September  2020, 19(9): 4545-4573. doi: 10.3934/cpaa.2020206

Finite dimensional global attractor for a damped fractional anisotropic Schrödinger type equation with harmonic potential

Laboratoire de Recherche: Analyse, Probabilité et Fractals, Université de Monastir, Faculté des Sciences, Avenue de l'environnement, 5019 Monastir, Tunisie

Received  January 2020 Revised  March 2020 Published  June 2020

We study the long time behaviour of the solutions for a class of nonlinear damped fractional Schrödinger type equation with anisotropic dispersion and in presence of a quadratic potential in a two dimensional unbounded domain. We prove that this behaviour is characterized by the existence of regular compact global attractor with finite fractal dimension.

Citation: Brahim Alouini. Finite dimensional global attractor for a damped fractional anisotropic Schrödinger type equation with harmonic potential. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4545-4573. doi: 10.3934/cpaa.2020206
References:
[1]

B. Alouini, Long-time behavior of a Bose-Einstein equation in a two dimensional thin domain, Commun. Pure Appl. Anal., 10 (2011), 1629-1643.  doi: 10.3934/cpaa.2011.10.1629.  Google Scholar

[2]

B. Alouini, Finite dimensional global attractor for a Bose-Einstein equation in a two dimensional unbounded domain, Commun. Pure Appl. Anal., 14 (2015), 1781-1801.  doi: 10.3934/cpaa.2015.14.1781.  Google Scholar

[3]

B. Alouini and O. Goubet, Regularity of the attractor for a Bose-Einstein equation in a two dimensional unbounded domain, Discrete Contin. Dyn. Syst. B, 19 (2014), 651-677.  doi: 10.3934/dcdsb.2014.19.651.  Google Scholar

[4]

A. H. ArdilaL. Cely and M. Squassina, Logarithmic Bose-Einstein condensates with harmonic potential, Asymptotic Anal., 116 (2020), 27-40.  doi: 10.3233/ASY-191538.  Google Scholar

[5]

A. H. Ardila, Existence and stability of standing waves for nonlinear fractional Schrödinger equation with logarithmic nonlinearity, Nonlinear Anal., 155 (2017), 52-64.  doi: 10.1016/j.na.2017.01.006.  Google Scholar

[6]

R. Askey and S. Wainger, Mean convergence of expensions in Laguerre and Hermite series, Amer. J. Math., 87 (1965), 695-708.  doi: 10.2307/2373069.  Google Scholar

[7]

Y. Bahri, S. Ibrahim and H. Kikuchi, Remarks on solitary waves and Cauchy problem for a half-wave Schrödinger equations, preprint, arXiv: math/1810.01385. Google Scholar

[8]

J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst. A, 10 (2004), 31-52.  doi: 10.3934/dcds.2004.10.31.  Google Scholar

[9]

B. Bongioanni and J. L. Torrea, Sobolev spaces associated to the harmonic oscillator, Proc. Indian. Acad. Sci. (Math. Sci.), 116 (2006), 337-360.  doi: 10.1007/BF02829750.  Google Scholar

[10]

R. Carles, Remarks on nonlinear Schrödinger equation with harmonic potential, Ann. Henri Poincare, 3 (2002), 757-772.  doi: 10.1007/s00023-002-8635-4.  Google Scholar

[11]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, Vol. 10, New York, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.  Google Scholar

[12]

M. Cheng, Bound state for the fractional Schrödinger equation with unbounded potential, J. Math. Phy., 53 (2012), Art. 043507. doi: 10.1063/1.3701574.  Google Scholar

[13]

A. Choffrut and O. Pocovnicu, Ill-posedness of the cubic nonlinear half-wave equation and other fractional NLS on the real line, Int. Math. Res. Notices, 2018 (2018), 699-738.  doi: 10.1093/imrn/rnw246.  Google Scholar

[14]

I. D. Chueshov, Introduction to The Theory of Infinite-Dimensional Dissipative Systems, University Lectures in Contemporary Mathematics, Vol. 19, ACTA, 2002.  Google Scholar

[15]

I. D. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations With Nonlinear Damping, Memoirs of the American Mathematical Society, Amer. Math. Soc., Vol. 195, 2008. doi: 10.1090/memo/0912.  Google Scholar

[16]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[17]

