December  2021, 41(12): 5825-5849. doi: 10.3934/dcds.2021097

Sharp well-posedness of the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili equation in anisotropic Sobolev spaces

1. 

School of Mathematics and Information Science, Henan Normal University, Xinxiang, Henan 453007, China

2. 

Center for Mathematical Sciences, Wuhan University of Technology, Wuhan, Hubei 430070, China

3. 

School of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, China

4. 

Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA

* Corresponding author: Yimin Zhang

Received  February 2021 Revised  May 2021 Published  December 2021 Early access  July 2021

Fund Project: This work is supported by the NSFC under grant numbers 11771127, 11571118 and 11471330. The first author is also supported by the education department of Henan province under the grant number 21A110014. The third author is also supported by the Fundamental Research Funds for the Central Universities of China under the grant number 2017ZD094

We consider the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili (RMKP) equation
$ \begin{align*} \partial_{x}\left(u_{t}-\beta\partial_{x}^{3}u +\partial_{x}(u^{2})\right)+\partial_{y}^{2}u-\gamma u = 0 \end{align*} $
in the anisotropic Sobolev spaces
$ H^{s_{1},s_{2}}(\mathbb{R}^{2}) $
. When
$ \beta <0 $
and
$ \gamma >0, $
we prove that the Cauchy problem is locally well-posed in
$ H^{s_{1}, s_{2}}(\mathbb{R}^{2}) $
with
$ s_{1}>-\frac{1}{2} $
and
$ s_{2}\geq 0 $
. Our result considerably improves the Theorem 1.4 of R. M. Chen, Y. Liu, P. Z. Zhang(Transactions of the American Mathematical Society, 364(2012), 3395–3425.). The key idea is that we divide the frequency space into regular region and singular region. We further prove that the Cauchy problem for RMKP equation is ill-posed in
$ H^{s_{1},0}(\mathbb{R}^{2}) $
with
$ s_{1}<-\frac{1}{2} $
in the sense that the flow map associated to the rotation-modified Kadomtsev-Petviashvili is not
$ C^{3} $
. When
$ \beta <0,\gamma >0, $
by using the
$ U^{p} $
and
$ V^{p} $
spaces, we prove that the Cauchy problem is locally well-posed in
$ H^{-\frac{1}{2},0}(\mathbb{R}^{2}) $
.
Citation: Wei Yan, Yimin Zhang, Yongsheng Li, Jinqiao Duan. Sharp well-posedness of the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili equation in anisotropic Sobolev spaces. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5825-5849. doi: 10.3934/dcds.2021097
References:
[1]

L. A. Abramyan and Y. A. Stepanyants, The structure of two-dimensional solitons in media with anomalously small dispersion, Sov. Phys. JETP., 61 (1985), 963-966. 

[2]

M. Ben-Artzi and J. C. Saut, Uniform decay estimates for a class of oscillatory integrals and applications, Diff. Int. Eqns., 12 (1999), 137-145. 

[3]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part II: The KdV equation, Geom. Funct. Anal., 3 (1993), 209-262.  doi: 10.1007/BF01895688.

[4]

J. Bourgain, On the Cauchy problem for the Kadomtsev-Petviashvili equation, Geom. Funct. Anal., 3 (1993), 315-341.  doi: 10.1007/BF01896259.

[5]

R. M. ChenV. Hur and Y. Liu, Solitary waves of the rotation-modified Kadomtsev-Petviashvili equation, Nonlinearity, 21 (2008), 2949-2979.  doi: 10.1088/0951-7715/21/12/012.

[6]

R. M. ChenY. Liu and P. Z. Zhang, Local regularity and decay estimates of solitary waves for the rotation-modified Kadomtsev-Petviashvili equation, Trans. Ameri. Math. Soc., 364 (2012), 3395-3425.  doi: 10.1090/S0002-9947-2012-05383-9.

[7]

J. CollianderC. E. Kenig and G. Staffilani, Low regularity solutions for the Kadomtsev-Petviashvili-I equation, Geom. Funct. Anal., 13 (2003), 737-794.  doi: 10.1007/s00039-003-0429-4.

