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On 1d Quadratic Klein–Gordon Equations with a Potential and Symmetries

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Abstract

This paper is a continuation of the previous work (Pusateri, in: Forum of mathematics, Cambridge University Press) by the first two authors. We focus on 1-dimensional quadratic Klein–Gordon equations with a potential, under some assumptions that are less general than (Pusateri, in: Forum of mathematics, Cambridge University Press), but that allow us to present some simplifications in the proof of the global existence with decay for small solutions. In particular, we can propagate a stronger control on a basic \(L^2\)-weighted-type norm while providing some shorter and less technical proofs for some of the arguments.

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Notes

  1. Smoothness and decay assumptions are stated like this for convenience, but a finite amount of regularity and algebraic decay is sufficient.

  2. These parity assumptions can be omitted for generic potentials.

  3. \(p_0\) can be chosen of the form \(C\varepsilon ^2\) for some absolute constant \(C>0\), this constraint arising from the Sobolev energy estimate.

  4. Compared to [5], we changed slightly the definitions of the singular and regular parts, in accordance with those in Proposition 4.1 and to simplify the notation in the forthcoming sections.

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Acknowledgements

While working on this project, PG was supported by the NSF Grant DMS-1501019, by the Simons collaborative grant on weak turbulence, by the Center for Stability, Instability and Turbulence (NYUAD), and by the Erwin Schrödinger Institute. FP was supported in part by a start-up Grant from the University of Toronto, NSERC Grant No. 06487, and a Connaught Foundation New Researcher Award. ZZ was supported by the Simons collaborative Grant on weak turbulence and by an AMS-Simons travel Grant.

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Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, 10012-1185, NY, USA

    Katherine Zhiyuan Zhang

  2. Department of Mathematics, University of Toronto, 40 St. George street, Toronto, M5S 2E4, Ontario, Canada

    Fabio Pusateri

  3. Department of Mathematics, Imperial College London, London, United Kingdom

    Pierre Germain

Authors
  1. Pierre Germain
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  2. Fabio Pusateri
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  3. Katherine Zhiyuan Zhang
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Correspondence to Katherine Zhiyuan Zhang.

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Communicated by N. Masmoudi.

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Germain, P., Pusateri, F. & Zhang, K.Z. On 1d Quadratic Klein–Gordon Equations with a Potential and Symmetries. Arch Rational Mech Anal 247, 17 (2023). https://doi.org/10.1007/s00205-023-01853-0

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