# 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.
Document type :
Preprints, Working Papers, ...

https://hal.archives-ouvertes.fr/hal-02228045
Contributor : Thierry Cazenave <>
Submitted on : Thursday, August 1, 2019 - 9:10:08 AM
Last modification on : Saturday, August 3, 2019 - 1:39:14 AM

### Identifiers

• HAL Id : hal-02228045, version 1
• ARXIV : 1907.13048

### Citation

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. 2019. ⟨hal-02228045⟩

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