[8]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on R and T, J. Amer. Math. Soc., 16 (2003), 705-749.  doi: 10.1090/S0894-0347-03-00421-1.

[9]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness for KdV in Sobolev spaces of negative index, Electron. J. Diff. Eqns., 26 (2001), 7 pp.

[10]

J. CollianderA. D. IonescuC. E. Kenig and G. Staffilani, Weighted low-regularity solutions of the KP-I initial-value problem, Discrete Contin. Dyn. Syst., 20 (2008), 219-258.  doi: 10.3934/dcds.2008.20.219.

[11]

A. Esfahani and S. Levandosky, Stability of solitary waves of the Kadomtsev–Petviashvili equation with a weak rotation, SIAM J. Math. Anal., 49 (2017), 5096-5133.  doi: 10.1137/16M1103865.

[12]

J. Ginibre, Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d'espace (d'après Bourgain), Astérisque, Séminaire Bourbaki, 1994/95 (1996), 163-187. 

[13]

R. Grimshaw, Evolution equations for weakly nonlinear, long internal waves in a rotating fluid, Stu. Appl. Math., 73 (1985), 1-33.  doi: 10.1002/sapm19857311.

[14]

R. H. J. GrimshawL. A. OstrovskyV. I. Shrira and Yu. A. Stepanyants, Long nonlinear surface and internal gravity waves in a rotating ocean, Surveys in Geophysics, 19 (1998), 289-338. 

[15]

A. Grünrock, New Applications of the Fourier Restriction Norm Method to Wellposedness Problems for Nonlinear Evolution Equations, Ph. D thesis, Universit$\ddot{a}$t Wuppertal in Dissertation, Germany, 2002.

[16]

Z. H. GuoL. Z. Peng and B. X. Wang, On the local regularity of the KP-I equation in anisotropic Sobolev space, J. Math. Pures Appl., 94 (2010), 414-432.  doi: 10.1016/j.matpur.2010.03.012.

[17]

M. Hadac, Well-posedness for the Kadomtsev-Petviashvili II equation and generalizations, Trans. Ameri. Math. Soc., 360 (2008), 6555-6572.  doi: 10.1090/S0002-9947-08-04515-7.

[18]

M. HadacS. Herr and H. Koch, Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. H. Poincaré-AN, 26 (2009), 917-941.  doi: 10.1016/j.anihpc.2008.04.002.

[19]

A. D. IonescuC. E. Kenig and D. Tataru, Global well-posedness of the KP-I initial-value problem in the energy space, Invent. Math., 173 (2008), 265-304.  doi: 10.1007/s00222-008-0115-0.

[20]

P. Isaza and J. Mejía, Local and global Cauchy problems for the Kadomtsev-Petviashvili (KP-II) equation in Sobolev spaces of negative indices, Comm. Partial Diff. Eqns., 26 (2001), 1027-1054.  doi: 10.1081/PDE-100002387.

[21]

C. E. Kenig, On the local and global well-posedness theory for the KP-I equation, Ann I. H. Poincaré-AN, 21 (2004), 827-838.  doi: 10.1016/j.anihpc.2003.12.002.

[22]

B. B. Kadomtsev and V. I. Petviashvili, On the stability of solitary waves in weakly dispersive media, Soviet. Phys. Dokl., 15 (1970), 539-541. 

[23]

C. E. KenigG. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71 (1993), 1-21.  doi: 10.1215/S0012-7094-93-07101-3.

[24]

C. E. KenigG. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603.  doi: 10.1090/S0894-0347-96-00200-7.

[25]

C. E. Kenig and S. N. Ziesler, Local well-posedness for modified Kadomtsev-Petviashvili equations, Diff. Int. Eqns., 10 (2005), 1111-1146. 

[26]

H. Koch and D. Tataru, Dispersive estimates for principally normal pseudo-differential operators, Commu. Pure. Appl. Math., 58 (2005), 217-284. doi: 10.1002/cpa. 20067.

[27]

H. Koch and D. Tataru, A priori bounds for the 1D cubic NLS in negative Sobolev spaces, Int. Math. Res. Not., 2007 (2007), article ID rnm053, 36pp. doi: 10.1093/imrn/rnm053.

[28]

H. Koch and J. F. Li, Global well-posedness and scattering for small data for the three-dimensional Kadomtsev-Petviashvili II equation, Commun. Partial Diff. Eqns., 42 (2017), 950-976.  doi: 10.1080/03605302.2017.1320410.

[29]

L. MolinetJ. C. Saut and N. Tzvetkov, Well-posedness and ill-posedness results for the Kadomtsev-Petviashvili-I equation, Duke Math. J., 115 (2002), 353-384.  doi: 10.1215/S0012-7094-02-11525-7.

[30]

L. MolinetJ. C. Saut and N. Tzvetkov, Global well-posedness for the KP-I equation, Math. Ann., 324 (2002), 255-275.  doi: 10.1007/s00208-002-0338-0.

[31]

L. MolinetJ. C. Saut and N. Tzvetkov, Global well-posedness for the KP-I equation on the background of a non localized solution, Comm. Math. Phys., 272 (2007), 775-810.  doi: 10.1007/s00220-007-0243-1.

[32]

L. C. MolinetJ. C. Saut and N. Tzvetkov, Global well-posedness for the KP-II equation on the background of a non-localized solution, Ann. Inst. H. Poincaré-AN, 28 (2011), 653-676.  doi: 10.1016/j.anihpc.2011.04.004.

[33]

H. Takaoka, Global well-posedness for the Kadomtsev-Petviashvili II equation, Discrete Contin. Dyn. Syst., 6 (2000), 483-499.  doi: 10.3934/dcds.2000.6.483.

[34]

H. Takaoka, Well-posedness for the Kadomtsev-Petviashvili II equation, Adv. Diff. Eqns., 5 (2000), 1421-1443. 

[35]

H. Takaoka and N. Tzvetkov, On the local regularity of the Kadomtsev-Petviashvili-II equation, Int. Math. Res. Not., 2001 (2001), 77-114.  doi: 10.1155/S1073792801000058.

[36]

N. Tzvetkov, On the Cauchy problem for Kadomtsev-Petviashvili equation, Comm. Partial Diff. Eqns., 24 (1999), 1367-1397.  doi: 10.1080/03605309908821468.

[37]

N. Tzvetkov, Global low-regularity solutions for Kadomtsev-Petviashvili equation, Diff. Int. Eqns., 13 (2000), 1289-1320. 

[38]

N. Wiener, The quadratic variation of a function and its Fourier coefficients, in Collected Works with Commentaries. Volume II: Generalized Harmonic analysis and Tauberian Theory; Classical Harmonic and Complex Analysis, Mathematicians of Our Time, 15, The MIT Press, Cambridge, MA-London, 1979.

[39]

Y. Zhang, Local well-posedness of KP-I initial value problem on torus in the Besov space, Comm. Partial Diff. Eqns., 41 (2016), 256-281.  doi: 10.1080/03605302.2015.1126733.

show all references

References:
[1]

L. A. Abramyan and Y. A. Stepanyants, The structure of two-dimensional solitons in media with anomalously small dispersion, Sov. Phys. JETP., 61 (1985), 963-966. 

[2]

M. Ben-Artzi and J. C. Saut, Uniform decay estimates for a class of oscillatory integrals and applications, Diff. Int. Eqns., 12 (1999), 137-145. 

[3]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part II: The KdV equation, Geom. Funct. Anal., 3 (1993), 209-262.  doi: 10.1007/BF01895688.

[4]

J. Bourgain, On the Cauchy problem for the Kadomtsev-Petviashvili equation, Geom. Funct. Anal., 3 (1993), 315-341.  doi: 10.1007/BF01896259.

[5]

R. M. ChenV. Hur and Y. Liu, Solitary waves of the rotation-modified Kadomtsev-Petviashvili equation, Nonlinearity, 21 (2008), 2949-2979.  doi: 10.1088/0951-7715/21/12/012.

[6]

R. M. ChenY. Liu and P. Z. Zhang, Local regularity and decay estimates of solitary waves for the rotation-modified Kadomtsev-Petviashvili equation, Trans. Ameri. Math. Soc., 364 (2012), 3395-3425.  doi: 10.1090/S0002-9947-2012-05383-9.

[7]

J. CollianderC. E. Kenig and G. Staffilani, Low regularity solutions for the Kadomtsev-Petviashvili-I equation, Geom. Funct. Anal., 13 (2003), 737-794.  doi: 10.1007/s00039-003-0429-4.

[8]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on R and T, J. Amer. Math. Soc., 16 (2003), 705-749.  doi: 10.1090/S0894-0347-03-00421-1.

[9]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness for KdV in Sobolev spaces of negative index, Electron. J. Diff. Eqns., 26 (2001), 7 pp.

[10]

J. CollianderA. D. IonescuC. E. Kenig and G. Staffilani, Weighted low-regularity solutions of the KP-I initial-value problem, Discrete Contin. Dyn. Syst., 20 (2008), 219-258.  doi: 10.3934/dcds.2008.20.219.

[11]

A. Esfahani and S. Levandosky, Stability of solitary waves of the Kadomtsev–Petviashvili equation with a weak rotation, SIAM J. Math. Anal., 49 (2017), 5096-5133.  doi: 10.1137/16M1103865.

[12]

J. Ginibre, Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d'espace (d'après Bourgain), Astérisque, Séminaire Bourbaki, 1994/95 (1996), 163-187. 

[13]

R. Grimshaw, Evolution equations for weakly nonlinear, long internal waves in a rotating fluid, Stu. Appl. Math., 73 (1985), 1-33.  doi: 10.1002/sapm19857311.

[14]

R. H. J. GrimshawL. A. OstrovskyV. I. Shrira and Yu. A. Stepanyants, Long nonlinear surface and internal gravity waves in a rotating ocean, Surveys in Geophysics, 19 (1998), 289-338. 

[15]

A. Grünrock, New Applications of the Fourier Restriction Norm Method to Wellposedness Problems for Nonlinear Evolution Equations, Ph. D thesis, Universit$\ddot{a}$t Wuppertal in Dissertation, Germany, 2002.

[16]

Z. H. GuoL. Z. Peng and B. X. Wang, On the local regularity of the KP-I equation in anisotropic Sobolev space, J. Math. Pures Appl., 94 (2010), 414-432.  doi: 10.1016/j.matpur.2010.03.012.

[17]

M. Hadac, Well-posedness for the Kadomtsev-Petviashvili II equation and generalizations, Trans. Ameri. Math. Soc., 360 (2008), 6555-6572.  doi: 10.1090/S0002-9947-08-04515-7.

[18]

M. HadacS. Herr and H. Koch, Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. H. Poincaré-AN, 26 (2009), 917-941.  doi: 10.1016/j.anihpc.2008.04.002.

[19]

A. D. IonescuC. E. Kenig and D. Tataru, Global well-posedness of the KP-I initial-value problem in the energy space, Invent. Math., 173 (2008), 265-304.  doi: 10.1007/s00222-008-0115-0.

[20]

P. Isaza and J. Mejía, Local and global Cauchy problems for the Kadomtsev-Petviashvili (KP-II) equation in Sobolev spaces of negative indices, Comm. Partial Diff. Eqns., 26 (2001), 1027-1054.  doi: 10.1081/PDE-100002387.

[21]

C. E. Kenig, On the local and global well-posedness theory for the KP-I equation, Ann I. H. Poincaré-AN, 21 (2004), 827-838.  doi: 10.1016/j.anihpc.2003.12.002.

[22]

B. B. Kadomtsev and V. I. Petviashvili, On the stability of solitary waves in weakly dispersive media, Soviet. Phys. Dokl., 15 (1970), 539-541. 

[23]

C. E. KenigG. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71 (1993), 1-21.  doi: 10.1215/S0012-7094-93-07101-3.

[24]

C. E. KenigG. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603.  doi: 10.1090/S0894-0347-96-00200-7.

[25]

C. E. Kenig and S. N. Ziesler, Local well-posedness for modified Kadomtsev-Petviashvili equations, Diff. Int. Eqns., 10 (2005), 1111-1146. 

[26]

H. Koch and D. Tataru, Dispersive estimates for principally normal pseudo-differential operators, Commu. Pure. Appl. Math., 58 (2005), 217-284. doi: 10.1002/cpa. 20067.

[27]

H. Koch and D. Tataru, A priori bounds for the 1D cubic NLS in negative Sobolev spaces, Int. Math. Res. Not., 2007 (2007), article ID rnm053, 36pp. doi: 10.1093/imrn/rnm053.

[28]

H. Koch and J. F. Li, Global well-posedness and scattering for small data for the three-dimensional Kadomtsev-Petviashvili II equation, Commun. Partial Diff. Eqns., 42 (2017), 950-976.  doi: 10.1080/03605302.2017.1320410.

[29]

L. MolinetJ. C. Saut and N. Tzvetkov, Well-posedness and ill-posedness results for the Kadomtsev-Petviashvili-I equation, Duke Math. J., 115 (2002), 353-384.  doi: 10.1215/S0012-7094-02-11525-7.

[30]

L. MolinetJ. C. Saut and N. Tzvetkov, Global well-posedness for the KP-I equation, Math. Ann., 324 (2002), 255-275.  doi: 10.1007/s00208-002-0338-0.

[31]

L. MolinetJ. C. Saut and N. Tzvetkov, Global well-posedness for the KP-I equation on the background of a non localized solution, Comm. Math. Phys., 272 (2007), 775-810.  doi: 10.1007/s00220-007-0243-1.

[32]

L. C. MolinetJ. C. Saut and N. Tzvetkov, Global well-posedness for the KP-II equation on the background of a non-localized solution, Ann. Inst. H. Poincaré-AN, 28 (2011), 653-676.  doi: 10.1016/j.anihpc.2011.04.004.

[33]

H. Takaoka, Global well-posedness for the Kadomtsev-Petviashvili II equation, Discrete Contin. Dyn. Syst., 6 (2000), 483-499.  doi: 10.3934/dcds.2000.6.483.

[34]

H. Takaoka, Well-posedness for the Kadomtsev-Petviashvili II equation, Adv. Diff. Eqns., 5 (2000), 1421-1443. 

[35]

H. Takaoka and N. Tzvetkov, On the local regularity of the Kadomtsev-Petviashvili-II equation, Int. Math. Res. Not., 2001 (2001), 77-114.  doi: 10.1155/S1073792801000058.

[36]

N. Tzvetkov, On the Cauchy problem for Kadomtsev-Petviashvili equation, Comm. Partial Diff. Eqns., 24 (1999), 1367-1397.  doi: 10.1080/03605309908821468.

[37]

N. Tzvetkov, Global low-regularity solutions for Kadomtsev-Petviashvili equation, Diff. Int. Eqns., 13 (2000), 1289-1320. 

[38]

N. Wiener, The quadratic variation of a function and its Fourier coefficients, in Collected Works with Commentaries. Volume II: Generalized Harmonic analysis and Tauberian Theory; Classical Harmonic and Complex Analysis, Mathematicians of Our Time, 15, The MIT Press, Cambridge, MA-London, 1979.

[39]

Y. Zhang, Local well-posedness of KP-I initial value problem on torus in the Besov space, Comm. Partial Diff. Eqns., 41 (2016), 256-281.  doi: 10.1080/03605302.2015.1126733.

