Local smooth solutions of the nonlinear Klein-gordon equation
| Autor: | Ivan Naumkin, Thierry Cazenave |
|---|---|
| Přispěvatelé: | Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Departamento de Física Matemática, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Universidad Nacional Autónoma de México (UNAM) |
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
Physics
Primary 35L70 secondary 35L60 35A01 35B65 Nonlinear Dirac equation Computer Science::Information Retrieval Applied Mathematics Operator (physics) Dirac (video compression format) 010102 general mathematics 01 natural sciences 010305 fluids & plasmas Combinatorics symbols.namesake Nonlinear system Mathematics - Analysis of PDEs Nonlinear wave equation 0103 physical sciences FOS: Mathematics symbols Discrete Mathematics and Combinatorics [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] 0101 mathematics Klein–Gordon equation Analysis Analysis of PDEs (math.AP) |
| Zdroj: | Discrete and Continuous Dynamical Systems-Series S Discrete and Continuous Dynamical Systems-Series S, American Institute of Mathematical Sciences, 2021, 14 (5), pp.1649-1672. ⟨10.3934/dcdss.2020448⟩ |
| ISSN: | 1937-1632 1937-1179 |
| DOI: | 10.3934/dcdss.2020448⟩ |
| Popis: | Given any \begin{document}$ \mu _1, \mu _2\in {\mathbb C} $\end{document} and \begin{document}$ \alpha >0 $\end{document} , we prove the local existence of arbitrarily smooth solutions of the nonlinear Klein-Gordon equation \begin{document}$ \partial _{ tt } u - \Delta u + \mu _1 u = \mu _2 |u|^\alpha u $\end{document} on \begin{document}$ {\mathbb R}^N $\end{document} , \begin{document}$ N\ge 1 $\end{document} , that do not vanish, i.e. \begin{document}$ |u (t, x) | >0 $\end{document} for all \begin{document}$ x \in {\mathbb R}^N $\end{document} and all sufficiently small \begin{document}$ t $\end{document} . 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. |
| Databáze: | OpenAIRE |
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