Nonrelativistic limit in the energy space for nonlinear Klein-Gordon equations. (English) Zbl 0991.35080

Machihara, Shuji; Nakanishi, Kenji; Ozawa, Tohru

Math. Ann. 322, No. 3, 603-621 (2002).

Summary: We study the nonrelativistic limit of the Cauchy problem for the nonlinear Klein-Gordon equation \[ \frac {\hbar}{2mc^2} \ddot u- \frac {\hbar}{2m} \Delta u+ \frac{mc^2}{2}u+ f(u)=0, \] where \(u=u(t,x): \mathbb{R}^{1+n}\to \mathbb{C}\), \(f(u)= \lambda|u|^p u\) with \(p>0\) and \(\lambda\in \mathbb{R}\) and prove that any finite energy solution converges to the corresponding solution of the nonlinear Schrödinger equation in the energy space, after the infinite oscillation in time is removed. We also derive the optimal rate of convergence in \(L^2\).

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
35B40	Asymptotic behavior of solutions to PDEs
81Q05	Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics

Keywords:

Strichartz-type estimates; \(L^2\) convergence rate; nonrelativistic limit; Cauchy problem; nonlinear Klein-Gordon equation; nonlinear Schrödinger equation