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This paper deals with the Cauchy problem of the semilinear wave equation with a small initial data in 2-dimensional space. When the nonlinearity is cubic, we can not expect the global existence of smooth solutions, in general. However, Godin [1] showed that if the nonlinearity has the null-form, the solution exists globally. In this paper, we will show the global solvability for the other type of nonlinearities which does not have null-form.

The existence of the global solutions to semilinear wave equations with a class of cubic nonlinearities in 2-dimensional space