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Blow-up for a semilinear Tricomi equation in the oscillatory regime at the critical Strauss-type exponent

  • Diego Marcon
  • Wanderley Nunes do Nascimento
  • Matheus C. Santos
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Abstract

We study finite-time blow-up for a semilinear shifted Tricomi equation with decreasing propagation speed and an oscillatory scale-invariant mass. We focus on the Strauss-type critical regime and prove that every weak solution with finite speed of propagation, arising from nonnegative nontrivial energy data satisfying a suitable localization condition, blows up in finite time. For sufficiently small initial data, we also obtain the corresponding critical exponential upper bound for the lifespan. The proof relies on a positive self-similar solution of the homogeneous adjoint equation represented by the Gauss hypergeometric function. Combined with a previously established lower bound for the nonlinear term and new estimates adapted to the critical case, this construction reduces the PDE problem to a nonlinear differential inequality. An ODE comparison argument then yields both finite-time blow-up and the lifespan estimate.

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Cite

BibTeXRIS
BibTeX
@misc{marcon2026blowup,
  title = {{Blow-up for a semilinear Tricomi equation in the oscillatory regime at the critical Strauss-type exponent}},
  author = {Diego Marcon and Wanderley Nunes do Nascimento and Matheus C. Santos},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.28389},
  eprint = {2609.28389},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.28389},
}
RIS
TY  - UNPB
TI  - Blow-up for a semilinear Tricomi equation in the oscillatory regime at the critical Strauss-type exponent
AU  - Diego Marcon
AU  - Wanderley Nunes do Nascimento
AU  - Matheus C. Santos
PY  - 2026
DA  - 2026/10/18
PB  - arXiv
DO  - 10.48550/arxiv.2609.28389
UR  - https://arxiv.org/abs/2609.28389
AB  - We study finite-time blow-up for a semilinear shifted Tricomi equation with decreasing propagation speed and an oscillatory scale-invariant mass. We focus on the Strauss-type critical regime and prove that every weak solution with finite speed of propagation, arising from nonnegative nontrivial energy data satisfying a suitable localization condition, blows up in finite time. For sufficiently small initial data, we also obtain the corresponding critical exponential upper bound for the lifespan. The proof relies on a positive self-similar solution of the homogeneous adjoint equation represented by the Gauss hypergeometric function. Combined with a previously established lower bound for the nonlinear term and new estimates adapted to the critical case, this construction reduces the PDE problem to a nonlinear differential inequality. An ODE comparison argument then yields both finite-time blow-up and the lifespan estimate.
ER  -