Given any $ \mu _1, \mu _2\in {\mathbb C} $ and $ \alpha >0 $, we prove the local existence of arbitrarily smooth solutions of the nonlinear Klein-Gordon equation $ \partial _{ tt } u - \Delta u + \mu _1 u = \mu _2 |u|^\alpha u $ on $ {\mathbb R}^N $, $ N\ge 1 $, that do not vanish, i.e. $ |u (t, x) | >0 $ for all $ x \in {\mathbb R}^N $ and all sufficiently small $ t $. We write the equation in the form of a first-order system associated with a pseudo-differential operator, then use a method adapted from [Commun. Contemp. Math. 19 (2017), no. 2, 1650038]. We also apply a similar (but simpler than in the case of the Klein-Gordon equation) argument to prove an analogous result for a class of nonlinear Dirac equations.
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