November  2021, 41(11): 5409-5437. doi: 10.3934/dcds.2021082

Local well-posedness for the inhomogeneous nonlinear Schrödinger equation

1. 

Université de Tunis El Manar, Faculté des Sciences de Tunis, Département de Mathématiques, 2092 Tunis, Tunisie

2. 

Université de Tunis El Manar, Faculté des Sciences de Tunis, Département de Mathématiques, Laboratoire Équations aux, Dérivées Partielles LR03ES04, 2092 Tunis, Tunisie

* Corresponding author

Received  October 2020 Revised  March 2021 Published  November 2021 Early access  May 2021

We consider the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation $ i\partial_t u +\Delta u = \mu |x|^{-b}|u|^\alpha u,\; u(0)\in H^s({\mathbb R}^N),\; N\geq 1,\; \mu\in {\mathbb C},\; \; b>0 $ and $ \alpha>0. $ Only partial results are known for the local existence in the subcritical case $ \alpha<(4-2b)/(N-2s) $ and much more less in the critical case $ \alpha = (4-2b)/(N-2s). $ In this paper, we develop a local well-posedness theory for the both cases. In particular, we establish new results for the continuous dependence and for the unconditional uniqueness. Our approach provides simple proofs and allows us to obtain lower bounds of the blowup rate and of the life span. The Lorentz spaces and the Strichartz estimates play important roles in our argument. In particular this enables us to reach the critical case and to unify results for $ b = 0 $ and $ b>0. $

Citation: Lassaad Aloui, Slim Tayachi. Local well-posedness for the inhomogeneous nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2021, 41 (11) : 5409-5437. doi: 10.3934/dcds.2021082
References:
[1]

L. Aloui and S. Tayachi, Global existence and scattering for the inhomogeneous nonlinear Schrödinger equation, Preprint, 2021. Google Scholar

[2]

L. Aloui and S. Tayachi, Local existence, global existence and scattering for the 3D inhomogeneous nonlinear Schrödinger equation, Preprint, 2021. Google Scholar

[3]

K. F. Andersen and R. T. John, Weighted inequalities for vector-valued maximal functions and singular integrals, Studia Math., 69 (1980/81), 19-31.  doi: 10.4064/sm-69-1-19-31.  Google Scholar

[4]

J. Bergh and J. Löfström, Interpolation Spaces, Springer-Verlag, Berlin-New York, 1976.  Google Scholar

[5]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lect. Notes Math., vol. 10, New York University, Courant Institute of Mathematical Sciences/Amer. Math. Soc., New York/Providence, RI, 2003. doi: 10.1090/cln/010.  Google Scholar

[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.  Google Scholar

[7]

T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal., 14 (1990), 807-836.  doi: 10.1016/0362-546X(90)90023-A.  Google Scholar

[8]

T. Cazenave and F. B. Weissler, The structure of solutions to the pseudo-conformally invariant nonlinear Schrödinger equation, Proc. Roy. Soc. Edinburgh Sect. A, 117 (1991), 251-273.  doi: 10.1017/S0308210500024719.  Google Scholar

[9]

Y. Cho, S. Hong and K. Lee, On the GWP of focusing energy-critical inhomogeneous NLS, Preprint, arXiv: 1905.10063. Google Scholar

[10]

F. M. Christ and M. I. 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.  Google Scholar

[11]

D. V. Cruz-Uribe, J. M. Martell and C. Pérez, Weights, Extrapolation and the Theory of Rubio de Francia, Operator Theory: Advances and Applications, 215. Birkhäuser/Springer Basel AG, Basel, 2011. doi: 10.1007/978-3-0348-0072-3.  Google Scholar

[12]

D. Cruz-Uribe and V. Naibo, Kato-Ponce inequalities on weighted and variable Lebesgue spaces, Differential Integral Equations, 29 (2016), 801-836.   Google Scholar

[13]

V. D. Dinh, Scattering theory in a weighted $L^2$ space for a class of the defocusing inhomegeneous nonlinear Schrödinger equation, Preprint, 2017. arXiv: 1710.01392. Google Scholar

[14]

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.  Google Scholar

[15]

J. Duoandikoetxea, Fourier Analysis, Graduate Studies in Mathematics, 29. American Mathematical Society, Providence, RI, 2001. Translated and revised from the 1995 Spanish original by David Cruz-Uribe. doi: 10.1090/gsm/029.  Google Scholar

