Local smooth solutions of the nonlinear Klein-gordon equation

Thierry Cazenave; Ivan Naumkin

doi:10.3934/dcdss.2020448

Article Contents

2021, Volume 14, Issue 5: 1649-1672. Doi: 10.3934/dcdss.2020448

This issue Previous Article Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $ Next Article The algorithmic numbers in non-archimedean numerical computing environments

Local smooth solutions of the nonlinear Klein-gordon equation

Thierry Cazenave^1, and
Ivan Naumkin^2, ,

1.
Sorbonne Université, CNRS, Université de Paris, Laboratoire Jacques-Louis Lions, B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05, France
2.
Departamento de Física Matemática, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Apartado Postal 20-126, Ciudad de México, 01000, México

^* Corresponding author: Ivan Naumkin
^* Corresponding author: Ivan Naumkin

Received: August 2019

Revised: May 2020

Early access: November 2020

Published: May 2021

Ivan Naumkin is a Fellow of Sistema Nacional de Investigadores. The research was partially supported by project PAPIIT IA101820

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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. 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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Mathematics Subject Classification: Primary: 35L70; secondary: 35L60, 35A01, 35B65.

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