Z. Ding and H. Hajaiej, On a fractional Schrödinger equation in the presence of harmonic potential, preprint, arXiv: math/1908.05719. Google Scholar

[18]

E. Elgart and B. Schlein, Mean field dynamics of boson stars, Commun. Pure Appl. Math., 60 (2017), 500-545.  doi: 10.1002/cpa.20134.  Google Scholar

[19]

A. Esfahani and A. Pastor, Sharp constant of an anisotropic Gagliardo-Nirenberg type inequality and applications, Bull. Braz. Math. Soc., New Series, 48 (2017), 175-185.  doi: 10.1007/s00574-016-0017-5.  Google Scholar

[20]

G. B. Folland, Fourier Analysis and Its Applications, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1992.  Google Scholar

[21]

P. Gérard and S. Grellier, The cubic Szegö equation, Ann. Sci. de L'école Norm. Super., 43 (2010), 761-810.  doi: 10.24033/asens.2133.  Google Scholar

[22]

O. Goubet, Regularity of the attractor for a weakly damped nonlinear Schrödinger equation in $ \mathbb{R}^2$, Adv. Differ. Equ., 3 (1998), 337-360.   Google Scholar

[23]

O. Goubet and E. Zahrouni, Finite dimensional global attractor for a fractional nonlinear Schrödinger equation, NoDEA, 24 (2017), 59. doi: 10.1007/s00030-017-0482-6.  Google Scholar

[24]

L. Grafakos and S. Oh, The Kato-Ponce inequality, Commun. Partial Differ. Equ., 39 (2014), 1128-1157.  doi: 10.1080/03605302.2013.822885.  Google Scholar

[25]

B. Guo and Z. Huo, Global well-posedness for the fractional nonlinear Schrödinger equation, Commun. Partial Differ. Equ., 36 (2011), 247-255.  doi: 10.1080/03605302.2010.503769.  Google Scholar

[26]

Y. Hong and Y. Sire, On fractional Schrödinger equations in Sobolev spaces, Commun. Pure Appl. Anal., 14 (2015), 2265-2282.  doi: 10.3934/cpaa.2015.14.2265.  Google Scholar

[27]

C. Huang and L. Dong, Beam propagation management in a fractional Schrödinger equation, Sci. Rep., 7 (2017), 5442. doi: 10.1038/s41598-017-05926-5.  Google Scholar

[28]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.  doi: 10.1353/ajm.1998.0039.  Google Scholar

[29]

K. Kirkpatrick and Y. Zhang, Fractional Schrödinger dynamics and decoherence, Physica D, 332 (2016), 41-54.  doi: 10.1016/j.physd.2016.05.015.  Google Scholar

[30]

H. Koch and D. Tataru, $L^p$ Eigenfunction bounds for the Hermite operator, Duke Math. J., 128 (2005), 369-392.  doi: 10.1215/S0012-7094-04-12825-8.  Google Scholar

[31]

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.  Google Scholar

[32]

N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), Art. 56108. doi: 10.1103/PhysRevE.66.056108.  Google Scholar

[33]

P. Laurençot, Long-time behavior for weakly damped driven nonlinear Schrödinger equations in $\mathbb{R}^N, \; N\leq 3$, NoDEA, 2 (1995), 357-369.  doi: 10.1007/BF01261181.  Google Scholar

[34]

Q. LiuY. ZhouJ. Zhang and W. Zhang, Sharp condition of global existence for nonlinear Schrödinger equation with a harmonic potential, Appl. Math. Comput., 177 (2006), 482-487.  doi: 10.1016/j.amc.2005.11.024.  Google Scholar

[35]

S. Longhi, Fractional Schrödinger equation in optics, Optics Lett., 40 (2015), 1117-1120.  doi: 10.1364/OL.40.001117.  Google Scholar

[36]

C. Martinez and M. Sanz, The Theory of Fractional Powers of Operators, North-Holland Mathematics Studies, North Holland, Vol. 187, 2001.  Google Scholar

[37]

F. Pinsker, W. Bao, Y. Zhang, H. Ohadi, A. Dreismann and J. Baumberg, Fractional quantum mechanics in polariton condensates with velocity dependent mass, Phys. Rev. B, 92 (2015), Art. 195310. doi: 10.1103/PhysRevB.92.195310.  Google Scholar

[38]

H. Pollard, The mean convergence of orthogonal series â…¡, Trans. Amer. Math. Soc., 63 (1948), 355-367.  doi: 10.2307/1990435.  Google Scholar

