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The Cauchy problem for nonlinear Klein-Gordon equations in the Sobolev spaces. (English) Zbl 1006.35068

This paper deals with the local and global well-posedness in the Sobolev space \(H^s(\mathbb{R}^n)= (\text{Id}-\Delta)^{-s/2} L^2(\mathbb{R}^n)\) of fractional order \(s\) with \(s>n/2\) for the Cauchy problem for nonlinear Klein-Gordon equations of the form \[ \partial^2_t u-\Delta u+u=f(u), \tag{1} \] in space-time \(I\times \mathbb{R}^n\), where \(I=[-T,T]\) with \(T>0\) for local solutions and \(I=\mathbb{R}\) for global solutions. Here \(u\) is a complex-valued function of \((t,x)\in I\times\mathbb{R}^n\), \(\partial^2_t= {\partial^2\over \partial t^2}\), \(\Delta\) is the Laplacian in \(\mathbb{R}^n\), and \(f(u)\) is a nonlinear interaction given by a complex-valued function \(f\) on \(\mathbb{C}\).

MSC:

35L70 Second-order nonlinear hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations
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