Local smooth solutions of the nonlinear Klein-gordon equation

Abstract

<p style='text-indent:20px;'>Given any <inline-formula><tex-math id="M1">\begin{document}$ \mu _1, \mu _2\in {\mathbb C} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ \alpha &gt;0 $\end{document}</tex-math></inline-formula>, we prove the local existence of arbitrarily smooth solutions of the nonlinear Klein-Gordon equation <inline-formula><tex-math id="M3">\begin{document}$ \partial _{ tt } u - \Delta u + \mu _1 u = \mu _2 |u|^\alpha u $\end{document}</tex-math></inline-formula> on <inline-formula><tex-math id="M4">\begin{document}$ {\mathbb R}^N $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ N\ge 1 $\end{document}</tex-math></inline-formula>, that do not vanish, i.e. <inline-formula><tex-math id="M6">\begin{document}$ |u (t, x) | &gt;0 $\end{document}</tex-math></inline-formula> for all <inline-formula><tex-math id="M7">\begin{document}$ x \in {\mathbb R}^N $\end{document}</tex-math></inline-formula> and all sufficiently small <inline-formula><tex-math id="M8">\begin{document}$ t $\end{document}</tex-math></inline-formula>. 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. <b>19</b> (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.