[39]

G. Raugel, Global Attractors in Partial Differential Equations, Handbook of dynamical systems, North-Holland, Vol. 2,885?82, North-Holland, Amsterdam, 2002. doi: 10.1016/S1874-575X(02)80038-8.  Google Scholar

[40]

J. C. Robinson, Infinite Dimensional Dynamical Systems, An Introduction to Dissipative Parabolic PDEs and The Theorie of Global Attractors, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2001. doi: 10.1115/1.1579456.  Google Scholar

[41]

E. Russ, Racine carrées d'opérateurs elliptiques et espaces de Hardy, Confl. Math., 3 (2011), 1-119.  doi: 10.1142/S1793744211000278.  Google Scholar

[42]

B. A. Stickler, Potential condensed-matter realisation of space-fractional quantum mechanics: the one dimensional Lévy crystal, Phys. Rev. E, 88 (2013), Art. 012120. doi: 10.1103/PhysRevE.88.012120.  Google Scholar

[43]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer applied mathmatical sciences, Vol. 68, Springer-Verlag, 2$^nd$ Edition, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[44]

X. Wang, An energy equation for the weakly damped driven nonlinear Schrödinger equations and its application to their attractors, Physica D, 88 (1995), 167-175.  doi: 10.1016/0167-2789(95)00196-B.  Google Scholar

[45]

H. Xu, Unbounded Sobolev trajectories and modified scattering theory for a wave guide nonlinear Schrödinger equation, Math. Z., 286 (2017), 443-489.  doi: 10.1007/s00209-016-1768-9.  Google Scholar

[46]

Y. ZhangH. ZhongM. BeliećN. AhmedY. Zhang and M. Xiao, Diffraction free beams in fractional Schrödinger equation, Sci. Rep., 6 (2016), 1-8.  doi: 10.1038/srep23645.  Google Scholar

show all references

References:
[1]

B. Alouini, Long-time behavior of a Bose-Einstein equation in a two dimensional thin domain, Commun. Pure Appl. Anal., 10 (2011), 1629-1643.  doi: 10.3934/cpaa.2011.10.1629.  Google Scholar

[2]

B. Alouini, Finite dimensional global attractor for a Bose-Einstein equation in a two dimensional unbounded domain, Commun. Pure Appl. Anal., 14 (2015), 1781-1801.  doi: 10.3934/cpaa.2015.14.1781.  Google Scholar

[3]

B. Alouini and O. Goubet, Regularity of the attractor for a Bose-Einstein equation in a two dimensional unbounded domain, Discrete Contin. Dyn. Syst. B, 19 (2014), 651-677.  doi: 10.3934/dcdsb.2014.19.651.  Google Scholar

[4]

A. H. ArdilaL. Cely and M. Squassina, Logarithmic Bose-Einstein condensates with harmonic potential, Asymptotic Anal., 116 (2020), 27-40.  doi: 10.3233/ASY-191538.  Google Scholar

[5]

A. H. Ardila, Existence and stability of standing waves for nonlinear fractional Schrödinger equation with logarithmic nonlinearity, Nonlinear Anal., 155 (2017), 52-64.  doi: 10.1016/j.na.2017.01.006.  Google Scholar

[6]

R. Askey and S. Wainger, Mean convergence of expensions in Laguerre and Hermite series, Amer. J. Math., 87 (1965), 695-708.  doi: 10.2307/2373069.  Google Scholar

[7]

Y. Bahri, S. Ibrahim and H. Kikuchi, Remarks on solitary waves and Cauchy problem for a half-wave Schrödinger equations, preprint, arXiv: math/1810.01385. Google Scholar

[8]

J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst. A, 10 (2004), 31-52.  doi: 10.3934/dcds.2004.10.31.  Google Scholar

[9]

B. Bongioanni and J. L. Torrea, Sobolev spaces associated to the harmonic oscillator, Proc. Indian. Acad. Sci. (Math. Sci.), 116 (2006), 337-360.  doi: 10.1007/BF02829750.  Google Scholar

[10]

R. Carles, Remarks on nonlinear Schrödinger equation with harmonic potential, Ann. Henri Poincare, 3 (2002), 757-772.  doi: 10.1007/s00023-002-8635-4.  Google Scholar

[11]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, Vol. 10, New York, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.  Google Scholar