[16]

F. Genoud, Bifurcation and stability of travelling waves in self-focusing planar waveguides, Adv. Nonlinear Stud., 10 (2010), 357-400.  doi: 10.1515/ans-2010-0207.  Google Scholar

[17]

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.  Google Scholar

[18]

J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case, J. Functional Analysis, 32 (1979), 1-32.  doi: 10.1016/0022-1236(79)90076-4.  Google Scholar

[19]

L. Grafakos, Classical Fourier Analysis, 2nd ed., Graduate texts in Mathematics, Vol. 249, Springer, New York, 2008.  Google Scholar

[20]

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

[21]

H. HajaiejX. Yu and Z. Zhai, Fractional Gagliardo-Nirenberg and Hardy inequalities under Lorentz norms, J. Math. Anal. Appl., 396 (2012), 569-577.  doi: 10.1016/j.jmaa.2012.06.054.  Google Scholar

[22]

I. Halperin, Uniform convexity in function spaces, Duke Math. J., 21 (1954), 195-204.   Google Scholar

[23]

R. A. Hunt, On $L^{p, q}$ spaces, Enseign. Math., 12 (1966), 249-276.   Google Scholar

[24]

T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 46 (1987), 113-129.   Google Scholar

[25]

T. Kato, On nonlinear Schrödinger equations. Ⅱ. $H^s$-solutions and unconditional well-posedness, J. Anal. Math., 67 (1995), 281-306.  doi: 10.1007/BF02787794.  Google Scholar

[26]

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

[27]

C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.  doi: 10.1007/s00222-006-0011-4.  Google Scholar

[28]

R. Killip and M. Vişan, Nonlinear Schrödinger equations at critical regularity, in Evolution Equations, Clay Math. Proc., 17, Amer. Math. Soc., Providence, RI (Edited by D. Ellwood, I. Rodnianski, G. Staffilani and J. Wunsch), Clay Mathematics Institute, Cambridge, MA, (2013), 325–437.  Google Scholar

[29]

J. Kim, Y. Lee and I. Seo, On well-posedness for the inhomogeneous nonlinear Schrödinger equation in the critical case, J. Differential Equations, 280 (2021), 179–202. arXiv: 1907.11871v1. doi: 10.1016/j.jde.2021.01.023.  Google Scholar

[30]

Y. Lee and I. Seo, The Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation, preprint, 2019. arXiv: 1911.01112v2. Google Scholar

[31]

P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman & Hall/CRC Research Notes in Mathematics., 431 (2002), Chapman & Hall/CRC, Boca Raton, FL. doi: 10.1201/9781420035674.  Google Scholar

[32]

G. G. Lorentz, Some new functional spaces, Ann. of Math., 51 (1950), 37-55.  doi: 10.2307/1969496.  Google Scholar

[33]

C. E. Mueller and F. B. Weissler, Single point blow-up for a general semilinear heat equation, Indiana Univ. Math. J., 34 (1985), 881-913.  doi: 10.1512/iumj.1985.34.34049.  Google Scholar

[34]

R. O'Neil, Convolution operators and $L(p, q)$ spaces, Duke Math. J., 30 (1963), 129-142.   Google Scholar

[35]

K. M. Rogers, Unconditional well-posedness for subcritical NLS in $H^s$, C. R. Math. Acad. Sci. Paris, 345 (2007), 395-398.  doi: 10.1016/j.crma.2007.09.003.  Google Scholar

[36]

E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidian Spaces, Princeton Mathematical Series, Princeton University Press, 1971.  Google Scholar

[37]

R. J. Taggart, Inhomogeneous Strichartz estimates, Forum Math., 22 (2010), 825-853.  doi: 10.1515/FORUM.2010.044.  Google Scholar

[38]

T. Tao and M. Visan, Stability of energy-critical nonlinear Schrödinger equations in high dimensions, Electron. J. Differential Equations, (2005), No. 118, 28 pp.  Google Scholar

[39]

L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces, Lecture Notes of the Unione Mathematica Italiana, Springer-Verlag, Berlin Heidelberg, 2007.  Google Scholar

[40]

L. Tartar, Imbedding theorems of Sobolev spaces into Lorentz spaces, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 1 (1998), 479-500.   Google Scholar

