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Existence of infinitely many singular self-similar solutions for the semilinear heat equation

  • Yūki Naito
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Abstract

We show the existence of infinitely many singular positive forward self-similar solutions for the semilinear heat equation ∂t u = Δu + up for x ∈ RN, t > 0, where N ≥ 3 and N/(N−2) ≤ p ≤ (N+2)/(N−2). We also investigate the asymptotic behavior and the bound of the profiles of the solutions as |x| → 0 and |x| → ∞, respectively. Furthermore, we show nonuniqueness of positive soutions of the Cauchy problem for the equation with p = N/(N−2) by using singular self-similar solutions.

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Cite

BibTeXRIS
BibTeX
@misc{naito2026existence,
  title = {{Existence of infinitely many singular self-similar solutions for the semilinear heat equation}},
  author = {Yūki Naito},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.36821},
  eprint = {2609.36821},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.36821},
}
RIS
TY  - UNPB
TI  - Existence of infinitely many singular self-similar solutions for the semilinear heat equation
AU  - Yūki Naito
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.36821
UR  - https://arxiv.org/abs/2609.36821
AB  - We show the existence of infinitely many singular positive forward self-similar solutions for the semilinear heat equation ∂_t u = Δu + u^p for x ∈ R^N, t > 0, where N ≥ 3 and N/(N−2) ≤ p ≤ (N+2)/(N−2). We also investigate the asymptotic behavior and the bound of the profiles of the solutions as |x| → 0 and |x| → ∞, respectively. Furthermore, we show nonuniqueness of positive soutions of the Cauchy problem for the equation with p = N/(N−2) by using singular self-similar solutions.
ER  -