American Institute of Mathematical Sciences

Advanced
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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

T. Cazenave, Semilinear Schrödinger Equations, Courant lecture notes, 10, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.  Google Scholar

[7]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, The Clarendon Press, Oxford University Press, New York 1998.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

J. Dieudonné, Éléments D'analyse. Tome Ⅰ: Fondements de L'analyse Moderne Gauthier-Villars, Paris 1968.  Google Scholar

[13]

L. Fanelli, Semilinear Schrödinger equation with time dependent coefficients, Math. Nachr., 282 (2009), 976-994.  doi: 10.1002/mana.200610784.  Google Scholar

[14]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin, 1981.  Google Scholar

[15]

T. Kato, Fractional powers of dissipative operators Ⅱ, J. Math. Soc. Japan, 14 (1962), 242-248.  doi: 10.2969/jmsj/01420242.  Google Scholar

[16]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations Springer, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[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.  Google Scholar

[18]

K. Taira, Analytic Semigroups and Semilinear Initial Boundary Value Problems Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511662362.  Google Scholar

[19]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[20]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Veb Deutscher, Berlin, 1978.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

R. A. Adams, Sobolev Spaces Academic Press, New York, 1975.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

T. Cazenave, Semilinear Schrödinger Equations, Courant lecture notes, 10, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.  Google Scholar

[7]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, The Clarendon Press, Oxford University Press, New York 1998.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

J. Dieudonné, Éléments D'analyse. Tome Ⅰ: Fondements de L'analyse Moderne Gauthier-Villars, Paris 1968.  Google Scholar

[13]

L. Fanelli, Semilinear Schrödinger equation with time dependent coefficients, Math. Nachr., 282 (2009), 976-994.  doi: 10.1002/mana.200610784.  Google Scholar

[14]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin, 1981.  Google Scholar

[15]

T. Kato, Fractional powers of dissipative operators Ⅱ, J. Math. Soc. Japan, 14 (1962), 242-248.  doi: 10.2969/jmsj/01420242.  Google Scholar

[16]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations Springer, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[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.  Google Scholar

[18]

K. Taira, Analytic Semigroups and Semilinear Initial Boundary Value Problems Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511662362.  Google Scholar

[19]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[20]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Veb Deutscher, Berlin, 1978.  Google Scholar

[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.  Google Scholar

[1]

Kun Cheng, Yinbin Deng. Nodal solutions for a generalized quasilinear Schrödinger equation with critical exponents. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 77-103. doi: 10.3934/dcds.2017004
[2]

Pavel I. Naumkin, Isahi Sánchez-Suárez. On the critical nongauge invariant nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 807-834. doi: 10.3934/dcds.2011.30.807
[3]

Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309
[4]

Mohamad Darwich. On the $L^2$-critical nonlinear Schrödinger Equation with a nonlinear damping. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2377-2394. doi: 10.3934/cpaa.2014.13.2377
[5]

Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037
[6]

Seunghyeok Kim. On vector solutions for coupled nonlinear Schrödinger equations with critical exponents. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1259-1277. doi: 10.3934/cpaa.2013.12.1259
[7]

Yongpeng Chen, Yuxia Guo, Zhongwei Tang. Concentration of ground state solutions for quasilinear Schrödinger systems with critical exponents. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2693-2715. doi: 10.3934/cpaa.2019120
[8]

Jingxue Yin, Chunhua Jin. Critical exponents and traveling wavefronts of a degenerate-singular parabolic equation in non-divergence form. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 213-227. doi: 10.3934/dcdsb.2010.13.213
[9]

Claude Bardos, François Golse, Peter Markowich, Thierry Paul. On the classical limit of the Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5689-5709. doi: 10.3934/dcds.2015.35.5689
[10]

Camille Laurent. Internal control of the Schrödinger equation. Mathematical Control & Related Fields, 2014, 4 (2) : 161-186. doi: 10.3934/mcrf.2014.4.161
[11]

D.G. deFigueiredo, Yanheng Ding. Solutions of a nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 563-584. doi: 10.3934/dcds.2002.8.563
[12]

Frank Wusterhausen. Schrödinger equation with noise on the boundary. Conference Publications, 2013, 2013 (special) : 791-796. doi: 10.3934/proc.2013.2013.791
[13]

Yuxia Guo, Zhongwei Tang. Multi-bump solutions for Schrödinger equation involving critical growth and potential wells. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3393-3415. doi: 10.3934/dcds.2015.35.3393
[14]

Jian Zhang, Shihui Zhu, Xiaoguang Li. Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential. Mathematical Control & Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119
[15]

Yinbin Deng, Yi Li, Xiujuan Yan. Nodal solutions for a quasilinear Schrödinger equation with critical nonlinearity and non-square diffusion. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2487-2508. doi: 10.3934/cpaa.2015.14.2487
[16]

Kenji Nakanishi, Tristan Roy. Global dynamics above the ground state for the energy-critical Schrödinger equation with radial data. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2023-2058. doi: 10.3934/cpaa.2016026
[17]

Benoît Pausader. The focusing energy-critical fourth-order Schrödinger equation with radial data. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1275-1292. doi: 10.3934/dcds.2009.24.1275
[18]

Daniela De Silva, Nataša Pavlović, Gigliola Staffilani, Nikolaos Tzirakis. Global well-posedness for the $L^2$ critical nonlinear Schrödinger equation in higher dimensions. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1023-1041. doi: 10.3934/cpaa.2007.6.1023
[19]

Miaomiao Niu, Zhongwei Tang. Least energy solutions for nonlinear Schrödinger equation involving the fractional Laplacian and critical growth. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3963-3987. doi: 10.3934/dcds.2017168
[20]

Van Duong Dinh. On blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 689-708. doi: 10.3934/cpaa.2019034

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (46)
  • HTML views (118)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences