Research

Research Interest:

I am currently interested in:
    • Strichartz estimates for the linear Schrodinger-type equations (fractional Schrodinger equation, half wave equation, higher-order Schrodinger equation) on manifolds.
    • Local and global well-posedness, blowup, scattering, stability and instability of standing waves for the nonlinear fractional Schrodinger equation, nonlinear half wave equation, nonlinear fourth-order Schrodinger equation, inhomogeneous nonlinear Schrodinger equation, nonlinear Schrodinger equation with inverse-square potential and system of nonlinear Schrodinger equations.
    • Mass concentration of prescribed minimizers for the energy functional related to nonlinear Schrodinger equations, nonlinear Hartree equations, fractional nonlinear Schrodinger equations and Schrodinger-Poisson systems.

Publications:


2019


  • Existence, stability of standing waves and the characterization of finite time blow-up solutions for a system NLS with quadratic interaction, Nonlinear Anal. 190 (2019). [arXiv] [Journal]
  • (with B. Feng) On fractional nonlinear Schrodinger equation with combined power-type nonlinearities, Discrete Contin. Dyn. Syst. 39 (2019), No. 8, pp. 4565-4612. [pdf] [Journal]
  • On instability of standing waves for the mass-supercritical fractional nonlinear Schrodinger equation, Z. Angew. Math. Phys. (2019) 70:58. [arXiv] [Journal]
  • (with A. Bensouilah and M. Majdoub) Scattering in the weighted L^2-space for a 2D nonlinear Schrödinger equation with inhomogeneous exponential nonlinearity, Commun. Pure Appl. Anal. 18 (2019), No. 5, pp. 2735-2755. [arXiv] [Journal]
  • Blow-up criteria for fractional nonlinear Schrodinger equations, Nonlinear Anal. Real World Appl. 48 (2019), pp. 117-140. [arXiv][Journal]
  • Energy Scattering for a class of the defocusing inhomogeneous nonlinear Schrodinger equation, J. Evol. Equ. 2019, pp. 1-24. [arXiv][Journal]
  • On blowup solutions to the focusing mass-critical nonlinear fractional Schrodinger equation, Commun. Pure Appl. Anal. 18 (2019), No. 2, pp. 689-708. [arXiv] [Journal]

2018


  • (with A. Bensouilah and S. Zhu) On stability and instability of standing waves for the nonlinear Schrodinger equation with inverse-square potential, J. Math. Phys. 59 (2018), 101505. [arXiv][Journal]
  • Well-posedness of nonlinear fractional Schrodinger and wave equations in Sobolev spaces, Int. J. Appl. Math. 31 (2018), No. 4, pp. 843–525. [arXiv][Journal]
  • Global existence and blowup for a class of the focusing nonlinear Schrodinger equation with inverse-square potential, J. Math. Anal. Appl. 468 (2018), No. 1, pp. 270-303. [arXiv] [Journal]
  • Global Strichartz estimates for the fractional Schrodinger equations on asymptotically Euclidean manifolds, J. Funct. Anal. 275 (2018), No. 8, pp. 1943-2014. [arXiv] [Journal]
  • On blowup solutions to the focusing intercritical nonlinear fourth-order Schrodinger equation, J. Dynam. Differential Equations 2018 (online first). [arXiv] [Journal]
  • On blowup solutions to the focusing L^2-supercritical nonlinear fractional Schrodinger equation, J. Math. Phys. 59 (2018), 071506. [arXiv] [Journal]
  • Blowup of H^1 solutions for a class of the focusing inhomogeneous nonlinear Schrodinger equation, Nonlinear Anal. 174 (2018), pp. 169-188. [arXiv] [Journal]
  • Global existence and scattering for a class of nonlinear fourth-order Schrodinger equation below the energy space, Nonlinear Anal. 172 (2018), pp. 115-140. [arXiv] [Journal]
  • On well-posedness, regularity and ill-posedness of the nonlinear fourth-order Schrodinger equation, Bull. Belg. Math. Soc. Simon Stevin, 25 (2018), No. 3, pp. 415-437. [arXiv] [Journal]
  • On the Cauchy problem for the nonlinear semirelativistic equation in Sobolev spaces, Discrete Contin. Dyn. Syst. 38 (2018), No. 3, pp. 1127-1143. [arXiv] [Journal]
  • Global well-posedness for a L2-critical nonlinear higher-order Schrodinger equation, J. Math. Anal. Appl. 458 (2018), No. 1, pp. 174-192. [arXiv] [Journal]

2017


  • On the focusing mass-critical nonlinear fourth-order Schrodinger equation below the energy space, Dyn. Partial Differ. Equ. 14 (2017), No. 3, pp. 295-320. [arXiv] [Journal]
  • Strichartz estimates for the fractional Schrodinger and wave equations on compact manifolds without boundary, J. Differential Equations 263 (2017), No. 12, pp. 8804-8837. [arXiv] [Journal]

Preprints:

  • Energy scattering for a class of inhomogeneous nonlinear Schrodinger equations in two dimensions, 2019. [arXiv]
  • Existence and blow-up behavior of standing waves for the Gross-Pitaevskii functional with a higher order interaction, 2019. [pdf]
  • Existence, non-existence and blow-up behavior of minimizers for the mass-critical fractional nonlinear Schrodinger equations with periodic potentials, 2019. [pdf]
  • On nonlinear Schrodinger equations with attractive inverse-power potentials, 2019. [arXiv]
  • (with A. H. Ardila) Some qualitative studies of the focusing inhomogeneous Gross-Pitaevskii equation, 2019. [arXiv]
  • On nonlinear Schrodinger equations with repulsive inverse-power potentials, submitted, 2018. [arXiv]
  • Strong instability of standing waves for a system NLS with quadratic interaction, submitted, 2018. [arXiv]
  • On instability of standing waves for the nonlinear Schrödinger equation with inverse-square potential, submitted, 2018. [arXiv]
  • Scattering in weighted L2 space for a class of the defocusing inhomogeneous nonlinear Schrodinger equation, submitted, 2017. [arXiv]

Unpublished papers:

  • Global existence for the defocusing mass-critical nonlinear fourth-order Schrodinger equation below the energy space, 2017. [arXiv
  • (with A. BensouilahMass concentration and characterization of finite time blow-up solutions for the nonlinear Schrodinger equation with inverse-square potential, 2018. [arXiv]

Doctoral thesis:

  • Thesis title: Strichartz estimates and the nonlinear Schrodinger-type equations on manifolds [pdf]
  • Supervisor: Prof. Jean-Marc Bouclet