August  2022, 27(8): 4143-4172. doi: 10.3934/dcdsb.2021221

Continuous dependence of the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation in $H^{s} (\mathbb R^{n})$

Faculty of Mathematics, Kim Il Sung University, Pyongyang, Democratic People's Republic of Korea

*Corresponding author: JinMyong Kim (jm.kim0211@ryongnamsan.edu.kp)

Received  April 2021 Published  August 2022 Early access  September 2021

We consider the Cauchy problem for the inhomogeneous nonlinear Schrödinger (INLS) equation
$ iu_{t} +\Delta u = |x|^{-b} f(u), \;u(0)\in H^{s} (\mathbb R^{n} ), $
where
$ n\in \mathbb N $
,
$ 0<s<\min \{ n, \; 1+n/2\} $
,
$ 0<b<\min \{ 2, \;n-s, \;1+\frac{n-2s}{2} \} $
and
$ f(u) $
is a nonlinear function that behaves like
$ \lambda |u|^{\sigma } u $
with
$ \sigma>0 $
and
$ \lambda \in \mathbb C $
. Recently, the authors in [1] proved the local existence of solutions in
$ H^{s}(\mathbb R^{n} ) $
with
$ 0\le s<\min \{ n, \; 1+n/2\} $
. However even though the solution is constructed by a fixed point technique, continuous dependence in the standard sense in
$ H^{s}(\mathbb R^{n} ) $
with
$ 0< s<\min \{ n, \; 1+n/2\} $
doesn't follow from the contraction mapping argument. In this paper, we show that the solution depends continuously on the initial data in the standard sense in
$ H^{s}(\mathbb R^{n} ) $
, i.e. in the sense that the local solution flow is continuous
$ H^{s}(\mathbb R^{n} )\to H^{s}(\mathbb R^{n} ) $
, if
$ \sigma $
satisfies certain assumptions.
Citation: JinMyong An, JinMyong Kim, KyuSong Chae. Continuous dependence of the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation in $H^{s} (\mathbb R^{n})$. Discrete and Continuous Dynamical Systems - B, 2022, 27 (8) : 4143-4172. doi: 10.3934/dcdsb.2021221
References:
[1]

J. An and J. Kim, Local well-posedness for the inhomogeneous nonlinear Schrödinger equation in $H^{s} \left(\mathbb R^{n} \right)$, Nonlinear Anal. Real World Appl., 59 (2021), 103268. doi: 10.1016/j.nonrwa.2020.103268.

[2]

O. V. BorovkovaY. V. KartashovB. A. Malomed and L. Torner, Algebraic bright and vortex solitons in defocusing media, Opt. Lett., 36) (2011), 3088-3090.  doi: 10.1364/OL.36.003088.

[3]

O. V. BorovkovaY. V. KartashovV. A. VysloukhV. E. LobanovB. A. Malomed and L. Torner, Solitons supported by spatially inhomogeneous nonlinear losses, Opt. Express, 20 (2012), 2657-2667. 

[4]

L. Campos, Scattering of radial solutions to the inhomogeneous nonlinear Schrödinger equation, Nonlinear Anal., 202 (2021), 112118. doi: 10.1016/j.na.2020.112118.

[5]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.

[6]

T. CazenaveD. Fang and Z. Han, Continuous dependence for NLS in fractional order spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 135-147.  doi: 10.1016/j.anihpc.2010.11.005.

[7]

M. Christ and M. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg–de Vries equation, J. Funct. Anal., 100 (1991), 87-109.  doi: 10.1016/0022-1236(91)90103-C.

[8]

V. Combet and F. Genoud, Classification of minimal mass blow-up solutions for an $L^{2} $ critical inhomogeneous NLS, J. Evol. Equ., 16 (2016), 483-500.  doi: 10.1007/s00028-015-0309-z.