[41]

S. Tayachi, Uniqueness and non–uniqueness of solutions for critical Hardy-Hénon parabolic equations, J. Math. Anal.Appl., 488 (2020), 123976, 51 pp. doi: 10.1016/j.jmaa.2020.123976.  Google Scholar

[42]

S. Tayachi and F. B. Weissler, The nonlinear heat equation involving highly singular initial values and new blowup and life span results, J. Elliptic Parabol. Equ., 4 (2018), 141-176.  doi: 10.1007/s41808-018-0014-5.  Google Scholar

[43]

S. Tayachi and F. B. Weissler, Some remarks on life span results, Preprint. Google Scholar

[44]

Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987), 115-125.   Google Scholar

[45]

F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math., 38 (1981), 29-40.  doi: 10.1007/BF02761845.  Google Scholar

show all references

References:
[1]

L. Aloui and S. Tayachi, Global existence and scattering for the inhomogeneous nonlinear Schrödinger equation, Preprint, 2021. Google Scholar

[2]

L. Aloui and S. Tayachi, Local existence, global existence and scattering for the 3D inhomogeneous nonlinear Schrödinger equation, Preprint, 2021. Google Scholar

[3]

K. F. Andersen and R. T. John, Weighted inequalities for vector-valued maximal functions and singular integrals, Studia Math., 69 (1980/81), 19-31.  doi: 10.4064/sm-69-1-19-31.  Google Scholar

[4]

J. Bergh and J. Löfström, Interpolation Spaces, Springer-Verlag, Berlin-New York, 1976.  Google Scholar

[5]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lect. Notes Math., vol. 10, New York University, Courant Institute of Mathematical Sciences/Amer. Math. Soc., New York/Providence, RI, 2003. doi: 10.1090/cln/010.  Google Scholar

[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.  Google Scholar

[7]

T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal., 14 (1990), 807-836.  doi: 10.1016/0362-546X(90)90023-A.  Google Scholar

[8]

T. Cazenave and F. B. Weissler, The structure of solutions to the pseudo-conformally invariant nonlinear Schrödinger equation, Proc. Roy. Soc. Edinburgh Sect. A, 117 (1991), 251-273.  doi: 10.1017/S0308210500024719.  Google Scholar

[9]

Y. Cho, S. Hong and K. Lee, On the GWP of focusing energy-critical inhomogeneous NLS, Preprint, arXiv: 1905.10063. Google Scholar

[10]

F. M. Christ and M. I. 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.  Google Scholar

[11]

D. V. Cruz-Uribe, J. M. Martell and C. Pérez, Weights, Extrapolation and the Theory of Rubio de Francia, Operator Theory: Advances and Applications, 215. Birkhäuser/Springer Basel AG, Basel, 2011. doi: 10.1007/978-3-0348-0072-3.  Google Scholar

[12]

D. Cruz-Uribe and V. Naibo, Kato-Ponce inequalities on weighted and variable Lebesgue spaces, Differential Integral Equations, 29 (2016), 801-836.   Google Scholar

[13]

V. D. Dinh, Scattering theory in a weighted $L^2$ space for a class of the defocusing inhomegeneous nonlinear Schrödinger equation, Preprint, 2017. arXiv: 1710.01392. Google Scholar

[14]

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.  Google Scholar

[15]

J. Duoandikoetxea, Fourier Analysis, Graduate Studies in Mathematics, 29. American Mathematical Society, Providence, RI, 2001. Translated and revised from the 1995 Spanish original by David Cruz-Uribe. doi: 10.1090/gsm/029.  Google Scholar

[16]

F. Genoud, Bifurcation and stability of travelling waves in self-focusing planar waveguides, Adv. Nonlinear Stud., 10 (2010), 357-400.  doi: 10.1515/ans-2010-0207.  Google Scholar

[17]

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.  Google Scholar

[18]

J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case, J. Functional Analysis, 32 (1979), 1-32.  doi: 10.1016/0022-1236(79)90076-4.  Google Scholar

[19]

L. Grafakos, Classical Fourier Analysis, 2nd ed., Graduate texts in Mathematics, Vol. 249, Springer, New York, 2008.  Google Scholar

[20]

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

[21]

