Local smooth solutions of the nonlinear Klein-gordon equation

Abstract

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. {\bf 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.