[9]

W. DaiW. Yang and D. Cao, Continuous dependence of Cauchy problem for nonlinear Schrödinger equation in $H^{s} $, J. Differential Equations, 255 (2013), 2018-2064.  doi: 10.1016/j.jde.2013.06.005.

[10]

V. D. Dinh, Energy scattering for a class of the defocusing inhomogeneous nonlinear Schrödinger equation, J. Evol. Equ., 19 (2019), 411-434.  doi: 10.1007/s00028-019-00481-0.

[11]

V. D. Dinh, Blowup of $H^{1} $ solutions for a class of the focusing inhomogeneous nonlinear Schrödinger equation, Nonlinear Anal., 174 (2018), 169-188.  doi: 10.1016/j.na.2018.04.024.

[12]

L. G. Farah, Global well-posedness and blow-up on the energy space for the inhomogeneous nonlinear Schrödinger equation, J. Evol. Equ., 16 (2016), 193-208.  doi: 10.1007/s00028-015-0298-y.

[13]

L. G. Farah and C. M. Guzmán, Scattering for the radial 3D cubic focusing inhomogeneous nonlinear Schrödinger equation, J. Differential Equations, 262 (2017), 4175-4231.  doi: 10.1016/j.jde.2017.01.013.

[14]

L. G. Farah and C. M. Guzmán, Scattering for the radial focusing inhomogeneous NLS equation in higher dimensions, Bull. Braz. Math. Soc. (N.S.), 51 (2020), 449-512.  doi: 10.1007/s00574-019-00160-1.

[15]

F. Genoud and C. A. Stuart, Schrödinger equations with a spatially decaying nonlinearity: existence and stability of standing waves, Discrete Contin. Dyn. Syst., 21 (2008), 137-186.  doi: 10.3934/dcds.2008.21.137.

[16]

F. Genoud, An inhomogeneous, $L^{2} $-critical, nonlinear Schrödinger equation, Z. Anal. Anwend., 31 (2012), 283-290.  doi: 10.4171/ZAA/1460.

[17]

T. S. Gill, Optical guiding of laser beam in nonuniform plasma, Pramana J. Phys., 55 (2000), 835-842.  doi: 10.1007/s12043-000-0051-z.

[18]

C. M. Guzmán, On well posedness for the inhomogeneous nonlinear Schrödinger equation, Nonlinear Anal. Real World Appl., 37 (2017), 249-286.  doi: 10.1016/j.nonrwa.2017.02.018.

[19]

C. E. KenigG. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620.  doi: 10.1002/cpa.3160460405.

[20]

F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, 2$^{nd}$ edition, Universitext. Springer, New York, 2015. doi: 10.1007/978-1-4939-2181-2.

[21]

C. S. Liu and V. K. Tripathi, Laser guiding in an axially nonuniform plasma channel, Phys. Plasmas, 1 (1994), 3100-3103.  doi: 10.1063/1.870501.

[22]

B. X. Wang, Z. Huo, C. Hao and Z. Guo, Harmonic Analysis Method for Nonlinear Evolution Equations, I, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011. doi: 10.1142/9789814360746.

show all references

References:
[1]

J. An and J. Kim, Local well-posedness for the inhomogeneous nonlinear Schrödinger equation in $H^{s} \left(\mathbb R^{n} \right)$, Nonlinear Anal. Real World Appl., 59 (2021), 103268. doi: 10.1016/j.nonrwa.2020.103268.

[2]

O. V. BorovkovaY. V. KartashovB. A. Malomed and L. Torner, Algebraic bright and vortex solitons in defocusing media, Opt. Lett., 36) (2011), 3088-3090.  doi: 10.1364/OL.36.003088.

[3]

O. V. BorovkovaY. V. KartashovV. A. VysloukhV. E. LobanovB. A. Malomed and L. Torner, Solitons supported by spatially inhomogeneous nonlinear losses, Opt. Express, 20 (2012), 2657-2667. 

