# MASAKI Satoshi Professor

### Research Activities

My major is the mathematical analysis of partial differential equations. I am especially interested in nonlinear dispersive equations. The dispersive equations are, roughly speaking, the equations describing various wave phenomena. In particular, I work on the nonlinear Schrodinger equation, the nonlinear Klein-Gordon equation, and the (generalized) KdV equation. One goal of my study is to understand the influence of nonlinearity on the global behavior of solutions of equations.

One topic I am specifically working on is the study of time-global dynamics of solutions. Typically, a nonlinear dispersive equation has a special solution called a soliton solution. This kind of equilibrium solution is caused by the balance between the linear dispersion effect, which tends to scatter the wave, and the nonlinear interaction, which tends to collect the wave. One interesting point about nonlinear equations is that a single equation can contain many solutions with quite different behavior. In addition to the soliton solution, in many cases, there are solutions (scattering solutions) in which linear dispersion effects dominate and solutions in which nonlinear interactions dominate and, as a result, blowup in finite time.

The question here is to study the structure of the phase space which these various solutions make. To work with infinite-dimensional vector spaces, we introduce various mathematical tools such as functional inequalities and variational analysis.

In addition, I am also investigating the effects of nonlinearity appearing in the scattering solution. Recently, I am interested in classifying systems of nonlinear equations from the perspective of Hamiltonian structure and conservation laws.

#### Papers:

- S. Masaki, Classication of a class of systems of cubic ordinary dierential equations, J. Dierential Equations 344 (2023), 471-508.
- S. Masaki, J. Segata, and K. Uriya, On Asymptotic behavior of solutions to cubic nonlinear Klein-Gordon systems in one space dimension. Trans. Amer. Math. Soc., series B, 9 (2022) 517-563.
- R. Killip, S, Masaki, J. Murphy, and M. Visan, The radial mass-subcritical NLS in negative order Sobolev spaces. Discrete Contin. Dyn. Syst. 39 (2019), no. 1, 553-583.
- S. Masaki, H. Miyazaki, and K. Uriya, Long-range scattering for nonlinear Schrödinger equations with critical homogeneous nonlinearity in three space dimensions. Trans. Amer. Math. Soc. 371 (2019), no. 11, 7925-7947.
- S. Masaki and J. Segata, Modified scattering for the quadratic nonlinear Klein-Gordon equation in two dimensions. Trans. Amer. Math. Soc. 370 (2018), no. 11, 8155-8170.
- S. Masaki and J. Segata, Existence of a minimal non-scattering solution to the masssubcritical generalized Korteweg-de Vries equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), no. 2, 283-326.
- S. Masaki, A Sharp scattering condition for focusing mass-subcritical nonlinear Schrödinger equation, Comm. in Pure and Applied Analysis 14 (2015), no.4, 1481-1531
- S. Masaki, Energy solution to Schrödinger-Poisson system in the two-dimensional whole space, SIAM J. Math. Anal. 43 (2011), no. 6, 2275-2295.

#### contact

masaki（at）math.sci.hokudai.ac.jp