G. Dore, M. Hieber, J. Prüss and A. Venni,
New thoughts on old results of R. T. Seeley, Math. Ann., 328 (2004), 545-583.
doi: 10.1007/s00208-003-0493-y. |
[12] |
J. Dieudonné,
Éléments D'analyse. Tome Ⅰ: Fondements de L'analyse Moderne Gauthier-Villars, Paris 1968. |
[13] |
L. Fanelli,
Semilinear Schrödinger equation with time dependent coefficients, Math. Nachr., 282 (2009), 976-994.
doi: 10.1002/mana.200610784. |
[14] |
D. Henry,
Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin, 1981. |
[15] |
T. Kato,
Fractional powers of dissipative operators Ⅱ, J. Math. Soc. Japan, 14 (1962), 242-248.
doi: 10.2969/jmsj/01420242. |
[16] |
A. Pazy,
Semigroups of Linear Operators and Applications to Partial Differential Equations Springer, Berlin, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[17] |
T. Saanouni,
Remarks on the semilinear Schrödinger equation, J. Math. Anal. Appl., 400 (2013), 331-344.
doi: 10.1016/j.jmaa.2012.11.037. |
[18] |
K. Taira,
Analytic Semigroups and Semilinear Initial Boundary Value Problems Cambridge University Press, Cambridge, 1995.
doi: 10.1017/CBO9780511662362. |
[19] |
R. Temam,
Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[20] |
H. Triebel,
Interpolation Theory, Function Spaces, Differential Operators, Veb Deutscher, Berlin, 1978. |
[21] |
W. von Wahl,
Global solutions to evolution equations of parabolic type,
in: Differential Equations in Banach Spaces, Proceedings, 1985 (Eds. A. Favini, E. Obrecht), Springer-Verlag, Berlin, 1223 (1986), 254-266.
doi: 10.1007/BFb0099198. |
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