H. HajaiejX. Yu and Z. Zhai, Fractional Gagliardo-Nirenberg and Hardy inequalities under Lorentz norms, J. Math. Anal. Appl., 396 (2012), 569-577.  doi: 10.1016/j.jmaa.2012.06.054.  Google Scholar

[22]

I. Halperin, Uniform convexity in function spaces, Duke Math. J., 21 (1954), 195-204.   Google Scholar

[23]

R. A. Hunt, On $L^{p, q}$ spaces, Enseign. Math., 12 (1966), 249-276.   Google Scholar

[24]

T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 46 (1987), 113-129.   Google Scholar

[25]

T. Kato, On nonlinear Schrödinger equations. Ⅱ. $H^s$-solutions and unconditional well-posedness, J. Anal. Math., 67 (1995), 281-306.  doi: 10.1007/BF02787794.  Google Scholar

[26]

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

[27]

C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.  doi: 10.1007/s00222-006-0011-4.  Google Scholar

[28]

R. Killip and M. Vişan, Nonlinear Schrödinger equations at critical regularity, in Evolution Equations, Clay Math. Proc., 17, Amer. Math. Soc., Providence, RI (Edited by D. Ellwood, I. Rodnianski, G. Staffilani and J. Wunsch), Clay Mathematics Institute, Cambridge, MA, (2013), 325–437.  Google Scholar

[29]

J. Kim, Y. Lee and I. Seo, On well-posedness for the inhomogeneous nonlinear Schrödinger equation in the critical case, J. Differential Equations, 280 (2021), 179–202. arXiv: 1907.11871v1. doi: 10.1016/j.jde.2021.01.023.  Google Scholar

[30]

Y. Lee and I. Seo, The Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation, preprint, 2019. arXiv: 1911.01112v2. Google Scholar

[31]

P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman & Hall/CRC Research Notes in Mathematics., 431 (2002), Chapman & Hall/CRC, Boca Raton, FL. doi: 10.1201/9781420035674.  Google Scholar

[32]

G. G. Lorentz, Some new functional spaces, Ann. of Math., 51 (1950), 37-55.  doi: 10.2307/1969496.  Google Scholar

[33]

C. E. Mueller and F. B. Weissler, Single point blow-up for a general semilinear heat equation, Indiana Univ. Math. J., 34 (1985), 881-913.  doi: 10.1512/iumj.1985.34.34049.  Google Scholar

[34]

R. O'Neil, Convolution operators and $L(p, q)$ spaces, Duke Math. J., 30 (1963), 129-142.   Google Scholar

[35]

K. M. Rogers, Unconditional well-posedness for subcritical NLS in $H^s$, C. R. Math. Acad. Sci. Paris, 345 (2007), 395-398.  doi: 10.1016/j.crma.2007.09.003.  Google Scholar

[36]

E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidian Spaces, Princeton Mathematical Series, Princeton University Press, 1971.  Google Scholar

[37]

R. J. Taggart, Inhomogeneous Strichartz estimates, Forum Math., 22 (2010), 825-853.  doi: 10.1515/FORUM.2010.044.  Google Scholar

[38]

T. Tao and M. Visan, Stability of energy-critical nonlinear Schrödinger equations in high dimensions, Electron. J. Differential Equations, (2005), No. 118, 28 pp.  Google Scholar

[39]

L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces, Lecture Notes of the Unione Mathematica Italiana, Springer-Verlag, Berlin Heidelberg, 2007.  Google Scholar

[40]

L. Tartar, Imbedding theorems of Sobolev spaces into Lorentz spaces, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 1 (1998), 479-500.   Google Scholar

[41]

S. Tayachi, Uniqueness and non–uniqueness of solutions for critical Hardy-Hénon parabolic equations, J. Math. Anal.Appl., 488 (2020), 123976, 51 pp. doi: 10.1016/j.jmaa.2020.123976.  Google Scholar

[42]

S. Tayachi and F. B. Weissler, The nonlinear heat equation involving highly singular initial values and new blowup and life span results, J. Elliptic Parabol. Equ., 4 (2018), 141-176.  doi: 10.1007/s41808-018-0014-5.  Google Scholar

[43]

S. Tayachi and F. B. Weissler, Some remarks on life span results, Preprint. Google Scholar

[44]

Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987), 115-125.   Google Scholar

[45]

F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math., 38 (1981), 29-40.  doi: 10.1007/BF02761845.  Google Scholar

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