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PreprintarXiv licencearXiv

Non-uniqueness for a differential equation

  • Brian Street
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Abstract

Let f(t,x),M(t,x)∈ C([0,1]2) with M(t,x)>0. We consider differential equations of the form (∂ f)/(∂ t)(t,x)=(M(t,x)f(t,x)−M(t,0)f(t,0))/x, x>0. For a fixed positive weight M, we ask whether the condition f(0,x)=0 forces f≡ 0. We show the answer is negative for smooth functions: there exist f(t,x),M(t,x)∈ C∞([0,1]2) with f(0,x)=0, f(t,0)not≡ 0, and M(t,x)>0 satisfying the above equation. However, we show that for a Borel-Laplace class of M(t,x), the equation does have uniqueness. We relate this to uniqueness/non-uniqueness theorems for weighted Laplace transforms. A key example originated in an output by ChatGPT-5.5-Pro.

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Cite

BibTeXRIS
BibTeX
@misc{street2026nonuniqueness,
  title = {{Non-uniqueness for a differential equation}},
  author = {Brian Street},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2605.04810},
  eprint = {2605.04810},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.04810},
}
RIS
TY  - UNPB
TI  - Non-uniqueness for a differential equation
AU  - Brian Street
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2605.04810
UR  - https://arxiv.org/abs/2605.04810
AB  - Let f(t,x),M(t,x)∈ C([0,1]^2) with M(t,x)>0. We consider differential equations of the form (∂ f)/(∂ t)(t,x)=(M(t,x)f(t,x)−M(t,0)f(t,0))/x, x>0. For a fixed positive weight M, we ask whether the condition f(0,x)=0 forces f≡ 0. We show the answer is negative for smooth functions: there exist f(t,x),M(t,x)∈ C^∞([0,1]^2) with f(0,x)=0, f(t,0)not≡ 0, and M(t,x)>0 satisfying the above equation. However, we show that for a Borel-Laplace class of M(t,x), the equation does have uniqueness. We relate this to uniqueness/non-uniqueness theorems for weighted Laplace transforms. A key example originated in an output by ChatGPT-5.5-Pro.
ER  -