[4]

L. Campos, Scattering of radial solutions to the inhomogeneous nonlinear Schrödinger equation, Nonlinear Anal., 202 (2021), 112118. doi: 10.1016/j.na.2020.112118.

[5]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.

[6]

T. CazenaveD. Fang and Z. Han, Continuous dependence for NLS in fractional order spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 135-147.  doi: 10.1016/j.anihpc.2010.11.005.

[7]

M. Christ and M. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg–de Vries equation, J. Funct. Anal., 100 (1991), 87-109.  doi: 10.1016/0022-1236(91)90103-C.

[8]

V. Combet and F. Genoud, Classification of minimal mass blow-up solutions for an $L^{2} $ critical inhomogeneous NLS, J. Evol. Equ., 16 (2016), 483-500.  doi: 10.1007/s00028-015-0309-z.

[9]

W. DaiW. Yang and D. Cao, Continuous dependence of Cauchy problem for nonlinear Schrödinger equation in $H^{s} $, J. Differential Equations, 255 (2013), 2018-2064.  doi: 10.1016/j.jde.2013.06.005.

[10]

V. D. Dinh, Energy scattering for a class of the defocusing inhomogeneous nonlinear Schrödinger equation, J. Evol. Equ., 19 (2019), 411-434.  doi: 10.1007/s00028-019-00481-0.

[11]

V. D. Dinh, Blowup of $H^{1} $ solutions for a class of the focusing inhomogeneous nonlinear Schrödinger equation, Nonlinear Anal., 174 (2018), 169-188.  doi: 10.1016/j.na.2018.04.024.

[12]

L. G. Farah, Global well-posedness and blow-up on the energy space for the inhomogeneous nonlinear Schrödinger equation, J. Evol. Equ., 16 (2016), 193-208.  doi: 10.1007/s00028-015-0298-y.

[13]

L. G. Farah and C. M. Guzmán, Scattering for the radial 3D cubic focusing inhomogeneous nonlinear Schrödinger equation, J. Differential Equations, 262 (2017), 4175-4231.  doi: 10.1016/j.jde.2017.01.013.

[14]

L. G. Farah and C. M. Guzmán, Scattering for the radial focusing inhomogeneous NLS equation in higher dimensions, Bull. Braz. Math. Soc. (N.S.), 51 (2020), 449-512.  doi: 10.1007/s00574-019-00160-1.

[15]

F. Genoud and C. A. Stuart, Schrödinger equations with a spatially decaying nonlinearity: existence and stability of standing waves, Discrete Contin. Dyn. Syst., 21 (2008), 137-186.  doi: 10.3934/dcds.2008.21.137.

[16]

F. Genoud, An inhomogeneous, $L^{2} $-critical, nonlinear Schrödinger equation, Z. Anal. Anwend., 31 (2012), 283-290.  doi: 10.4171/ZAA/1460.

[17]

T. S. Gill, Optical guiding of laser beam in nonuniform plasma, Pramana J. Phys., 55 (2000), 835-842.  doi: 10.1007/s12043-000-0051-z.

[18]

C. M. Guzmán, On well posedness for the inhomogeneous nonlinear Schrödinger equation, Nonlinear Anal. Real World Appl., 37 (2017), 249-286.  doi: 10.1016/j.nonrwa.2017.02.018.

[19]

C. E. KenigG. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620.  doi: 10.1002/cpa.3160460405.

[20]

F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, 2$^{nd}$ edition, Universitext. Springer, New York, 2015. doi: 10.1007/978-1-4939-2181-2.

[21]

C. S. Liu and V. K. Tripathi, Laser guiding in an axially nonuniform plasma channel, Phys. Plasmas, 1 (1994), 3100-3103.  doi: 10.1063/1.870501.

[22]

B. X. Wang, Z. Huo, C. Hao and Z. Guo, Harmonic Analysis Method for Nonlinear Evolution Equations, I, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011. doi: 10.1142/9789814360746.

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