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Local smooth solutions of the nonlinear Klein-Gordon equation. (English) Zbl 1479.35570

Summary: 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 [the authors, Commun. Contemp. Math. 19, No. 2, Article ID 1650038, 20 p. (2017; Zbl 1365.35149)]. 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.

MSC:

35L71 Second-order semilinear hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations
35L60 First-order nonlinear hyperbolic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35B65 Smoothness and regularity of solutions to PDEs

Citations:

Zbl 1365.35149
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References:

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