NLS-like equations in bounded domains: Parabolic approximation procedure

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January 2018, 23(1): 57-77. doi: 10.3934/dcdsb.2018005

NLS-like equations in bounded domains: Parabolic approximation procedure

Alexandre N. Carvalho ^1, and Jan W. Cholewa ^2,,

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

* Corresponding author: Jan W. Cholewa

Received September 2016 Published January 2018

Fund Project: A.N.C. partially supported by CNPq # 303929/2015-4 and by FAPESP # 2003/10042-0, Brazil.
J.C. partially supported by grant MTM2012-31298 from Ministerio de Economia y Competitividad, Spain.

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)$.

Keywords: Schrödinger equation, parabolic equation, critical exponents.

Mathematics Subject Classification: Primary: 35Q55, 35K55; Secondary: 35B33.

Citation: Alexandre N. Carvalho, Jan W. Cholewa. NLS-like equations in bounded domains: Parabolic approximation procedure. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 57-77. doi: 10.3934/dcdsb.2018005

References:

[1]	R. A. Adams, Sobolev Spaces Academic Press, New York, 1975.
[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]	R. A. Adams, Sobolev Spaces Academic Press, New York, 1975.
[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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