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Global Solutions with Asymptotic Self-Similar Behaviour for the Cubic Wave Equation

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Abstract

We construct a two-parameter family of explicit solutions to the cubic wave equation on \({\mathbb {R}}^{1+3}\). Depending on the value of the parameters, these solutions either scatter to linear, blow-up in finite time, or exhibit a new type of threshold behaviour which we characterize precisely.

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Notes

  1. See [25, Appendix A.4] Alternatively, see [36, Lemma A.1] for a source that does not rely on tools from conformal geometry.

  2. We will give an explicit expression for these solutions at the end of this subsection.

  3. Denoting as usual \(\frac{\partial }{\partial t}\) the derivative with respect to t at fixed r.

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Authors and Affiliations

  1. LAGA (UMR 7539), Université Sorbonne Paris Nord, Institut Galilée, 99 avenue Jean-Baptiste Clément, 93430, Villetaneuse, France

    Thomas Duyckaerts

  2. Instituto Superior Técnico, Avenida Rovisco Pais, 1049-001, Lisbon, Portugal

    Giuseppe Negro

Authors
  1. Thomas Duyckaerts
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  2. Giuseppe Negro
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Correspondence to Giuseppe Negro.

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Communicated by K. Nakanishi.

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Duyckaerts, T., Negro, G. Global Solutions with Asymptotic Self-Similar Behaviour for the Cubic Wave Equation. Commun. Math. Phys. 405, 84 (2024). https://doi.org/10.1007/s00220-024-04962-3

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