The wave equation is one of the typical partial differential equations and has a long history. Although the wave equation looks like so simple, its mathematical structure is quite rich. In my research the effect from some perturbation such as the nonlinear perturbation, the presence of an obstacle, and so on are analyzed. The main issue is to compare the leading term of a solution to the unperturbed system and that to the perturbed system. For instance, the scattering theory is nothing else but the comparison between the behavior of solutions to these systems as time goes to infinity. We use functional analysis and real analysis for studying the scattering theory. But heavy computations based on calculus are the core of our analysis. It is of special interest to consider the case where the effect from the perturbation is balanced with that from the unperturbed system, because such consideration enables us to see the essential feature of the unperturbed and perturbed systems. Recently, I’m also interested in systems appeared in mathematical physics which are reduced to the wave equation and in the non-commutative structure of some partial differential equations.
- H. Kubo,
Modification of the vector-field method related to quadratically perturbed wave equations in two space dimensions,
Advanced Studies in Pure Mathematics 81, “Asymptotic Analysis for Nonlinear Dispersive and Wave Equations”, (2019), 139-172.
- V. Georgiev, H. Kubo, K. Wakasa,
Critical exponent for nonlinear damped wave equations with non-negative potential in 3D,
J. Differential Equations 267(2019), 3271 – 3288.
- H. Kubo, T. Ogawa, T. Suguro,
Beckner type of the logarithmic Sobolev and a new type of Shannon’s inequalities and an application to the uncertainty principle,
Proceedings of the American Mathematical Society Vol. 147 (4), (2019) 1511 – 1518.