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Blow Up for Klein–Gordon Equation with Nonlocal Operator and Trudinger–Moser Nonlinearity

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Abstract

In this paper we prove a blow up result for a class of nonlocal scalar Klein–Gordon equation. We assume that the nonlinearity has critical exponential growth. Additionally, we prove that the ground state solution of the elliptic problem associated to the original problem is unstable. The strategy is to adapt the recent ideas of Carrião et al. (J Dyn Differ Equ 2023. https://doi.org/10.1007/s10884-023-10281-3) to find two regions (stable and unstable regions). Since the one dimensional case combined with the 1/2 Laplacian operator cause lack of the control on the \(L^2(\mathbb {R})\) norm of \((-\Delta )^{\frac{1}{4}}u\), new delicate calculations are necessary. We prove also that there exists a subset of \(H^{\frac{1}{2}}(\mathbb {R})\times L^2(\mathbb {R})\) such that the solution is global when the initial data is taken into this set.

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Funding

Research of O. H. Miyagaki is partially supported by the CNPq Grant 303256/2022-2. Research of A. Vicente is partially supported by the CNPq Grant 306771/2023-3.

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Authors and Affiliations

  1. Department of Mathematics, Universidade Federal de Minas Gerais, Belo Horizonte, Minas Gerais, Brazil

    Paulo Cesar Carrião

  2. Department of Mathematics, Universidade Federal de São Carlos, São Carlos, São Paulo, Brazil

    Olimpio Hiroshi Miyagaki

  3. Centro de Ciências Exatas e Tecnológicas, Universidade Estadual do Oeste do Paraná, Cascavel, Paraná, Brazil

    André Vicente

Authors
  1. Paulo Cesar Carrião
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  2. Olimpio Hiroshi Miyagaki
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  3. André Vicente
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The authors contributed equally to this work.

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Correspondence to André Vicente.

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Carrião, P.C., Miyagaki, O.H. & Vicente, A. Blow Up for Klein–Gordon Equation with Nonlocal Operator and Trudinger–Moser Nonlinearity. J Dyn Diff Equat (2024). https://doi.org/10.1007/s10884-024-10366-7

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  • DOI: https://doi.org/10.1007/s10884-024-10366-7

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