[1]

Sergey Degtyarev. Cauchy problem for a fractional anisotropic parabolic equation in anisotropic Hölder spaces. Evolution Equations and Control Theory, 2022  doi: 10.3934/eect.2022029

[2]

Van Duong Dinh. On the Cauchy problem for the nonlinear semi-relativistic equation in Sobolev spaces. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1127-1143. doi: 10.3934/dcds.2018047

[3]

Yongsheng Mi, Chunlai Mu, Pan Zheng. On the Cauchy problem of the modified Hunter-Saxton equation. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 2047-2072. doi: 10.3934/dcdss.2016084

[4]

Ying Fu. A note on the Cauchy problem of a modified Camassa-Holm equation with cubic nonlinearity. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 2011-2039. doi: 10.3934/dcds.2015.35.2011

[5]

Paolo Antonelli, Daniel Marahrens, Christof Sparber. On the Cauchy problem for nonlinear Schrödinger equations with rotation. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 703-715. doi: 10.3934/dcds.2012.32.703

[6]

Hao Tang, Zhengrong Liu. On the Cauchy problem for the Boltzmann equation in Chemin-Lerner type spaces. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 2229-2256. doi: 10.3934/dcds.2016.36.2229

[7]

Shouming Zhou. The Cauchy problem for a generalized $b$-equation with higher-order nonlinearities in critical Besov spaces and weighted $L^p$ spaces. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4967-4986. doi: 10.3934/dcds.2014.34.4967

[8]

V. Varlamov, Yue Liu. Cauchy problem for the Ostrovsky equation. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 731-753. doi: 10.3934/dcds.2004.10.731

[9]

Adrien Dekkers, Anna Rozanova-Pierrat. Cauchy problem for the Kuznetsov equation. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 277-307. doi: 10.3934/dcds.2019012

[10]

Belkacem Said-Houari. Global well-posedness of the Cauchy problem for the Jordan–Moore–Gibson–Thompson equation with arbitrarily large higher-order Sobolev norms. Discrete and Continuous Dynamical Systems, 2022, 42 (9) : 4615-4635. doi: 10.3934/dcds.2022066

[11]

Rudong Zheng, Zhaoyang Yin. The Cauchy problem for a generalized Novikov equation. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3503-3519. doi: 10.3934/dcds.2017149

[12]

M. Nakamura, Tohru Ozawa. The Cauchy problem for nonlinear wave equations in the Sobolev space of critical order. Discrete and Continuous Dynamical Systems, 1999, 5 (1) : 215-231. doi: 10.3934/dcds.1999.5.215

[13]

Pierre-Étienne Druet. Higher $L^p$ regularity for vector fields that satisfy divergence and rotation constraints in dual Sobolev spaces, and application to some low-frequency Maxwell equations. Discrete and Continuous Dynamical Systems - S, 2015, 8 (3) : 475-496. doi: 10.3934/dcdss.2015.8.475

[14]

Arturo de Pablo, Guillermo Reyes, Ariel Sánchez. The Cauchy problem for a nonhomogeneous heat equation with reaction. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 643-662. doi: 10.3934/dcds.2013.33.643

[15]

Shaoyong Lai, Yong Hong Wu. The asymptotic solution of the Cauchy problem for a generalized Boussinesq equation. Discrete and Continuous Dynamical Systems - B, 2003, 3 (3) : 401-408. doi: 10.3934/dcdsb.2003.3.401

[16]

Guillermo Reyes, Juan-Luis Vázquez. The Cauchy problem for the inhomogeneous porous medium equation. Networks and Heterogeneous Media, 2006, 1 (2) : 337-351. doi: 10.3934/nhm.2006.1.337

[17]

Binhua Feng, Xiangxia Yuan. On the Cauchy problem for the Schrödinger-Hartree equation. Evolution Equations and Control Theory, 2015, 4 (4) : 431-445. doi: 10.3934/eect.2015.4.431

[18]

Binhua Feng, Dun Zhao. On the Cauchy problem for the XFEL Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4171-4186. doi: 10.3934/dcdsb.2018131

[19]

Defu Chen, Yongsheng Li, Wei Yan. On the Cauchy problem for a generalized Camassa-Holm equation. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 871-889. doi: 10.3934/dcds.2015.35.871

[20]

Hongwei Wang, Amin Esfahani. On the Cauchy problem for a nonlocal nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022039

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (148)
  • HTML views (197)
  • Cited by (0)

Other articles
by authors

[Back to Top]