研究内容


専門は偏微分方程式論で, 非線形Schrödinger方程式や非線形Klein-Gordon方程式などの非線形分散型方程式について研究を行っています. 非線形分散型方程式は分散性と非線形性の相互作用によって多様な振る舞いを示します. 中でも非線形分散型方程式の解が時間が十分経過した後, どのように振る舞うかについて研究しています. この問題は非線形項の次数が低い場合や空間次元が高くなった場合の解析で困難があることが知られており，30年以上未解決となっています．
最近は空間1次元で3次の非線形項をもつ非線形Schrödinger方程式系とKlein-Gordon方程式系について，非線形項の係数に応じた解の漸近挙動の分類を行っています．



論文 (査読付き)

  1. Ogawa, T., Uriya, K.,
    Asymptotic behavior of solutions to a quadratic nonlinear Schrödinger system with mass resonance,
    RIMS Kokyuroku Bessatsu B42 (2013), 153--170.
  2. Ogawa, T., Uriya, K.,
    Final state problem for a quadratic nonlinear Schrödinger system in two space dimensions with mass resonance,
    J. Differential Equations 258 (2015), 483--503.
  3. Iwabuchi, T., Uriya, K.,
    Ill-posedness for the quadratic nonlinear Schrödinger equation with nonlinearity $|u|^2$,
    Commun. Pure Appl. Anal. 14 (2015), 1395--1405.
  4. Uriya, K.,
    Final state problem for a system of nonlinear Schrödinger equations with three wave interaction,
    J. Evol. Equ. 16 (2016), 173--191.
  5. Iwabuchi, T., Ogawa, T., Uriya, K.,
    Ill-posedness for a system of quadratic nonlinear Schrödinger equations in two dimensions,
    J. Funct. Anal. 271 (2016), 136--163.
  6. Uriya, K.,
    Final state problem for systems of cubic nonlinear Schrödinger equations in one dimension,
    Annales Henri Poincaré 18 (2017), 2523--2542.
  7. Masaki, S., Miyazaki, H., Uriya, K.,
    Long range scattering for nonlinear Schrödinger equation with critical homogeneous nonlinearity in three dimensions,
    Trans. Amer. Math. Soc. 371 (2019), 7925--7947.
  8. Okamoto, M., Uriya, K.,
    Final state problem for the nonlocal nonlinear Schröodinger equation with dissipative nonlinearity,
    Differ. Equ. Appl. 11 (2019), 481--494.
  9. Masaki, S., Segata, J., Uriya, K.,
    Long range scattering for the complex-valued Klein-Gordon equation with quadratic nonlinearity in two dimensions,
    J. Math. Pures Appl. 139 (2020), 177--203.
  10. Okamoto, M., Uriya, K.,
    Long-time behavior of solutions to a fourth-order nonlinear Schrödinger equation with critical nonlinearity,
    J. Evol. Eqn. 21 (2021), 4897--4929.
  11. Aoki, K., Inui, T., Miyazaki, H., Mizutani, H., Uriya, K.,
    Asymptotic behavior for the long-range nonlinear Schrödinger equation on star graph with the Kirchhoff boundary condition,
    Pure Appl. Anal. 4 (2022), 287--311.
  12. Masaki, S., Segata, J., Uriya, K.,
    On asymptotic behavior of solutions to cubic nonlinear Klein-Gordon systems in one space dimension,
    Trans. Amer. Math. Soc. Ser. B 9 (2022), 517--563.
  13. Kita, N., Masaki, S., Segata, J., Uriya, K.,
    Polynomial deceleration for a system of cubic nonlinear Schrödinger equations in one space dimension,
    Nonlinear Anal. 230 (2023), Paper No. 113216, 22 pp.
  14. Masaki, S., Segata, J., Uriya, K.,
    Asymptotic behavior in time of solution to system of cubic nonlinear Schrödinger equations in one space dimension,
    Mathematical physics and its interactions, 119–180, Springer Proc. Math. Stat., 451.

プレプリント

  1. Aoki, K., Inui, T., Miyazaki, H., Mizutani, H., Uriya, K.,
    Modified scattering for inhomogeneous nonlinear Schrödinger equations with and without inverse-square potential,
    submitted.
  2. Masaki, S., Segata, J., Uriya, K.,
    Non-polynomial conserved quantities for ODE systems and its application to the long-time behavior of solutions to cubic NLS systems,
    submitted.

研究費

  1. JSPS 科研費 若手研究,
    高次元における非線形分散型方程式の解の漸近挙動の解明,
    2019年度〜2022年度.
  2. JSPS 科研費 基盤研究C,
    非線形分散型方程式系の漸近解析,
    2024年度〜2027年度.