Scattering for the one-dimensional Klein–Gordon equation with exponential nonlinearity
| Autor: | Masahiro Ikeda, Mamoru Okamoto, Takahisa Inui |
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| Rok vydání: | 2020 |
| Předmět: |
Physics
Scattering Computer Science::Information Retrieval General Mathematics 010102 general mathematics Astrophysics::Instrumentation and Methods for Astrophysics Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) Exponential nonlinearity 01 natural sciences 010101 applied mathematics symbols.namesake Nonlinear system Mathematics - Analysis of PDEs Dimension (vector space) FOS: Mathematics symbols Computer Science::General Literature Initial value problem 0101 mathematics Klein–Gordon equation Analysis Analysis of PDEs (math.AP) Mathematical physics |
| Zdroj: | Journal of Hyperbolic Differential Equations. 17:295-354 |
| ISSN: | 1793-6993 0219-8916 |
| DOI: | 10.1142/s0219891620500083 |
| Popis: | We consider the asymptotic behavior of solutions to the Cauchy problem for the defocusing nonlinear Klein-Gordon equation (NLKG) with exponential nonlinearity in the one spatial dimension with data in the energy space $H^1(\mathbb{R}) \times L^2(\mathbb{R})$. We prove that any energy solution has a global bound of the $L^6_{t,x}$ space-time norm, and hence scatters in $H^1(\mathbb{R}) \times L^2(\mathbb{R})$ as $t\rightarrow\pm \infty$. The proof is based on the argument by Killip-Stovall-Visan (Trans. Amer. Math. Soc. 364 (2012), no. 3, 1571--1631). However, since well-posedness in $H^{1/2}(\mathbb{R}) \times H^{-1/2}(\mathbb{R})$ for NLKG with the exponential nonlinearity holds only for small initial data, we use the $L_t^6 W^{s-1/2,6}_x$-norm for some $s>\frac{1}{2}$ instead of the $L_{t,x}^6$-norm, where $W_x^{s,p}$ denotes the $s$-th order $L^p$-based Sobolev space. 52 pages. arXiv admin note: text overlap with arXiv:1008.2712 by other authors |
| Databáze: | OpenAIRE |
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