[12]

M. Cheng, Bound state for the fractional Schrödinger equation with unbounded potential, J. Math. Phy., 53 (2012), Art. 043507. doi: 10.1063/1.3701574.  Google Scholar

[13]

A. Choffrut and O. Pocovnicu, Ill-posedness of the cubic nonlinear half-wave equation and other fractional NLS on the real line, Int. Math. Res. Notices, 2018 (2018), 699-738.  doi: 10.1093/imrn/rnw246.  Google Scholar

[14]

I. D. Chueshov, Introduction to The Theory of Infinite-Dimensional Dissipative Systems, University Lectures in Contemporary Mathematics, Vol. 19, ACTA, 2002.  Google Scholar

[15]

I. D. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations With Nonlinear Damping, Memoirs of the American Mathematical Society, Amer. Math. Soc., Vol. 195, 2008. doi: 10.1090/memo/0912.  Google Scholar

[16]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[17]

Z. Ding and H. Hajaiej, On a fractional Schrödinger equation in the presence of harmonic potential, preprint, arXiv: math/1908.05719. Google Scholar

[18]

E. Elgart and B. Schlein, Mean field dynamics of boson stars, Commun. Pure Appl. Math., 60 (2017), 500-545.  doi: 10.1002/cpa.20134.  Google Scholar

[19]

A. Esfahani and A. Pastor, Sharp constant of an anisotropic Gagliardo-Nirenberg type inequality and applications, Bull. Braz. Math. Soc., New Series, 48 (2017), 175-185.  doi: 10.1007/s00574-016-0017-5.  Google Scholar

[20]

G. B. Folland, Fourier Analysis and Its Applications, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1992.  Google Scholar

[21]

P. Gérard and S. Grellier, The cubic Szegö equation, Ann. Sci. de L'école Norm. Super., 43 (2010), 761-810.  doi: 10.24033/asens.2133.  Google Scholar

[22]

O. Goubet, Regularity of the attractor for a weakly damped nonlinear Schrödinger equation in $ \mathbb{R}^2$, Adv. Differ. Equ., 3 (1998), 337-360.   Google Scholar

[23]

O. Goubet and E. Zahrouni, Finite dimensional global attractor for a fractional nonlinear Schrödinger equation, NoDEA, 24 (2017), 59. doi: 10.1007/s00030-017-0482-6.  Google Scholar

[24]

L. Grafakos and S. Oh, The Kato-Ponce inequality, Commun. Partial Differ. Equ., 39 (2014), 1128-1157.  doi: 10.1080/03605302.2013.822885.  Google Scholar

[25]

B. Guo and Z. Huo, Global well-posedness for the fractional nonlinear Schrödinger equation, Commun. Partial Differ. Equ., 36 (2011), 247-255.  doi: 10.1080/03605302.2010.503769.  Google Scholar

[26]

Y. Hong and Y. Sire, On fractional Schrödinger equations in Sobolev spaces, Commun. Pure Appl. Anal., 14 (2015), 2265-2282.  doi: 10.3934/cpaa.2015.14.2265.  Google Scholar

[27]

C. Huang and L. Dong, Beam propagation management in a fractional Schrödinger equation, Sci. Rep., 7 (2017), 5442. doi: 10.1038/s41598-017-05926-5.  Google Scholar

[28]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.  doi: 10.1353/ajm.1998.0039.  Google Scholar

[29]

K. Kirkpatrick and Y. Zhang, Fractional Schrödinger dynamics and decoherence, Physica D, 332 (2016), 41-54.  doi: 10.1016/j.physd.2016.05.015.  Google Scholar

[30]

H. Koch and D. Tataru, $L^p$ Eigenfunction bounds for the Hermite operator, Duke Math. J., 128 (2005), 369-392.  doi: 10.1215/S0012-7094-04-12825-8.  Google Scholar

[31]

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.  Google Scholar

[32]

N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), Art. 56108. doi: 10.1103/PhysRevE.66.056108.  Google Scholar

[33]

P. Laurençot, Long-time behavior for weakly damped driven nonlinear Schrödinger equations in $\mathbb{R}^N, \; N\leq 3$, NoDEA, 2 (1995), 357-369.  doi: 10.1007/BF01261181.  Google Scholar

[34]

