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Existence of a separating norm for the semilinear heat equation
Abstract
We consider the initial value problem of the semilinear heat equation in the Fujita supercritical range for which global-in-time solutions may exist. The main result shows the existence of a separating norm |·|X in the sense that, for nonnegative initial data u0∈ L∞(Rn), the smallness of |u0|X implies global-in-time existence, whereas the largeness of |u0|X implies finite-time blow-up. The separating norm is unique up to equivalence on nonnegative data.
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BibTeX
@misc{takahashi2026existence,
title = {{Existence of a separating norm for the semilinear heat equation}},
author = {Jin Takahashi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05286},
eprint = {2610.05286},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05286},
}
RIS
TY - UNPB
TI - Existence of a separating norm for the semilinear heat equation
AU - Jin Takahashi
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05286
UR - https://arxiv.org/abs/2610.05286
AB - We consider the initial value problem of the semilinear heat equation in the Fujita supercritical range for which global-in-time solutions may exist. The main result shows the existence of a separating norm |·|_X in the sense that, for nonnegative initial data u_0∈ L^∞(R^n), the smallness of |u_0|_X implies global-in-time existence, whereas the largeness of |u_0|_X implies finite-time blow-up. The separating norm is unique up to equivalence on nonnegative data.
ER -