In light of recent work of the third author, we revisit a classic example given by Fritz John of a semi-linear wave equation which exhibits finite in time blow up for all compactly supported data. We present the construction of future global solutions from asymptotic data given in (Yu, D.: Nontrivial global solutions to some quasilinear wave equations in three space dimensions. arXiv preprint arXiv:2204.12870 (2022) 232) for this specific example, and clarify the relation of this result of Yu to John’s theorem in (Comm. Pure Appl. Math. 34:29–51). Furthermore we present a novel blow up result for finite energy solutions satisfying a sign condition due to the first author, and invoke this result to show that the constructed backwards in time solutions blow up in the past.

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John’s Blow up Examples and Scattering Solutions for Semi-Linear Wave Equations

  • Louie Bernhardt,
  • Volker Schlue,
  • Dongxiao Yu

摘要

In light of recent work of the third author, we revisit a classic example given by Fritz John of a semi-linear wave equation which exhibits finite in time blow up for all compactly supported data. We present the construction of future global solutions from asymptotic data given in (Yu, D.: Nontrivial global solutions to some quasilinear wave equations in three space dimensions. arXiv preprint arXiv:2204.12870 (2022) 232) for this specific example, and clarify the relation of this result of Yu to John’s theorem in (Comm. Pure Appl. Math. 34:29–51). Furthermore we present a novel blow up result for finite energy solutions satisfying a sign condition due to the first author, and invoke this result to show that the constructed backwards in time solutions blow up in the past.