Q. LiuY. ZhouJ. Zhang and W. Zhang, Sharp condition of global existence for nonlinear Schrödinger equation with a harmonic potential, Appl. Math. Comput., 177 (2006), 482-487.  doi: 10.1016/j.amc.2005.11.024.  Google Scholar

[35]

S. Longhi, Fractional Schrödinger equation in optics, Optics Lett., 40 (2015), 1117-1120.  doi: 10.1364/OL.40.001117.  Google Scholar

[36]

C. Martinez and M. Sanz, The Theory of Fractional Powers of Operators, North-Holland Mathematics Studies, North Holland, Vol. 187, 2001.  Google Scholar

[37]

F. Pinsker, W. Bao, Y. Zhang, H. Ohadi, A. Dreismann and J. Baumberg, Fractional quantum mechanics in polariton condensates with velocity dependent mass, Phys. Rev. B, 92 (2015), Art. 195310. doi: 10.1103/PhysRevB.92.195310.  Google Scholar

[38]

H. Pollard, The mean convergence of orthogonal series â…¡, Trans. Amer. Math. Soc., 63 (1948), 355-367.  doi: 10.2307/1990435.  Google Scholar

[39]

G. Raugel, Global Attractors in Partial Differential Equations, Handbook of dynamical systems, North-Holland, Vol. 2,885?82, North-Holland, Amsterdam, 2002. doi: 10.1016/S1874-575X(02)80038-8.  Google Scholar

[40]

J. C. Robinson, Infinite Dimensional Dynamical Systems, An Introduction to Dissipative Parabolic PDEs and The Theorie of Global Attractors, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2001. doi: 10.1115/1.1579456.  Google Scholar

[41]

E. Russ, Racine carrées d'opérateurs elliptiques et espaces de Hardy, Confl. Math., 3 (2011), 1-119.  doi: 10.1142/S1793744211000278.  Google Scholar

[42]

B. A. Stickler, Potential condensed-matter realisation of space-fractional quantum mechanics: the one dimensional Lévy crystal, Phys. Rev. E, 88 (2013), Art. 012120. doi: 10.1103/PhysRevE.88.012120.  Google Scholar

[43]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer applied mathmatical sciences, Vol. 68, Springer-Verlag, 2$^nd$ Edition, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[44]

X. Wang, An energy equation for the weakly damped driven nonlinear Schrödinger equations and its application to their attractors, Physica D, 88 (1995), 167-175.  doi: 10.1016/0167-2789(95)00196-B.  Google Scholar

[45]

H. Xu, Unbounded Sobolev trajectories and modified scattering theory for a wave guide nonlinear Schrödinger equation, Math. Z., 286 (2017), 443-489.  doi: 10.1007/s00209-016-1768-9.  Google Scholar

[46]

Y. ZhangH. ZhongM. BeliećN. AhmedY. Zhang and M. Xiao, Diffraction free beams in fractional Schrödinger equation, Sci. Rep., 6 (2016), 1-8.  doi: 10.1038/srep23645.  Google Scholar

[1]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345

[2]

Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021015

[3]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247

[4]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376

[5]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276

[6]

Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259

[7]

Van Duong Dinh. Random data theory for the cubic fourth-order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020284

[8]

Xiaorui Wang, Genqi Xu, Hao Chen. Uniform stabilization of 1-D Schrödinger equation with internal difference-type control. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021022

[9]

Fang Li, Bo You. On the dimension of global attractor for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021024

[10]

Xinyu Mei, Yangmin Xiong, Chunyou Sun. Pullback attractor for a weakly damped wave equation with sup-cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 569-600. doi: 10.3934/dcds.2020270

[11]

Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264

[12]

Norman Noguera, Ademir Pastor. Scattering of radial solutions for quadratic-type Schrödinger systems in dimension five. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021018

[13]

Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $ L^2 $-supercritical case. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298

[14]

Oleg Yu. Imanuvilov, Jean Pierre Puel. On global controllability of 2-D Burgers equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 299-313. doi: 10.3934/dcds.2009.23.299

[15]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081

[16]

Wenjun Liu, Hefeng Zhuang. Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 907-942. doi: 10.3934/dcdsb.2020147

[17]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243

[18]

Zheng Han, Daoyuan Fang. Almost global existence for the Klein-Gordon equation with the Kirchhoff-type nonlinearity. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020287

[19]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461

[20]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (124)
  • HTML views (86)
  • Cited by (0)

Other articles
by authors

[Back to Top]